Dynamic image and shape reconstruction in undersampled MRIDynamic image and shape reconstruction in undersampled MRI Iason Kastanis ... Reconstruction of images and shapes from measured

Dynamic image and shape reconstruction inundersampled MRI

Iason Kastanis

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

of the

University of London.

Department of Computer Science

University College London

February 3, 2007

2

Statement of intellectual contribution

The work carried out in this thesis is my own work with the exception of some preliminary

phantom studies, which was conducted in collaboration with Avi Silver, who was working in

the Computational Imaging Science Group, Department of Imaging Sciences, Guys Hospital,

Kings College. Clinical data was provided by Dr Michael Schaft Hansen, who was employed

in the Center of Medical Image Computing, UCL.

4 Statement of intellectual contribution

Abstract

Reconstruction of images and shapes from measured data is nowadays an essential requirement

for medicine. Medical imaging enhances the ability of clinicians to perform diagnosis non-

invasively.

In Magnetic Resonance Imaging, as well as other imaging modalities, data for a single

image frame requires more time than the object can be considered to be static. Therefore anal-

ysis of dynamic objects directly implies the need for fast data acquisition schemes in order to

represent motion in an adequate manner. A necessary condition for this is the collection of data

being limited to a bare minimum. The majority of available methods are designed to deal with

complete data sets. This thesis presents a novel methodology for the reconstruction of very

limited data sets from sparse angular samples. It takes advantage of the dynamic nature of the

reconstruction problem using the theory of inverse problems, as well as statistical analysis. A

model is used to represent the distribution of intensities in the image, as well as the shape of the

object of interest.

The novel reconstruction approach can be used to form both shapes and images directly

from measured data, avoiding some of the constraints of traditional methods, presenting both

qualitative and quantitative results for further analysis by clinicians. The clinical application

of interest is cardiac imaging, where fast imaging, not reliant on periodicity assumptions, is

essential. The method is demonstrated in simulations, phantom and clinical studies for static

and dynamic data sets. The method offers a degree of flexibility in the data collection pro-

cess, opening up the possibility of an intelligent acquisition scheme, where parameters can be

adjusted during the collection of data from patients.

6 Abstract

Acknowledgements

First of all, I would like to thank Prof. Simon Arridge and Prof. Derek Hill. It was their

ideas that initiated this exciting project. Simon Arridge has helped me from the beginning

to understand the mathematical nature of the problem. Derek Hill suggested directions and

applications for our methods. His knowledge on MR imaging has been invaluable.

Dr Daniel Alexander also reserves my gratitude for being my second supervisor, providing

useful comments and suggestions in internal examinations and various presentations.

I would like to acknowledge EPRSC MIAS-IRC for funding this work.

The quality of work is always dependant on its surroundings. I have found working in

the medical imaging group at the computer science department in UCL a great learning and

productive environment. For these reasons I would like to thank, Martin Schweiger for many

suggestions on mathematics and numerics, Rachid Elafouri and Abdel Douiri for their com-

ments and attention on my questions and Thanasis Zacharopoulos for the many discussions on

a variety of subjects.

I would like to thank Dr Michael Hansen and Avi Silver for their collaboration and for

providing data to be used in the experiments.

Without stating their names I would like to thank my friends, for their moral support

throughout this period of my life. Finally, I would like to thank my parents, Nikos and Ioanna,

for their continuous support in every imaginable way, financial, emotional, advices on cooking

properly and many things words cannot describe. They have even listened to me complain and

explain very specific problems about my research as I am sure they had little idea what I was

talking about.

8 Acknowledgements

Contents

1 Prologue 23

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

1.2 Problem statement - Contribution . . . . . . . . . . . . . . . . . . . . . . . . .23

1.3 Overview of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

2 Magnetic Resonance Imaging 27

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

2.2 Principles of MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

2.3 Image reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

2.4 Dynamic imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

2.4.1 Gated imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

2.4.2 Parallel imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

2.4.3 k-t imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38

3 Shape reconstruction background 41

3.1 Snake methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

3.2 Level set methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

4 Numerical optimization: Inverse problem theory 47

4.1 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

4.2 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50

4.2.1 Image parametrization . . . . . . . . . . . . . . . . . . . . . . . . . .50

4.2.2 Shape parametrization . . . . . . . . . . . . . . . . . . . . . . . . . .52

4.3 Data discrepancy functionals . . . . . . . . . . . . . . . . . . . . . . . . . . .55

4.4 Least squares approximation . . . . . . . . . . . . . . . . . . . . . . . . . . .55

10 Contents

4.4.1 Linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56

4.4.2 Nonlinear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

4.5 Constrained optimization: The method of Lagrange . . . . . . . . . . . . . . .60

4.6 Tikhonov regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

4.6.1 Linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

4.6.2 Nonlinear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

4.7 Statistical estimation: Kalman filters . . . . . . . . . . . . . . . . . . . . . . .65

4.7.1 Linear case: Discrete Kalman filters . . . . . . . . . . . . . . . . . . .66

4.7.2 Nonlinear case: Extended Kalman filters . . . . . . . . . . . . . . . .70

4.7.3 Fixed interval smoother . . . . . . . . . . . . . . . . . . . . . . . . .71

4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71

5 Image reconstruction method 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73

5.2 Forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74

5.3 Inverse problem: Direct solution . . . . . . . . . . . . . . . . . . . . . . . . .75

5.3.1 Least squares estimation . . . . . . . . . . . . . . . . . . . . . . . . .75

5.3.2 Damped least squares estimation . . . . . . . . . . . . . . . . . . . . .77

5.4 Inverse problem: Iterative solution . . . . . . . . . . . . . . . . . . . . . . . .78

5.4.1 Lagged diffusivity fixed point iteration . . . . . . . . . . . . . . . . . .80

5.4.2 Primal-dual Newton method . . . . . . . . . . . . . . . . . . . . . . .82

5.4.3 Constrained optimisation . . . . . . . . . . . . . . . . . . . . . . . . .85

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86

5.5.1 Simulated cardiac data . . . . . . . . . . . . . . . . . . . . . . . . . .87

5.5.2 Measured data from MRI . . . . . . . . . . . . . . . . . . . . . . . . .88

5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93

6 Shape reconstruction method 97

6.1 Forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97

6.2 Inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101

6.3.1 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102

6.3.2 Measured data from MRI . . . . . . . . . . . . . . . . . . . . . . . . .108

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111

Contents 11

7 Combined reconstruction method 113

7.1 Forward and inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . .113

7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115

7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122

8 Temporally correlated combined reconstruction method 125

8.1 Forward and inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . .125

8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126

8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138

9 Conclusions and future directions 141

A Acronyms 145

B Table of notation 147

C Difference imaging 149

12 Contents

List of Figures

2.1 (Left) Cartesian sampling. (Right) Radial sampling. . . . . . . . . . . . . . .28

2.2 From image to Radon projections. (Top left) Line integrals overlaid on an image

at θ = 45o. (Top right) A line integral forτ = 32. (Bottom left) The Radon

transform of the image atθ = 45o. (Bottom right) The Radon transform at four

angles. The purple circle indicates the location of the line integral. . . . . . . .30

2.3 A normal ECG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

2.4 Sheared sampling pattern in k-t space. Thet-axis represents time and theky-

axis the sampled locations in the phase encoding direction. Each point denotes

a completekx line in the read out direction. . . . . . . . . . . . . . . . . . . .37

2.5 Plot of an aliased function. Theq-axis is the temporal frequency and theF -axis

is the spatial frequency. Due to the temporal underampling the function has

been shifted in the temporal frequency dimension. This can be corrected with

the application of an appropriate low pass filter [91]. . . . . . . . . . . . . . .38

3.1 Level set function and corresponding shape boundary on the zero level set. . .44

3.2 Level set function and two corresponding shape boundaries on the zero level set.45

4.1 Regular3 × 3 grid. Thex andy axes represent the spatial location inR2 and

thez axis represents the intensity. . . . . . . . . . . . . . . . . . . . . . . . .51

4.2 Surface plot of the Kaiser-Bessel blob basis in 2D with support radius1.45 and

α = 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

4.3 Plot of radial profiles of linear(solid), Gauss(dashed), Wendland(dash-dotted)

and Kaiser-Bessel(dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

4.4 Plot of Fourier basis functions withNγ = 7. Dashed curves are thecos (even)

terms and solid curves are thesin (odd) terms. . . . . . . . . . . . . . . . . . 53

4.5 Plot of B-spline basis functions withNγ = 7. . . . . . . . . . . . . . . . . . . 54

4.6 From left to right. N. Wiener, A. Kolmogorov and R. Kalman. . . . . . . . . .66

14 List of Figures

5.1 Radon data. A sinogram with 8 projections each with 185 line integrals. . . . .74

5.2 Radial profile of the Kaiser-Bessel blob in Fourier space (Left) and Radon space

(Right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75

5.3 The system matrixJ . Each column corresponds to the vectorised basis function

in the Radon space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76

5.4 Ground truth image. Shepp-Logan phantom. . . . . . . . . . . . . . . . . . .76

5.5 8 projections. (Left) Filtered back-projectionrms = 1.2521. (Right) Least

squares reconstruction8× 8 grid rms = 0.73092. . . . . . . . . . . . . . . . 77

5.6 8 projections. (Left) Filtered back-projectionrms = 1.2521. (Right) Damped

least squares reconstruction 64x64 gridrms = 0.61756. . . . . . . . . . . . . 78

5.7 The solid line represents the absolute function|t| and the dashed line represents

the approximationψ(t) =√

t2 + β2 with β = 0.1. . . . . . . . . . . . . . . . 79

5.8 TheTV ′ block tridiagonal matrix. . . . . . . . . . . . . . . . . . . . . . . . .80

5.9 8 projections. (Left) Initial (damped least squares)rms = 0.61756. (Right)

Fixed point reconstructionrms = 0.5975. . . . . . . . . . . . . . . . . . . . 81

5.10 8 projections. (Left)rms error over iteration plot. (Right) Gradient norm plot. 81

5.11 8 projections. (Left) Initial (damped least squares)RMS = 0.61756. (Right)

Primal-dual reconstructionRMS = 0.5975. . . . . . . . . . . . . . . . . . . 84

5.12 8 projections. (Left)RMS error over iteration plot. (Right) Gradient norm plot.84

5.13 8 projections. (Left) Initial (damped least squares)rms = 0.61756. (Right)

Projected primal-dual reconstructionrms = 0.4833. . . . . . . . . . . . . . . 86

5.14 8 projections. (Left)rms error over iteration plot. (Right) Gradient norm plot. 86

5.15 Ground truth image. Fully sampled cardiac image. . . . . . . . . . . . . . . .87

5.16 Simulated data reconstructions. The numbers on the left column indicate the

number of profiles. (Left) Filtered backprojection. (Right) Projected primal-

dual reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88

5.17 (Left)Simulated cardiacrms plot over the number of profiles. The dashed

line represents the filtered backprojection method and the solid the primal-dual

method. (Right) Comparison of central lines of the ground truth and recon-

structed images for the case of 8 radial profiles. . . . . . . . . . . . . . . . . .89

5.18 Coil 1 reconstructions from measured data. The numbers on the left column

indicate the number of profiles. (Left) Gridding. (Right) Projected primal-dual

reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90

5.19 Coil 1. Fully sampled gridding reconstruction used as ground truth image. . .91

List of Figures 15

5.20 (Left) Coil 1rms plot over the number of profiles. The dashed line represents

the gridding method and the solid the primal-dual method. (Right) Comparison

of central lines of the ground truth and reconstructed images for the case of 8

radial profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91

5.21 Multiple coil. Fully sampled LS gridding reconstruction used as ground truth

image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92

5.22 Multiple coil reconstructions from measured data. The numbers on the left

column indicate the number of profiles. (Left) LS gridding. (Right) Projected

primal-dual reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .93

5.23 (Left) Multiple coilrms plot over the number of profiles. The dashed line rep-

resents the LS gridding method and the solid the primal-dual method. (Right)

Comparison of central lines of the ground truth and reconstructed images for

the case of 8 radial profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . .94

6.1 (Right) Contour with self-intersection at parametric pointse. (Left) Corrected

contour with the small loop removed. . . . . . . . . . . . . . . . . . . . . . .98

6.2 Exact parametric pointss1 ands2 of the intersection of the curve with a pixel. 101

6.3 Ground truth image. Cartoon heart. . . . . . . . . . . . . . . . . . . . . . . .102

6.4 Simulated data with no background. (Top Left) Initial superimposed to ground

truth image. (Top Right) Initial predicted image. (Bottom Left) Final superim-

posed to ground truth image. (Bottom Right) Final predicted image. . . . . . .103

6.5 Simulated data with no background. Gradient norm plot over iteration. . . . .103

6.6 Simulated data with no background and 15% added Gaussian noise. (Top Left)

Initial superimposed to ground truth image. (Top Right) Initial predicted image.

(Bottom Left) Final superimposed to ground truth image. (Bottom Right) Final

predicted image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104

6.7 Simulated data with no background and 15% added Gaussian noise. Gradient

norm plot over iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104

6.8 Ground truth image with multiple shapes. . . . . . . . . . . . . . . . . . . . .105

6.9 Simulated data with no background. (Top Left) Initial superimposed to ground

truth image. (Top Right) Initial predicted image. (Bottom Left) Final superim-

posed to ground truth image. (Bottom Right) Final predicted image. . . . . . .105

6.10 Simulated data with no background. Gradient norm plot over iteration. . . . .106

6.11 Ground truth image. Simulated cardiac phantom. . . . . . . . . . . . . . . . .106

16 List of Figures

6.12 Simulated data with known background. (Top Left) Initial superimposed to

ground truth image. (Top Right) Initial predicted image. (Bottom Left) Final

superimposed to ground truth image. (Bottom Right) Final predicted image. .107

6.13 Simulated data with known background. Gradient norm plot over iteration. . .107

6.14 Ground truth image calculated from a fully sampled single coil data set. . . . .108

6.15 Measured single coil data with known background. (Top Left) Initial super-

imposed to ground truth image. (Top Right) Initial predicted image. (Bottom

Left) Final superimposed to ground truth image. (Bottom Right) Final predicted

image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

6.16 Measured single coil data with known background. Gradient norm plot over

iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

6.17 Ground truth image calculated from a fully sampled multiple coil data set. . .110

6.18 Measured multiple coil data with known background. (Top Left) Initial super-



image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110

6.19 Measured multiple coil data with known background. Gradient norm plot over

iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111

7.1 Plot of the derivative ofψ(t) for different values ofβ. These values are as-

signed according to the classification of intensity coefficients as background

(solid line), interior (dotted line) and boundary (dashed line). . . . . . . . . . .115

7.2 Ground truth image for the simulated experiments. . . . . . . . . . . . . . . .116

7.3 Simulated data with unknown background. (Top Left) Initial superimposed to

ground truth image. (Top Right) Initial predicted image. (Bottom Left) Final

superimposed to ground truth image. (Bottom Right) Final predicted image.

The error for the reconstructed image isrms = 0.40217. . . . . . . . . . . . . 117

7.4 Simulated data with unknown background. (Left) Enhanced reconstructed im-

age. (Right) Plot of the gradient norm of the shape reconstruction over iteration.117

7.5 Ground truth image from fully sampled single coil data. . . . . . . . . . . . .118

7.6 Measured data with unknown background. Coil 5. (Top Left) Initial super-



image. The error for the reconstructed image isrms = 0.6509. . . . . . . . . 118

List of Figures 17

7.7 Measured data with unknown background. Coil 5. (Left) Enhanced recon-

structed image. (Right) Plot of the gradient norm of the shape reconstruction

over iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

7.8 Ground truth image from fully sampled multiple coil data. . . . . . . . . . . .120

7.9 Measured data with unknown background. Multiple coils. (Top Left) Initial

superimposed to ground truth image. (Top Right) Initial predicted image. (Bot-

tom Left) Final superimposed to ground truth image. (Bottom Right) Final

predicted image. The error for the reconstructed image isrms = 0.56808. . . 120

7.10 Measured data with unknown background. Multiple coils. (Left) Enhanced re-

constructed image. (Right) Plot of the gradient norm of the shape reconstruction

over iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121

8.1 Interleaved sampling pattern. . . . . . . . . . . . . . . . . . . . . . . . . . .127

8.2 Reconstructions from simulated data. The numbers on the left column indi-

cate the time point in the sequence. (Left) Reconstructed shapes superimposed

on ground truth images. (Right) Reconstructed images with restricted interior

intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128

8.3 Reconstructions from simulated data. The numbers on the left column indi-

cate the time point in the sequence. (Left) Filtered back-projection. (Right)

Reconstructed images using shape specificTVβ approach. . . . . . . . . . . .129

8.4 Error plots from simulated data reconstructions. (Left) Plot of the Dice simi-

larity coefficient over time (Middle) Plot ofrms over time. Filtered backpro-

jection (solid line) and temporally correlated combined approach (dotted line).

(Right) Predicted and ground truth areas over time. . . . . . . . . . . . . . . .130

8.5 x-t plots of the centralrx line in the image over time. The thick arrows point to

the papillary muscle. (Left) Ground truth. (Middle Left) Filtered backprojec-

tion. (Middle Right) Shape specific total variation method. (Right) Combined

shape and image method. . . . . . . . . . . . . . . . . . . . . . . . . . . . .130

8.6 Reconstructions from measured single coil data. The numbers on the left col-

umn indicate the time point in the sequence. (Left) Reconstructed shapes super-

imposed on ground truth images. (Right) Reconstructed images with restricted

interior intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132

18 List of Figures

8.7 Reconstructions from measured single coil data. The numbers on the left col-

umn indicate the time point in the sequence. (Left) Gridding. (Right) Recon-

structed images using shape specificTVβ approach. . . . . . . . . . . . . . .133

8.8 Error plots from measured single coil data reconstructions. (Left) Plot of the

Dice similarity coefficient over time (Middle) Plot ofrms over time. Gridding

(solid line) and temporally correlated combined approach (dotted line). (Right)

Predicted and ground truth areas over time. . . . . . . . . . . . . . . . . . . .134


the papillary muscle. (Left) Ground truth. (Middle Left) Gridding reconstruc-



8.10 Reconstructions from measured multiple coil data. The numbers on the left

column indicate the time point in the sequence. (Left) Reconstructed shapes

superimposed on ground truth images. (Right) Reconstructed images with re-

stricted interior intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

8.11 Reconstructions from measured multiple coil data. The numbers on the left

column indicate the time point in the sequence. (Left) Gridding. (Right) Re-

constructed images using shape specificTVβ approach. . . . . . . . . . . . .136

8.12 Error plots from measured multiple coil data reconstructions. (Left) Plot of the

Dice similarity coefficient over time (Middle) Plot ofrms over time. Gridding

(solid line) and temporally correlated combined approach (dotted line). (Right)

Predicted and ground truth areas over time. . . . . . . . . . . . . . . . . . . .137


the papillary muscle. (Left) Ground truth. (Middle Left) Gridding reconstruc-



C.1 Difference imaging approach with stationary background. (Top Left) Phantom

image at time point 1. (Top Middle) Phantom image at time point 8. (Top Right)

Image difference between time point 1 and 8. (Bottom Left) Phantom sinogram

data at time point 1. (Bottom Middle) Phantom sinogram data at time point 8.

(Bottom Right) Sinogram difference between time point 1 and 8. . . . . . . .149

List of Figures 19

C.2 Difference imaging reconstructions. The numbers on the left column indicate

the time point in the sequence. (Left) Ground truth images. (Right) Recon-

structed shapes superimposed on groundtruth. . . . . . . . . . . . . . . . . . .151

C.3 (Left) Plot of the Dice similarity coefficient over time (Right) Predicted and

ground truth areas over time. . . . . . . . . . . . . . . . . . . . . . . . . . . .152

C.4 Difference imaging approach with stationary background. (Top Left) Phantom

image at time point 1. (Top Middle) Phantom image at time point 8. (Top Right)

Image difference between time point 1 and 8. (Bottom Left) Phantom sinogram

data at time point 1. (Bottom Middle) Phantom sinogram data at time point 8.

(Bottom Right) Sinogram difference between time point 1 and 8. . . . . . . .153

20 List of Figures

Publications

Conference contributions

A.M.S. Silver,I. Kastanis, D.L.G. Hill and S.R. Arridge, Fourier snakes for the reconstruction

of massively undersampled MRI,Proc. MIUA 2003, Sheffield, 2003

I. Kastanis, S.R. Arridge, A.M.S. Silver, D.L.G. Hill and R. Razavi, Reconstruction of the

Heart Boundary from Undersampled Cardiac MRI using Fourier Shape Descriptors and Local

Basis Functions,Proc. ISBI 2004, pp. 1063-1066, 2004

A.M.S. Silver, D.L.G. Hill andI. Kastanis, Analysis of Variability of Cardiac MRI Data,Proc.

MIUA 2005, Bristol, pp. 59-62, 2005

I. Kastanis, S.R. Arridge, A.M.S. Silver and D.L.G. Hill, Reconstruction of Cardiac Images in

Limited Data MRI,Proc. AIP 2005, Cirencester, 2005

I. Kastanis, S.R. Arridge and D.L.G. Hill, Image reconstruction with basis functions: Applica-

tion to real-time radial cardiac MRI,Proc. MIUA 2006, Manchester, pp. 156-161, 2006

22 Publications

Chapter 1

Prologue

1.1 Introduction

As the World Health Organization states on their web site1 : “Although many cardiovascular

diseases (CVDs) can be treated or prevented, an estimated 17 million people die of CVDs each

year.” The need for detection and therefore prevention of heart disease is a major medical imag-

ing need, a need of clinicians who require better and faster tools to diagnose cardiovascular

disease. Methods have been developed and cardiac imaging is now a reality. Yet the problem

of imaging the heart is still far from being completely solved. The majority of methods require

a substantial amount of time and effort in order to obtain and analyse cardiac images. While

these methods assume that the measured data is complete, the proposed approach aims to re-

construct both images and shapes from limited data sets. This combined reconstruction reduces

the scanning time and simplifies the diagnostic procedure by offering qualitative and quantita-

tive results. This novel method, based on the physical reality of the cardiac imaging problem,

escapes some of the assumptions previous methods have made. The next section will give a

more precise idea of the problem in question.

1.2 Problem statement - Contribution

The problem of cardiac imaging is to capture the movement of a dynamic organ. Capturing

the movement of the heart has meant so far to reconstruct images for each phase of the cardiac

cycle. In the analysis of these images it is typical to delineate the left ventricle at each phase

of the cardiac cycle. This is performed manually for every image taking considerable time and

effort. The collection of data for these fully reconstructed images also takes a fair amount of

time, as it will be explained next.

The heart is moving at frequencies approximately between 1 - 3.3 Hz, that is 60 - 200 beats

per minute (bpm). Dynamic imaging is the imaging of objects, that are moving while the data is

1www.who.int

24 Chapter 1. Prologue

being acquired. In the case of cardiac Magnetic Resonance Imaging (MRI), the term dynamic

does not only refer to the motion of the heart, but also to the data acquisition. The data is being

collected sequentially while the heart is beating. The idea of a ‘snapshot’, an image captured in

an instance, does not hold in many medical imaging modalities especially not in MRI. In MRI

the data for a single image of the moving heart requires a lot more time than the time the heart

is considered to be stationary. In biological terms the heart is never stationary and that is a key

property of cardiac imaging.

Given only a small amount of data, where the heart can be considered to be stationary, the

problem becomes ill-posed. In broad terms a problem is called ill-posed when the data is not

sufficient for the solution of the problem and an approximation is the best that can be achieved.

In this thesis we present methodology based on inverse problem theory for both image and

shape reconstruction of limited data sets. While our novel approach is applicable in a variety

of tomographic and Fourier imaging problems, we concentrate on the reconstruction of radially

sampled cardiac MR images. The proposed method does not make any assumptions about the

periodicity of cardiac motion, making it suitable for free-breathing cardiac MRI, as well as for

patients suffering from arrythmia. The substantially small amount of data used by this novel

reconstruction approach also offers the ability of real-time imaging. Even though we do not

consider the presented method as a final solution for cardiac imaging, we believe that it is a step

in the correct direction, escaping the assumptions of current methodology.

Taking advantage of the ideas of inverse problem theory, cardiac imaging becomes a two-

part problem. The first part, forward model, is to parameterise the heart and predict how it would

look under an MRI scanner. Predictions are then compared with data collected from the scanner.

The second part of the problem is to transform this comparison, using the inverse model, to the

chosen representation of the heart. These two-parts are iterated until the parameterised solution

is acceptable.

It is desirable to obtain an analysis of cardiac movement. Using a model-based approach

the heart and the surrounding structures are represented with small set of parameters. This

compact representation makes the problem essentially smaller and therefore easier to solve.

A compact representation is in the mathematical sense a reduction of the dimensionality of

the problem. This parameterised model of the heart automatically separates the heart from

surrounding structures and cardiac motion can be further analysed.

Cardiac imaging is in these terms the problem of choosing the representation of the heart

model, simulating the MR scanner in the forward model and transforming the difference be-

tween the prediction and the data, in the inverse problem, to the parameters of the representa-

1.3. Overview of thesis 25

tion.

In this thesis we present methods for image and shape reconstruction using an inverse

problem approach. The proposed methods are not considered to be at this stage clinically

applicable, but are aimed to prove that the concept is valid. The model-based approaches that

will be presented in this thesis are a significant contribution to the reconstruction of images and

shapes from limited data sets, which are typically encountered in dynamic imaging applications.

Standard methods typically assume that data has been fully sampled, while in the presented

approach this assumption is removed and the reconstruction is stated as a minimisation problem.

In the next section, an overview of the thesis is given.

1.3 Overview of thesis

In chapter§2 we give an introduction to image reconstruction in MRI. We explain the basic ideas

in Magnetic Resonance imaging and overview the current methodology for the reconstruction of

both static and dynamic images. Shape reconstruction methods are discussed in chapter§3. In

chapter§4 the mathematical foundations for the proposed reconstruction method are explained.

Inverse problem theory is discussed from a deterministic and a statistical point of view. Chapter

§5 presents a reconstruction method for images that are uncorrelated in time. The data collection

is considered to be instantaneous. In chapter§6 we discuss the method for reconstructing shapes

directly from measured data. We assume that the background and interior intensities in the

image and shape are known. The combination of image and shape reconstruction is the subject

of chapter§7. The detection of cardiac boundaries can be used to adjust parameters of the image

reconstruction method. In the combined method both the background and interior intensities are

considered to be unknowns in the problem and they are reconstructed from the data. In chapter

§8 the method is developed further for the time correlated case. While the methodology of

the previous chapters§5 - 7 considers the reconstructed parameters to be uncorrelated in time,

in this chapter we assume that there is such correlation. This temporal variation is modelled

as a Markov process using the Kalman filter approach. In the final chapter of this thesis we

draw some conclusions on the methodology used and the results obtained. We propose future

directions of the inverse problem approach to dynamic reconstruction in cardiac MRI.

26 Chapter 1. Prologue

Chapter 2

Magnetic Resonance Imaging

2.1 Introduction

2.2 Principles of MRI

MRI [103] is based on the phenomenon of nuclear magnetic resonance that the nuclei of certain

elements exhibit. This phenomenon can be observed in elements that have an odd number of

protons or neutrons or both in their nucleus. The most important element for the MRI of human

tissue is hydrogenH. Hydrogen has odd atomic number and weight, a half-integral valued

spin, and is found in water moleculesH2O. Human tissue consists of 60% to 80% water [172,

p. 268], making MR ideal for imaging biological structures.

To collect information for MRI there is a need for spatial localisation of the data. The

magnetic field becomes spatially dependant through the use of three magnetic field gradients.

They are small perturbations to the main magnetic field. The three physical gradients are in

orthogonal directions labelled x,y and z. They are assigned by the operating software to three

logical gradients, the slice selection, the readout or frequency encoding and the phase encoding.

The MR image is simply a phase and frequency map collected from the spatially localised mag-

netic fields at each point of the image. The slice selection is the initial step in 2D MRI, it is the

localisation of the radiofrequency excitation to a region of space. This is accomplished through

a frequency selective pulse and the physical gradient corresponding to the logical slice selection

gradient. When the pulse is sent and at the same time the gradient is applied to a small region,

a slice of the object realises the resonance condition. The gradient orientation is perpendicular

to the slice so that the application of the gradient field is the same on every proton on the slice

regardless of its position within the slice. The readout gradient provides spatial localisation

within the slice in one of the two dimensions. It is applied perpendicular to the selected slice

and the protons begin to precess at different frequencies according to the dimension selected

by the gradient. There are two parameters associated with the readout gradient, the Field Of

28 Chapter 2. Magnetic Resonance Imaging

View (FOV) and the number of readout data points in each line of the resulting image matrix.

These data points are obtained without a change in the gradients. To move to a new data line the

gradient has to be changed, which requires substantially more time than to read out points on a

line. The Nyquist frequency [128] depends on both of these parameters. Finally the second di-

mension in the selected slice is defined with the help of the phase encoding gradient. The phase

encoding gradient is perpendicular to both the slice selection and the readout gradients. It is the

only gradient that varies its amplitude with time. This is based on the fact that the precession

of protons is periodical. Similarly to the readout gradient there are two parameters to define for

the phase encoding gradient, the FOV and the number of phase encoding steps. These two will

determine the spatial resolution in the final image. After all the data is collected in the Fourier

space often referred to as k-space, the image is most commonly reconstructed by a 2D Fourier

transform. If the data has been acquired radially (fig. 2.1 (Right)) instead of by Cartesian sam-

pling (fig. 2.1 (Left)), the image can be reconstructed using the Fourier central slice theorem

[126, p. 11]. It states that the 1D Fourier transform of the projection of a 2D function is the

central slice of the Fourier transform of that function. Lines in k-spaces collected in a radial

manner are referred to as radial profiles or simply profiles. For a complete discussion on MRI

principles refer to [152] and [172]. In the next section, we present the current methodology for

the reconstruction of images in MRI.

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

ky

kx−8 −6 −4 −2 0 2 4 6 8

−8

−6

−4

−2

0

2

4

6

8

kx

ky

Figure 2.1: (Left) Cartesian sampling. (Right) Radial sampling.

2.3 Image reconstruction

The foundations for tomographic reconstructions were laid by Johann Radon in 1917 [140].

Radon stated the following integral transform for a functionf(r) of the vector variabler ∈ Rn,

2.3. Image reconstruction 29

now known as the Radon transform

g(θ, τ) = (Rf) (θ, τ) =∫ ∞

−∞f(τuθ + svθ)ds, (2.1)

whereθ ∈ [0, 2π) is the slope of a line,τ ∈ R is its intercept,uθ is the vector defining the

direction of the line andvθ is its normal. In the 2D case (n = 2) uθ = (cos θ, sin θ) and

vθ = (− sin θ, cos θ). The Radon transformR maps a functionf ∈ Rn into the set of its

integrals over the hyperplanes ofRn. In the case wheref ∈ R2, thenf will be mapped into the

set of its line integrals at angleθ. In fig. 2.2 a description of the steps involved in the 2D Radon

transform is shown. Radon also introduced an inversion formula; first we define:

Fr(t) =12π

∫ 2π

0Rf(θ, 〈r,uθ〉+ t)dθ, (2.2)

where〈r,uθ〉 is the inner product. In the 2D case the inverse transform is

f(r) = − 1π

∫ ∞

0

dFr(t)t

. (2.3)

While this formula is elegant, it suffers from the singularity att = 0. An alternative derivation

uses the Hilbert transform, which is defined as follows:

fH(y) = H[f(x)] =1π

∫ ∞

−∞

f(x)x− y

dx. (2.4)

This is essentially a convolution operatorfH(y) = (h ∗ f)(y) where the convolution kernel

h(x) = 1/πx. The equivalent Radon inversion formula is

f(r) =12π

∫ ∞

−∞

∂gH(θ, ry − θrx)∂ry

dθ, (2.5)

wherer = rx, ry. The singularity is still present in the above integral, but it can be handled

as a Cauchy principal value. Apart from eqs. (2.3) and (2.5), other inversion formulas can be

derived. For more information refer to [126], [79] and for a modern treatise on the subject see

[29].

As the theory for tomographic reconstruction already existed, Magnetic Resonance Imag-

ing initially used these available techniques. When data is acquired radially in MRI, it is trivial

to convert it to a set of projections by means of a 1D inverse Fourier transform according to the

Fourier central slice theorem

F1Rf(ω, α) = F2f(k), (2.6)

where the n-dimensional Fourier transformFn and inverse Fourier transformF−n for a func-

tion f(r), r ∈ Rn are


rx

ry

10 20 30 40 50 60

10

20

30

40

50

60θ

sτ

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s

f

(Rf)(θ , τ ) =∑

s

fs

(Rf)(θ, τ ) =

∫f(s)ds

θ = 45o

τ = 32

0 10 20 30 40 50 60 700

2

4

6

8

10

12

14

16

18

20

Rf

τ

θ =45o

020

4060

80

0

50

100

1500

5

10

15

20

25

θ =0o

θ =45o

θ =90o

θ =135o

Rf

θτ

Figure 2.2: From image to Radon projections. (Top left) Line integrals overlaid on an image

at θ = 45o. (Top right) A line integral forτ = 32. (Bottom left) The Radon transform of

the image atθ = 45o. (Bottom right) The Radon transform at four angles. The purple circle

indicates the location of the line integral.

F (k) = (2π)−n/2

∫

Rn

f(r)e−ir·kdr (2.7)

f(r) = (2π)−n/2

∫

Rn

F (k)eik·rdk. (2.8)

Using this theorem the problem of reconstruction in radially sampled MRI is similar to the

Computed Tomography (CT) problem. In the early days [103] of MRI data was acquired radi-

ally and MRI borrowed much of the theory from CT. Quickly though it took its own path.

Algebraic Reconstruction Techniques (ART) existed from the early 1970’s, [59], [58] and

[79]. It is the application of Kaczmarz’s method to Radon’s integral equations [126]. The main

idea of these methods was to state the reconstruction problem as a system of linear equations

g = Rf . (2.9)

ART approximatesRf ≈ cU f and the previous equation becomes

2.3. Image reconstruction 31

g = cU f , (2.10)

whereU is a matrix indicating the locations each line integral intercepts pixels in the image

f(r) andc is an approximate correction factor. The predicted datagtj for thej-th line integral

is calculated as:

gtj = cjUjf t, (2.11)

wheref t is thet-th estimated image vector,Uj is a matrix (with a single row) with thei locations

corresponding to thej-th line integral equal to 1 andcj is a correction factor for that line

integral. The size of the linear system in eq. (2.10) prohibited the direct solution and ART is

essentially an iterative solver. The updated estimate of the image vectorf t+1 is given by:

f t+1 = max

[0, f t +

(gj

cj− gt

j

cj

)/Nj

], (2.12)

whereNj is the total number of intercepts of thej-th line integral withf(r). ART can be

initialised with all the image elements equal to the mean density of the object [58].

A more recent variant of ART methodology is to use basis functions to approximate the dis-

tribution of intensities in the image by replacing matrixU with the matrix of the basis functions.

Hanson and Wecksung [70] used local radially symmetric basis functions for image reconstruc-

tion in CT. To solve this linear system they used ART. In 1990 Lewitt [106] improved on the

method with the use of more general basis functions. Again Lewitt used an iterative method for

the solution of the large linear system. Schweiger and Arridge [147] compared different basis

functions for image reconstruction in optical tomography using an iterative nonlinear conjugate

gradient solver. Garduno and Herman [52] presented a method for surface reconstruction of

biological molecules using 3D basis functions.

Returning back to the early days of MRI and CT, filtered back-projection was originally

discovered by Bracewell and Riddle [15]. The filtered back-projection is a discrete approxi-

mation to the analytic formula in eq. (2.5), where the derivative and the Hilbert transform are

replaced with a ramp or a similar filter

f(r) =π

Nθ

Nθ∑

i=1

Qθi(r · uθi), (2.13)

whereNθ is the number of projections,uθi = (cos θi, sin θi) andQθi is the filtered data at angle

θi

Qθi(r · uθi) = gθi ∗ h, (2.14)


wheregθi is the projection at angleθi, h is a high pass filter and∗ denotes convolution. The high

pass filter enhances high frequency components, such as edge information and noise. The cal-

culation of the filter and the convolution can be performed directly in Fourier space to decrease

computational costs

Qθi(r · uθi) = F−1(F1(gθi)×F1(h)

). (2.15)

In 1971 the method was independently re-discovered by Ramanchandran and Lakshmi-

narayanan [141]. By 1973, when Lauterbur published the first paper [103] on MRI, using a

back-projection method to reconstruct the image of two glass tubes containing water, it was

already widely accepted that filtered back-projection methods were superior to algebraic re-

construction techniques. In 1974 Shepp and Logan [150] compared filtered back-projection to

ART. They used the now famous Shepp-Logan phantom and concluded that the filtered back-

projection method was superior to ART.

In 1975 Kumar et al [100] described an imaging method which took advantage of a se-

quence of orthogonal linear field gradients. They were able to obtain Fourier data on a Cartesian

grid. For image reconstruction a direct Fourier inversion was used instead of the iterative solu-

tions of large systems of linear equations. The fast Fourier transform (FFT) was known at that

time [30]. Edelstein et al [41] extended the method of Kumar et al [100] in 1980 with the use of

varied strength gradients instead of the constant ones Kumar et al had previously suggested. In

this manner they were capable of overcoming the field inhomogeneities problems of Kumar’s

method, making their method applicable to whole-body imaging.

While the inversion of Cartesian Fourier samples by means of an FFT algorithm is fast and

computationally not very demanding, the inversion of radial samples requires interpolation in

to a regular grid. Interpolation is in general a computationally expensive operation, especially if

it is to be precise. The reason for this is that it requires convolution with asinc function, which

is the ideal interpolation function. Thesinc function has infinite support making it prohibitive

for numerical implementations. It was not until 1981 that the groundwork was laid for what

is now the standard method for image reconstruction in radially sampled MRI. In [158] Stark

et al presented various methods for interpolating from polar to Cartesian samples. O’Sullivan

[130] used a Kaiser-Bessel function for this task to improve on the efficiency and quality of

the reconstruction. Jackson et al [82] further extended this methodology and compared various

convolution functions. If we define the data in MRI to be

gfr(k) =(F2f(r)

)×Ar(k), (2.16)

2.4. Dynamic imaging 33

whereAr is a sampling function

Ar(k) =N∑

i=1

δ(k− ki), (2.17)

with N being the number of samples andδ the Dirac delta function. The aim is to interpolate

the signalgfr as follows:

gfi(k) = gfr(k) ∗ h(k), (2.18)

whereh(k) is the convolution kernel. To compensate for the non-uniform sampling, a density

weighting functionw(k) = Ar(k) ∗ h(k) is introduced and the previous equation becomes

gfwi(k) =gfr(k)w(k)

∗ h(k). (2.19)

Re-sampling at Cartesian coordinates

gfwc(k) = gfwi(k)×Ac(k), (2.20)

whereAc(k) =∑

i=1

∑

j=1

δ(kx − i,ky − j) is a comb functionIII (k). Combining eqs. (2.18),

(2.19) and (2.20), we obtain

gfwc(k) =(

gfr(k)w(k)

∗ h(k))×Ac(k). (2.21)

These methods are commonly referred to as gridding.

2.4 Dynamic imaging

Dynamic imaging has emerged as an important research area in the last couple of decades. It is

desirable to be able to image moving or dynamic parts of the human anatomy, like the brain and

the heart. Often this is not easy, since the dynamic object is moving faster than the data can be

collected in a scan. Ideally data for each different image must be collected faster than the object

is moving. In the case of cardiac MRI, if the scanning is done in a purely sequential manner, the

data cannot be collected fast enough to represent different phases of the cardiac cycle clearly.

If the images are formed with enough data to satisfy the Nyquist spatial rate, then the collected

data will only be enough for a very small number of cardiac phases and the images of these

phases will be corrupted by motion artifacts. On the other hand, if more images, corresponding

to more phases, are formed then the data will not be enough for each separate image causing

heavy artifacts and rendering them clinically useless.

Much research has been done in the area of sequence design and as Weiger et al mentions

“ ..., the time efficiency of collecting data by mere gradient encoding seems to be approaching a


Figure 2.3: A normal ECG.

fundamental limitation.” [180, p. 177]. This means that new methods that explore other dimen-

sions of dynamic imaging in MR have to be investigated, other than just using magnetisation

techniques. Some work has been done in Fourier techniques to reduce the scanning time. An

example of this is Feinberg et al[44], who decreased the imaging time to half by compromising

the quality of the image.

2.4.1 Gated imaging

One of the most commonly used techniques to image the heart is gated cardiac imaging. This

method uses the electrocardiogram (ECG) signal to gate the cardiac cycle. When the heart is

contracting it exhibits electrical activity, this is exactly what the ECG measures. The electrical

activity of the heart can be used to determine the phase of the cardiac cycle. As seen in fig. (2.3),

the various letters represent different stages of the heart cycle. The most important is the interval

between the two highest peaks (RR interval), which represents the duration of the cardiac cycle.

Assuming that the ECG is exact in determining the phase of the cardiac cycle and that each

cardiac beat has the same duration, data lines that belong on to the same phase of the cardiac

cycle are collected in different beats of the heart at equal time intervals. This implies that the

data lines required to reconstruct an image, representing one phase of the cardiac cycle, are

collected with one heart beat difference each. The ECG signal provides a means to determine

in which phase of the cardiac cycle the collection of the data is done. This way there is enough

information to reconstruct clear images of various phases of the heart. To extend this idea of

gated imaging, it can be considered that instead of collecting one k-space profile for a phase

at each heart beat, more profiles could be collected. This assumes that while these data lines

are being collected in one heart beat for one phase, the heart is almost stationary. It should be


noted that gated cardiac imaging is performed on a single breath hold to reduce motion in the

surrounding structures due to the breathing process. Examples of gated cardiac imaging can be

found in Lanzer et al[101] who used different techniques to gate the cardiac motion. In [56],

Go et al study volumetric and planar cardiac imaging. In [47], Fletcher et al are using gated

cardiac imaging to study congenital heart malformations. An early system to reconstruct and

display gated cardiac movies was developed in [6].

2.4.2 Parallel imaging

Another approach for the solution of the dynamic imaging problem is the use of partial parallel

imaging. In parallel imaging an array of coils is used instead of just one. Data is collected

for each coil and combined to form one image. The benefit of using multiple coils is that the

data can be undersampled. Using information from each coil, artifacts due to undersampling

can be reduced in the reconstruction. There are two main methods for parallel imaging in MRI,

SMASH [155] , Simultaneous Acquisition of Spatial Harmonics and SENSE [139], Sensitivity

Encoding for fast MRI. Both methods work by approximating the sensitivity information for

each coil. SMASH uses the sensitivity variations to replace some of the phase encoding. Sensi-

tivity information is approximated by fitting linear combinations of sensitivity matrices to form

spatial harmonics. The MR signal in the phase encoding direction at coilj can be expressed as:

gj(ky) =∫

f(ry)Sj(ry)eikyrydry, (2.22)

wheref(ry) is the signal andSj(ry) is the coil sensitivity at each phase encoded line. Sensi-

tivity values are expressed as a linear combination to generate values from all coils

Sm(r) =Nc∑

j=1

wmj Sj(r) ≈ eim∆kyry , (2.23)

whereNc is the number of receiver coils,∆ky = 2π/FOV , FOV is a scalar representing

the field of view andm ∈ Z is the order of the spatial harmonics. This can be solved for the

weightswmj by fitting the coil sensitivitiesSj to the spatial harmonicseim∆kyry . Using eqs.

(2.22) and (2.23), an expression for the calculation of shifted k-space linesg(ky+m∆ky) using

the measured sensitivity matricesSj can be derived

Nc∑

j=1

wmj gj(ky) ≈ g(ky + m∆ky). (2.24)

Using eq. (2.24) missing k-space lines can be generated. In the SENSE approach data is reduced

by decreasing the size of the FOV for each separate receiver coil. Samples are located further

away in k-space. This creates folding artifacts. Sensitivity matrices are calculated in the spatial


domain, unlike SMASH which works in k-space. The full FOV image is calculated as a linear

combination of all the receiver coils by resolving for the superimposed image locations

fn =∑

j,k

Rj,kgj,k, (2.25)

wherefn is the vector of images values,j is the coil index,k is the k-space position index and

R is the reconstruction, or unfolding, matrix of then superimposed image positions and it is

calculated as follows:

R =(SHC−1S

)−1SHC−1, (2.26)

whereS is theNc × Ns coil sensitivity matrix withNc being the total number of coils and

Ns the total number of samples,C is theNc ×Nc receiver noise matrix and the superscriptH

denotes the conjugate transpose. Eq. (2.25) is solved for every position in the reduced FOV

image to produce the full FOV image. Both techniques in their original formulation require the

collection of extra data to be used for the sensitivity calculations. Initially SMASH imaging was

restricted to specific coil design [64] and imaging geometries [84]. Some recent developments

[19], [153], [78] have extended the coil combinations and coil geometry. Bydder et al [19]

reversed eq. (2.23) to express the coil sensitivity matricesSj as linear combinations of the

spatial harmonics

Sj(r) ≈p∑

m=−q

wmj eim∆kyry , (2.27)

whereq, p ∈ Z are integers defining the number of Fourier coefficientswmj for thejth coil. This

allowed the construction of a linear system not as restrictive as the original SMASH formula-

tion. Sodickson et al [153] included an extra termS0 in eq. (2.23) to account for sensitivity

variations in the phase encode direction

Nc∑

j=1

wmj Sj(r) ≈ S0e

im∆kyry . (2.28)

Another very recent variant of SMASH imaging named GRAPPA [63], an extension of [78],

provides unaliazed images for each coil, which can then be combined to produce even higher

Signal-to-Noise Ratio (SNR) than the original SMASH. An analysis of the SNR in SMASH

can be found in [154]. Extensions of the SENSE method are also popular. In [138] Pruessmann

et al extended the original SENSE formulation to arbitrary k-space trajectories using gridding

operations to improve the numerical efficiency of the reconstruction method. Kellman et al

combined SENSE with UNFOLD [114] in [90], which will discussed in the following section.

A detailed review of parallel MR imaging was presented in [14].


0 2 4 6 8 10 12 14 16

−8

−6

−4

−2

0

2

4

6

8

ky

t

Figure 2.4: Sheared sampling pattern in k-t space. Thet-axis represents time and theky-axis

the sampled locations in the phase encoding direction. Each point denotes a completekx line

in the read out direction.

2.4.3 k-t imaging

One of the most important recently developed methods for dynamic imaging is UNFOLD. It

uses the idea of k-t space. Even though it was not stated in these terms in the original UNFOLD

paper [114], it has been re-described in more recent papers by Tsao et al [166], [169]. UNFOLD

works by encoding information in the temporal dimension. Especially after the k-t framework

was introduced by Tsao in [166], it has been understood that the data collection in MRI is in a

spectro - temporal space. The main idea of the k-t space methods is that signals are modulated

by collecting data in an interleaved manner and that for dynamic imaging it makes sense to

investigate the Fourier Transform in the temporal dimension.

As seen in fig. 2.4, only one of every four samples is taken. This interleaved sampling

pattern drastically reduces scanning time up to a fourthfold. When the FT is taken in time, the

modulation of the data will push aliased signals to the end of the spectrum (fig. 2.5), which

allows the removal of ghost artifacts in the image with a low pass filter. Information about low

pass filter design for UNFOLD can be found in [91]. The concept behind this approach is that

modulation caused by the sheared sampling pattern is a shift in the phase encoding direction.

According to the Fourier shift theorem, a shift in the frequency domain results to a linear phase

shift in the time domain. In the x-f space the signals that are static will have little frequency in

time, implying that more bandwidth can be dedicated to the dynamic part.

Intuitively speaking this idea tries to pack the x-f space and therefore reduce scanning

times. The idea of using more bandwidth for the dynamic part is ideal for cardiac imaging,

where the main motion present is the heart beating, while everything else surrounding it is


−100 −50 0 50 1000

0.2

0.4

0.6

0.8

1

1.2

q

F

Aliased signal

Low pass filter

Figure 2.5: Plot of an aliased function. Theq-axis is the temporal frequency and theF -axis is

the spatial frequency. Due to the temporal underampling the function has been shifted in the

temporal frequency dimension. This can be corrected with the application of an appropriate low

pass filter [91].

static or close to static in single breath hold imaging. The basic idea of the UNFOLD method

can be summarised in the following concepts, the interleaved pattern, which reduces scanning

time and combined with the low pass filter that removes artifacts and allows more bandwidth

to the dynamic part of the image. There has been much interest in the UNFOLD method. One

of the most interesting extensions is the combination of BLAST (Broad-use Linear Acqusition

Speed-up Technique) [167] and SENSE with the k-t framework in [169] and [168]. BLAST is

a unification of prior-information methods for fast scanning

f(r, q) =(SHC−1

n|k,tS + C−1s|r,q

)−1SHC−1

n|k,tgk,t, (2.29)

whereS is the Fourier transform, from x-f to k-t spaceFrq→kt, of the sensitivity encoding

matrixS, Cn is the noise covariance matrix andCs is the signal covariance matrix. It provides a

method to accelerate imaging as well as a common equation for the most important accelerating

methods. Other parallel imaging combinations with the k-t ideas exist. In [113] UNFOLD is

combined with partial-Fourier imaging and SENSE. Hansen et al [66] presented a k-t BLAST

method applied to non-Cartesian sampling. An extension of UNFOLD to 3D is presented in

[186], as well as a different method to apply the UNFOLD technique by comparing spectral

energy.

2.5 Discussion

The majority of reconstruction methods in MRI is intended for data sets that satisfy or are

close to the Nyquist limit. When these methods are applied to limited data problems the re-

2.5. Discussion 39

construction produces severe artifacts, usually corrupting the image to a degree unacceptable

for analysis. In dynamic imaging there is a need for finer temporal resolution. To increase the

acquisition speed in MRI, the data available for each frame is necessarily reduced.

To overcome the problem of limited data in cardiac MRI, the common approach is to

use, as mentioned previously, ECG gating. ECG gated cardiac imaging makes two important

assumptions, the first one is that the ECG signal is exact in giving the location of the heart cycle

and repeats itself in an exact manner and the second one is that the heart is beating in precisely

the same way. The first assumption is a good approximation of the truth, but the second is

not necessary valid. Typically each monitored cardiac cycle is shrunk or stretched to fit an

average cardiac cycle. This becomes a problem especially in the case of patients with heart

abnormalities and examinations under stress. In examinations under stress the heart is beating

a lot faster than normally, it is therefore important to reduce the scanning to a bare minimum

in order to avoid having the patient under stress for a long time. If more than one data line is

collected for each phase in each heart cycle, the reconstructed image will have blurring artifacts

due to the motion of the heart. Gated imaging can be thought of as time averaged, in the sense

that a single image is formed by data from many time points at theoretically equal intervals.

Nevertheless it is not desirable to form an averaged image, the effort is to record the motion of

the heart.

Another drawback of this technique is that obtaining high resolution images requires more

data lines, implying longer scanning times. Gated cardiac imaging is a compromise between

resolution or quality, both spatial and temporal, and scanning time. Increasing the spatial res-

olution would imply capturing less phases of the heart cycle or more scanning time. If the

temporal resolution was increased the spatial resolution would have to be decreased or again

the scanning time would have to be longer.

Further to that the single breath hold approach limits the total imaging time, implying that

the spatial and temporal resolution are bounded. For the quantification of ventricular function

typical cardiac MRI often requires the collection of data over many heart beats and also for

more than one breath hold. The long times consumed inside the MRI scanner are stressful and

certainly not desired for patients. Extended breath holds lead to poorly understood flow and

pressure changes within the cardiac region [122]. It is also desirable though to image objects,

which do not behave in a periodic manner and gated imaging cannot be applied.

The vast majority of methods, with the main exception of the k-t approach, do not take

advantage of the dynamic nature of the problem. They consider the problem of reconstructing

a temporal sequence of images as a series of static problems. Some information in the image


can be recovered taking advantage of areas which are not in motion. Statistical properties of the

motion of the object can also be taken into account to improve results.

In the next chapter we will discuss the current approaches in shape reconstruction.

Chapter 3

Shape reconstruction background

Shape reconstruction has been a subject which has received much interest in the image pro-

cessing community. For many machine vision tasks and generally for quantitative analysis a

segmented shape of interest is required. In this chapter we will introduce basic approaches for

the reconstruction of shapes. In the first section, methods based on an explicit formulation of

the shape will be discussed. Following that the discussion will be on a more modern approach,

which has an implicit formulation of the shape.

3.1 Snake methods

Kass et al introduced in [89] the Active Contour Models, more commonly known as snakes.

Snakes are a specific case of the deformable model theory of Terzopoulos [163]. The de-

formable model theory is based on Fischler and Elschlager’s spring loaded templates [46] and

Widrow’s rubber mask technique [184] and [120, p. 92]. Snakes are 2D contours, that approx-

imate locations and shapes of structures in an image. This is done by minimizing an energy

functionalEsnake, that depends on the image and the smoothness or elasticity of the snake

Esnake(v) =∫ 1

0Eint(v(s)) + Eext(v(s)) + Eimage(v(s))ds, (3.1)

wherev(s) =

x(s)

y(s)

is a parametric contour withs ∈ [0, 1) with x(s) andy(s) defining

the x and y coordinates respectively. In the original snake formulation, these were defined as

parametric splines.Eint is the internal energy of the snake, which controls its smoothness.

Eext is an external force used for automatic initialisation and user-intervention. FinallyEimage

is the force defined by the image, usually using image gradients, edge locations or other image

features of interest, to drive the snake closer to the desired segmentation.

Many researchers have extended the original snake formulation in a variety of ways.

Staib and Duncan [157] presented a method based on Fourier parameterisation for the contour.

42 Chapter 3. Shape reconstruction background

Fourier representations are global representations, while splines depend on control points, im-

plying that they are local representations of closed curves on the plane. Fourier parameterisation

is more compact and usually only a few parameters are enough to define complex shapes. The

idea of representing shapes with Fourier descriptors dates back at least to the 1970’s, where var-

ious researchers used them for shape discrimination. In 1982 Kuhl [99] determined the Fourier

coefficients of chain-encoded contours. In this work Kuhl presented properties of Fourier de-

scriptors, such as normalisation and invariants. Further to that he discussed a recognition system

for arbitrary shaped, solid objects. In 1987 Lin [109] presented new invariants based on Fourier

descriptors with application to pattern recognition. In [133] shape discrimination was discussed

with applications in skeleton finding, character and machine parts recognition. In the same line

of research Aquado et al [5] used Fourier descriptors to parameterize shapes by extraction with

the Hough transform. Shapes were not restricted to closed curves, the parameterisation was

extended to open curves as well. Fourier parameterisations have also been used in an inverse

problem framework for the recovery of region boundaries. Kolehmainen [94] et al used mul-

tiple Fourier contours to reconstruct shapes with known internal intensity directly from optical

tomography measurements. In a similar methodology Zacharopoulos et al [189] reconstructed

3D surfaces using a spherical harmonics representation. Battle et al [10] reconstructed a trian-

gulated surface with constant interior density directly from tomographic measurements using

a Bayesian approach. Further development of this Bayesian methodology was presented in

[9], applied to lung images. They defined two homogeneous regions, one for each lung, and

then determined the internal density and location of the boundaries by a Newton minimization

method. Instead of deforming the surfaces directly, they use free-form deformation models to

warp the space surrounding them.

Other recent extensions of the original snake method include extensions that work in color

image space instead of gray scale. Sclaroff and Isidoro [149] presented a method which uses

both shape and color texture information. This definition differs significantly from most other

snake approaches, it resembles more the Active Appearance Models (AAM) approach of Cootes

et al [31]. AAM are a combination of Active Shape Models (ASM) [32] with a grey-level

appearance. ASM are statistical models of the shape of interest obtained using a training set.

The images in the training set are aligned with a modified Procrustes method and their main

modes of variation are calculated using eigenanalysis

CCpj = λjpj , (3.2)

whereCC is the covariance matrix of the aligned shapes,λj is thej-th eigenvalue andpj is the

3.1. Snake methods 43

j-th eigenvector. The eigenvectorspj provide a way of defining the possible ways a shape can

vary

C = C + Pw, (3.3)

whereC is the mean of the aligned shapes,P is the matrix of the firstn eigenvectors andw is

a vector of weights. They have the advantage and at the same time disadvantage of being based

on a training set. This set is the prior knowledge. In some cases this might prove to be limiting

the possible shapes and therefore forcing the algorithm to find a shape, which might not be the

real one. Initial applications of ASM were in hand gesture recognition tasks, while AAM were

targeting face recognition. Stegmann et al [159] used AAM to segment cardiac MR images.

Returning to the work of Sclaroff and Isidoro [149], shapes were defined using a triangu-

lar mesh model, based on a Delaunay triangular meshing algorithm. Their aim was to detect

the motion of objects and the registration process requires minimization of the residual error

with respect to the parameters of their snake model. For this optimisation problem, Sclaroff

and Isidoro use the Levenberg-Marquardt method. A different approach to color snakes was

presented in [54]. Their method is interactive in the sense that it allows the user to choose

subimages, where the object of interest lies. The image segmentation is performed with the use

of snakes that are based on color invariants.

Another recent paradigm of snakes is that of geodesic snakes [24]. Geodesic snakes are

based on the ideas of curve evolution in a metric space with minimal distance curves. The

connection between the calculation of minimal distance curves in the space induced from the

image and the snakes is shown in that work. To calculate the geodesic curve a level set approach

is used. One of the benefits of level set approaches is that curves are topologically adaptive.

Level set methods will be discussed in the next section. An application of the geodesic snakes

can be found in [144]. In that work geodesic snakes were combined with Gabor analysis.

A method for topologically adaptive shapes was presented by McInerney and Terzopoulos

[121]. The curves were defined using nodes connected with edges. The role of the affine cell

image decomposition (ACID) comes in the step of the re-parametarisation of the contour. Using

a particular kind of cell decomposition, simplicial, the space is subdivided into triangles. The

triangles can be of any size, offering fine detail or possibly a multi-scale approach. The inter-

sections of these triangles with the contour are then detected at every M steps of the iteration

and every intersection point gets assigned with an inside or outside value. By tracking the in-

terior vertices of the intersected triangles at every M steps, the contour can be re-parametirised

including topological changes, such as splitting, merging and self-intersecting. This approach is


referred to as Topologically adaptive snakes, T-snakes. An extension of T-snakes is developed

by Giraldi et al in [55]. Giraldi et al used dual snakes, one snake for the outside of the edge

and one for the inside. This was implemented in a Dynamic Programming framework. Evans

et al [43] used T-snakes to segment livers from CT images. An interesting new paradigm of

snakes by McInerney et al is presented in [119]. Following the general concepts of Alife [162]

McInerney et al develop the idea of using artificially intelligent snakes.

3.2 Level set methods

McInerney and Terzopoulos based their decision to use ACID, in the grounds that level set

higher order implicit formulations are not as convenient as the explicit, particularly when it

comes to defining the internal deformation energy term, controlling the snake via user inter-

action and imposing arbitrary geometric or topological constraints [121, p. 74-75]. Level set

methods though are becoming increasingly popular since their introduction in 1988 by Osher

and Sethian [129]. Paragios [131] used a level set method for the segmentation of the left

cardiac ventricle in 2 dimensions. Whitaker and Elangovan [182] reconstructed both 2D con-

tours and 3D surfaces directly from limited tomographic data. In diffusion optical tomography

Schweiger et al [148] reconstructed both the shape and the contrast values of the homogeneous

objects using two level set functions for the absorption and the diffusion values.

Figure 3.1: Level set function and corresponding shape boundary on the zero level set.

Level set methods are based on the ideas of front propagation. The boundary of a shape is

embedded on a higher dimensional function. For a boundary inR2 the level set function will

3.2. Level set methods 45

be a surface inR3. Next we give a brief introduction to the level set approach along the lines of

[145]. The boundary of the region of interestΩ ⊂ Rn is described by a functionφ(r)

∂Ω = r : φ(r) = 0. (3.4)

The level set function is build as a sequence of functionsφt(r) which approach the real region

Ω ast increasesΩt → Ω with ∂Ωt = r : φt(r) = 0. Assuming that the imagef(r) with

r ∈ Rn can be modelled as

f(r) =

fint(r) if r ∈ Ω

fext(r) if r /∈ Ω, (3.5)

then the level set functionφ(r) (fig. 3.1) is tied together with the image function as follows:

f(r) =

fint(r) if φ(r) < 0

fext(r) if φ(r) > 0. (3.6)

The boundary of the region is given by the zero level set,φ(r) = 0. While topological changes,

such as splitting and merging, are rather difficult to deal inR2, the level set function, a surface

in R3, can incorporate these naturally without changing the topology of the surface inR3 (fig.

3.2). The same is true for any dimensionRn.

Figure 3.2: Level set function and two corresponding shape boundaries on the zero level set.

In the case where the topology is known in advance, artificial constraints have to be introduced

in the level set representation to maintain this topology. A detailed review of level-set methods


is given in [38].

3.3 Discussion

Most shape reconstruction methods work in the image domain, thus require image reconstruc-

tion. In this two step approach, first reconstruction and then segmentation, the quality of the

segmentation is dependant on the quality of the reconstruction. If the image reconstruction is

ill-posed, and this is the case in many dynamic imaging problems, the reconstructed image will

contain a large amount of noise. The quality of the segmentation, which is the goal of the anal-

ysis, is ultimately dependant on the reconstruction. If the image reconstruction is of low quality

than segmentation will not be accurate nor robust.

Direct shape reconstruction methods do not use an image to reconstruct the shape, it is

created directly from measured data. These methods typically assume that the object and back-

ground are homogenous and clearly distinguishable. This is not true for cardiac MRI, as well

as many other dynamic imaging applications. The problem of reconstructing a shape with in-

homogeneous interior in an inhomogeneous background is much more complex. A promising

approach was presented by Ye et al [187], which does not require the region of interest to have

a smooth intensity distribution. It also capable of dealing with known inhomogeneous back-

ground. In addition their level set function does not require re-initialisation, which typically

has a high computational cost.

Chapter 4

Numerical optimization: Inverse problem

theory

4.1 Inverse Problems

Inverse problems are very common in Physics and image analysis among other areas of science.

In image analysis the application of inverse problem theory is fairly new since this brand of

science has only existed for around half a century. In the theory of inverse problems, a problem

is separated in to two parts, the forward part and the inverse part. The forward or direct part

of a problem is the prediction of the observable data given the parameters of a model. In this

sense a direct problem would be to calculate the position of a moving object at given time, with

the assumption that the velocity vector is known. The inverse part of a problem is to predict the

model parameters given the observable data. An inverse part problem would be to calculate the

forces acting on a planet given the observation of its trajectory. The terms forward and inverse

are tied together with the definition of what the model is and what the observations of the system

are. For the example of the moving object, if the initial and current positions are considered to

be the observations, then the estimation of the velocity becomes an inverse problem. Therefore

it is important to give a more formal definition of the model and data spaces.

The model spacePm will contain a minimal parameterisationp ∈ Pm that completely

describes the system, wherePm is typically an m-dimensional Hilbert spaceH or in the more

general case a Banach spaceB. A Hilbert space is a vector space with an embedded norm

defined by an inner product. A Banach space is a generalisation of a Hilbert space in the sense

that the norm need not be defined by an inner product. All Hilbert spaces are Banach spaces,

but the converse does not always hold. The data spaceYn will contain the observationsg ∈ Yn

that can be made about a particular system. The notions of forward modelling and inverse

modelling are of major importance to the concept of inverse problems. Forward modelling is

the discovery of the mappingZ from the model space to the data set

48 Chapter 4. Numerical optimization: Inverse problem theory

Z : Pm 7→ Yn. (4.1)

Intuitively, it is the set of laws that allows the prediction of the observationsg, which can

be made on the system, given the model parametersp. Inverse modelling is the mappingZ†

from the observations to the parameters of the model

Z† : Yn 7→ Pm. (4.2)

This implies that by observing a particular system, the parameters of the model can be deduced.

If we assume that there is no measurement noise, the model is

g = Z(p). (4.3)

For the operatorZ, we define the range to be the set of all values thatZ can take asp varies in

Pm

Range(Z) = g ∈ Yn | g = Z(p) for somep ∈ Pm (4.4)

and the nullspace, as the set of all vectorsp that solve the equationZ(p) = 0

Null(Z) = p ∈ Pm | Z(p) = 0. (4.5)

Assuming that the operatorZ is a linear mappingZ : Pm 7→ Yn, then the noise-free model of

eq. (4.3) can be expressed as a matrix multiplication

g = Zp, (4.6)

whereZ ∈ Rn×m. The singular value decomposition (SVD) of a matrixZ ∈ Rn×m with

n > m is given by:

Z = UDV T , (4.7)

where

4.1. Inverse Problems 49

D =

Σr 0

0 0

∈ Rn×m.

Σr = diag(σ1, σ2, ..., σr), σ1 ≥ σ2 ≥ ... ≥ σr > 0 are the singular values ofZ andr =

minm,n is the smallest dimension of the matrix. MatricesU = (u1,u2, ...,un) ∈ Rn×n

andV = (v1,v2, ...,vm) ∈ Rm×m are orthonormal, i.e.UUT = UT U = I andV V T =

V T V = I and the vectorsui andvi are the left and right singular vectorsσjuj = Zvj . If the

matrixZ ∈ Rn×m has more columns than rowsm > n, then the matrixD in eq. (4.1) will not

have any zeros on the diagonal. The condition number of a matrix is equal to the ratio of the

smallest singular value divided by the largestcond(Z) = σ1σr

.

Eq. (4.3) expresses a forward problem. The inverse problem is then to calculate the pa-

rametersp from the measurementsg. In the idealized case, where the forward mappingZ is

exact and there is no noise, this will be

Z(p)− g = 0. (4.8)

Typically though this is not the case. The system of eq. (4.3) is said to be well-posed in the

Hadamard sense [174], provided the following are true:

(i) ∀ g ∈ Yn, there exists a solutionp ∈ Pm for whichg = Z(p) holds.

(ii) the solutionp is unique

(iii) the solution is stable, i.e. ifg0 = Z(p0) andg = Z(p), thenp → p0 wheng → g0

If the system does not fulfill all three of the above requirements it is said to be ill-posed.

Hadamard was convinced that ill-posed problems are not motivated by the physical reality [12,

p. 7]. This dismissal of ill-posed problems can be reasoned in the general scientific mentality

of the early 20th century. This scientific mentality in the early 1900’s of absolute truth and

formality are characteristic in the works of Bertrand Russell and David Hilbert [33].

Definition of ill-conditioning

(i) If the forward mappingZ is rank-deficient, then eq. (4.3) has not got a unique solution. A

simple case where this happens is that the number of unknowns is more than the number

of equations. In other words the system is underdetermined.


(ii) Undetermined systems can also appear if the rankr of the n × m matrix is less than

its smallest dimensionr < m ≤ n. The rank of a matrix is the dimension of its range

r = dim(Range(Z)). In this case some of the columns ofZ are linearly dependant.

Equivalently, the null space ofZ will not be the empty setNull(Z) 6= 0.

(iii) Z can be numerically rank-deficient. Even thoughNull(Z) = 0, there can be a few

columns which are very close to being linearly dependant. This can be seen by a clear

gap in the decay of the singular valuesσ1, σ2, ..., σr, for somek > 1 the spectrum of

the singular values will drop suddenlyσk+1 ¿ σk. The matrixZ will then containk

numerically linearly independent columns and its effective rank will be equal tok.

(iv) The spectrum of the singular values is connected with the oscillations in the singular

vectorsuj ,vj . The smaller the singular valuesσj are, the more oscillatory the singular

vectors are [67], causing the amplification of noise, due to the discretezation, measure-

ment noise and inexactness of the forward model, in the solution.

The condition number in both numerically rank-deficient and discrete ill-posed problems is

typically very large, and the problem is effectively underdetermined.

If the system is ill-posed then the idealized solution in eq. (4.8) is unobtainable. In such

cases one seeks to minimize some discrepancy functional between the predicted and measured

data

pmin = arg minp

Φ(p), (4.9)

whereΦ(p) is the objective function, also referred to as cost function and is typically defined

as a norm||g −Z(p)||. In the next section the details of model selection will be discussed.

4.2 Model selection

4.2.1 Image parametrization

If the intensity variation across spatial locations is considered to be sufficiently smooth, an

imagef(r), with r ∈ Rn, can be approximated using local basis functions

f(r) =Nζ∑

k=1

Bk(r)ζk, (4.10)

whereζ1, ζ2, ..., ζNζ = pζ ∈ P forms the parameter vector,Nζ is number of basis functions,

the grid resolution andP ⊂ RNζ is the parameter vector space. Thek-th basis function is

defined on a regular grid (fig. 4.1)

4.2. Model selection 51

Bk(r) = B0(r + rk) = B0(r) ∗ δ(r− rk) , r ∈ Rn, (4.11)

0

20

40

60

80

0

20

40

60

80

0

0.5

1

x

y

z

Figure 4.1: Regular3× 3 grid. Thex andy axes represent the spatial location inR2 and thez

axis represents the intensity.

whererj is a set of grid points in the n-dimensional spatial domainRn andB0 is the central

basis function of general form:

B0(r) =

b(r) if d ≤ a

0 if d ≥ a, (4.12)

whered is the Euclidean distance from the center of thej-th basis functiond = ||r − rj ||2.

A typical choice for the basis functionsb(r) are the Kaiser-Bessel window function [106] (fig.

(4.2))

b(r) =1

Im(α)(√

1− (d/a)2)mIm(α√

1− (d/a)2), (4.13)

whereIm is the modified Bessel function of the first kind,m is the degree,a is the support and

α is a shape parameter. Kaiser-Bessel functions will be used for the reconstruction of images


Figure 4.2: Surface plot of the Kaiser-Bessel blob basis in 2D with support radius1.45 and

α = 6.4.

throughout the thesis. Another option for the basis functions are the Wendland functions ([181]

and [17]). These are defined forC6 continuity as follows:

b(r) = (1− d)8(32d3 + 25d2 + 8d + 1). (4.14)

A computationally faster alternative are the linear basis functions

b(r) = (1− d/a) (4.15)

and the Gauss radial basis functions [147]

b(r) = e−d2/σ2. (4.16)

whereσ controls the width.

Let B = (B1, B2, ..., BNζ) be the matrix whosek-th column is the vectorized image of

thek-th basis functionBk, then eq. (4.10) can be re-written as a matrix equation

f = Bpζ . (4.17)

Applications of local basis functions in 3D image reconstructions from tomographic data

can be found in [52], [107], [115], [116] and [118].

4.2.2 Shape parametrization

The boundary of a regionC(s) can be represented with global basis functions as follows:

4.2. Model selection 53

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1LinearGauss σ = a/2

Wendland C6

Bessel α = 1.45

Figure 4.3: Plot of radial profiles of linear(solid), Gauss(dashed), Wendland(dash-dotted) and

Kaiser-Bessel(dotted).

C(s) =

x(s)

y(s)

=

Nγ∑

n=1

γx

nθn(s)

γynθn(s)

, s ∈ [0, 1], (4.18)

whereθn are periodic and differentiable basis functions,Nγ is the number of basis functions

andγxn, γy

n ∈ R are the weights ofθn. Let γ denote the vector of all boundary coefficients, i.e.

γ = γx1 , γy

1 , γx2 , γy

2 , ..., γxNγ

, γyNγ ∈ P. It is γ that controls the shape of the regionC(s).

The trigonometric basis functionsθn (fig. 4.4) are defined as follows:

θ1(s) = 1

θn(s) = sin(2π(n

2)s), if n is even (4.19)

θn(s) = cos(2π(n− 1

2)s), if n is odd.

Figure 4.4: Plot of Fourier basis functions withNγ = 7. Dashed curves are thecos (even) terms

and solid curves are thesin (odd) terms.


Staib and Duncan [157] describe this Fourier parameterisation as rotating phasors defined by

groups of four parameters, two for each axis, where the parameters are the weights of the basis

functions. In a simplistic way, the generation of a contour using these functions can be thought

of as the movement of a robotic arm, made out of several parts all rotating continuously at

different speeds, with a pencil on the end. Generally this reminds the idea of the epicycles of

Epimenedes, that described planetary motion.

An alternative to the trigonometric functions are parametric splines. These are local basis

functions, in the sense that each one influences the curve locally. They are formed as overlap-

ping functions between different control points.Nγ in eq. (4.18) is in this case the total number

of control points andθn(s) is defined as follows:

θ1(s) = θ0(s)

θ2(s) = θ0(s) + d (4.20)

θn(s) = θ0(s) + (n− 1)d,

whered is the distance between neighboring control points andθ0 can be defined by cubic

polynomial segments [83] (fig. 4.5)

θ0(s) =

s3

6, if s < 0.25d

1 + 3s + 3s2 − 3s3

6, if 0.25d ≤ s < 0.5d

4− 6s2 + 3s3

6, if 0.5d ≤ s < 0.75d

1− 3s + 3s2 − s3

6, if 0.75d ≤ s < d

. (4.21)

A further example of the spline approach can be found in [50].

Figure 4.5: Plot of B-spline basis functions withNγ = 7.

4.3. Data discrepancy functionals 55

4.3 Data discrepancy functionals

Continuing from eq. (4.9), a variety of data discrepancy functionals can be used for the objective

function. The purpose of these functionals is to describe how close the predicted solutiony =

Zp is to the measured datag. The Kullback-Leibler distance is a statistical measure

Dkl(g,y) =n∑

j=1

gj loggj

yj. (4.22)

An example of minimization of the Kullback-Leibler distance under certain conditions, is

the expectation maximization (EM) method of Richardson[142]-Lucy[110]. More commonly

found are data discrepancy functionals based on theLp norm. For a functionf(x) in a measure

space(X, L), whereX is a measure space andL is a metric, theLp norm is

Lp = ||f ||p =(∫

X|f(x)|pdx

)(1/p)

. (4.23)

The discrete version of this norm is for a vectorf ∈ Bn, whereBn is an appropriate1 n-

dimensional Banach space

lp = ||f ||p =

n∑

j=1

|fj |p

(1/p)

. (4.24)

The next data discrepancy functional is based on thel1 norm, the taxi-cab distance

Dl1(g,y) = ||g − y||1 =n∑

j=1

|gj − yj |. (4.25)

For examples on the use ofl1 norm in optimization problems refer to [37]. The most common

of all, the least squares (LS) functional is based on the squaredl2 norm, the Euclidean distance

D2l2(g,y) = ||g − y||22 =

n∑

j=1

|gj − yj |2. (4.26)

4.4 Least squares approximation

The solution to the ill-posed nature of certain problems has been studied by mathematicians

like Pierre-Simon Laplace in 1799, who used theL1 and theL∞ norms, ‘least-absolute values’

1By appropriate, we mean that anlp norm can be embedded. For example it is not a Tsirelson space [170]


and ‘minimax’ [161] criterions in his own words. It was right in the beginning of the 19th cen-

tury that the ‘least-squares’ criterion was established by Adrien Marie Legendre in 1805[104],

Robert Adrain in 1808[2] (possibly aware of Legendre’s work [160]) and Johann Carl Friedrich

Gauss in 1809[53]. Details on the history of the ‘least-squares’ method can be found in the

following works of Harter [71, 72, 74, 75, 73, 76, 77].

In most real casesg 6∈ Range(Z) and the idealized solutiong − Z(p) = 0 cannot be

obtained. The aim is to minimize the error between predictions and measured data, in this

approach in a least squares sense

Φ(p) = D2l2(g,y), (4.27)

whereg are the measurements andy are the predictions, which are a function of the model

parametersy = Z(p). Let us first examine the case where the forward modelZ is linear.

4.4.1 Linear case

If the mappingZ is linear then it can be expressed as a matrixZ : P 7→ Y, whereP ⊂ Hm

andY ⊂ Hn. Note that a Hilbert space is an example of a Banach spaceB with the l2 norm

embedded. Thel2 norm is the onlylp norm that can be embedded in a Hilbert space, as it can

be defined by an inner product. The objective functional is going to be

Φ(p) = ||g − Zp||22. (4.28)

Expanding the above equation

Φ(p) = (g − Zp)T (g − Zp)

Φ(p) = gTg − (Zp)Tg − gT (Zp) + (Zp)T (Zp).

Note that(Zp)Tg = gT (Zp)

Φ(p) = gTg − 2pT ZTg + pT ZT Zp.

To obtain the minimum solutionpmin = arg minp

Φ(p), the derivative of the objective func-

tional is set to zero∂Φ(p)∂p = 0

4.4. Least squares approximation 57

∂Φ(p)∂p

= 0

∂(gTg − 2pT ZTg + pT ZT Zp

)

∂p= 0

∂gTg∂p

− 2∂pT ZTg

p+

∂pT ZT Zpp

= 0.

Using the matrix derivative identities∂pT a∂p = a and ∂pT Ap

∂p = (A + AT )p, wherep anda are

column vectors andA is a square matrix; we obtain

0− 2ZTg +((ZT Z) + (ZT Z)T

)p = 0. (4.29)

Noting that the matrixZT Z is symmetric, i.e.ZT Z = (ZT Z)T , we derive the set of normal

equations

ZT Zp = ZTg. (4.30)

The solution can be obtained by inversion

p =(ZT Z

)−1ZTg, (4.31)

where(ZT Z

)−1ZT is the Moore-Penrose generalized inverse, also referred to as pseudoin-

verse, discovered independently by Moore in 1920 [124] and by Penrose in 1955 [134]. The

Moore-Penrose inverseA† is uniquely determined by the following four conditions:

(1) AA†A = A, (2) A†AA† = A†,

(3) (AA†)H = AA†, (4) (A†A)H = A†A .

These properties of the Moore-Penrose inverse are also true for the proper inverseA−1. The

pseudoinverseA† is equivalent to the proper inverse if matrixA is square and non-singular. The

system in eq. (4.30) is typically solved by more efficient methods than matrix inversion, such

as QR decomposition [13].

In some cases the matrixA will not have full rankNull(A) 6= 0 and the least squares

problem in eq. (4.28) will not have a unique solution. More often though some columns will be

numerically close to being linearly dependant. The SVD spectrum of the theoretically rank-r

matrix will have a clear jump in its decay. Some singular values will be very close to zero.

Formally, for a small numberε there exist singular values such thatσk+1, ..., σr ≤ ε, while the

rest of the singular values will be far larger thanε, that isε À σ1, ..., σk. It is this k that is


the effective rank of matrixA. Inversion of this numerically rank-deficient matrix will result in

very unstable solutions. The remedy to this problem is to truncate the singular values that are

belowε by setting them equal to zero. This will reduce the variance in the solutions.

A†k = V

Σ−1

k 0

0 0

︸︷︷︸∈Rm×n

UT ,

whereΣk = diagσ1, ..., σk ∈ Rk×k. This approach is named Truncated Singular Value

Decomposition (TSVD). The solution to the TSVD modified problem is given bypTSV D =

Z†kg, whereZk is the best rank-k approximation ofZ.

4.4.2 Nonlinear case

For the nonlinear observation model, as in eq. (4.3), we define the weighted least squares func-

tional:

Φ(p) = ||Lw(g −Z(p))||22, (4.32)

whereLw is a weight matrix2. To minimize this functional, we approximate it with a linear

model in the local neighborhood of a given predictionpk.Under the assumption that the func-

tional is continuously differentiable in the local neighborhood ofpk it can be expanded into a

Taylor series

Φ(p) =∞∑

n=0

Φ(n)(pk)n!

(p− pk)n. (4.33)

It is typical in numerical methods to take the first two terms of its Taylor series expansion, the

quadratic approximation, around the current estimatepk

Φ(p) = Φ(pk) +(

∂Φ∂p

(pk) +12(p− pk)T ∂2Φ

∂p2(pk)

)(p− pk). (4.34)

Setting the derivative of the quadratic functionalΦ(p) equal to zero, we obtain the updated

estimatepk+1 as the minimiser ofΦ(p)

∂Φ∂p

(pk+1) =∂Φ∂p

(pk) + (pk+1 − pk)T ∂2Φ∂p2

(pk) = 0. (4.35)

2In the linear case the weighted least squares solution can be found simply by replacingZ with Z = LwZ and

g with g = Lwg in eq. (4.31)

4.4. Least squares approximation 59

Assuming that∂2Φ∂p2

, the Hessian matrix, is invertible

pk+1 = pk −(

∂2Φ∂p2

(pk))−1

∂Φ∂p

(pk). (4.36)

Eq. (4.36) is iterated until it converges or the stopping criteria are met. The derivative of the

objective functional∂Φ∂p

∈ Rm is

∂Φ∂p

= −2(

∂Z∂p

(pk))T

LTwLw(g −Z(pk)), (4.37)

where∂Z∂p (pk) = Jk is the Jacobian matrix. The Hessian matrix

∂2Φ∂p2

∈ Rm×m will then be

∂2Φ∂p2

= −2

N∑

j=1

(gj −Z(pk))LTwLw

∂2Zj

∂p2

+ 2JT

k LTwLwJk. (4.38)

Setting∑N

j=1(gj − Z(pk))LTwLw

∂2Zj

∂p2 = Kk and substituting eqs. (4.37) and (4.38) into

eq. (4.36), we obtain the Newton-Raphson iteration formula

pk+1 = pk + sk

(Kk + JT

k LTwLwJk

)−1JT

k LTwLw (g −Z(pk)) , (4.39)

wheresk is the step parameter, controlling the convergence of the iterations [13]. It is the

distance to be travelled on the minimizing direction. The computation of the step size can

be done with a line search algorithm ([11],[85]), that solves the 1D minimization problem of

finding the correct distance to be travelled on the downhill direction, so the minimum will not

be missed.

Computation of the Hessian matrix is usually a slow process, which can be a prohibiting

factor for many applications. Various approximations are often used as an alternative to the

Newton-Raphson method, these are referred to as quasi-Newton methods since they lack second

derivative information. ReplacingKk + JTk LT

wLwJk with the identity matrix we obtain the

steepest descent method

pk+1 = pk + skJTk LT

wLw (g −Z(pk)) . (4.40)

Another common approximation of eq. (4.39) is the Gauss-Newton method, where the termKk

is ignored


pk+1 = pk + sk

(JT

k LTwLwJk

)−1JT

k LTwLw (g −Z(pk)) . (4.41)

Each iteration of the above method solves the linearised problem

pmin = arg minp||Lw(g − (Z(pk) + Jk(p− pk)))||22, (4.42)

which will be used to update the estimated solutionspk+1 = pk + pmin. At each step of the

iterationspmin represents the error between the predicted dataZ(pk) and the measured datag

in the model space. The Gauss-Newton method can be thought of as a series of approximated

linear problems, which improve the estimated solutions at each iteration.

Finally, replacingKk with a control termλI and assuming that the step paremeter is fixed

sk = 1, we arrive at the Levenberg[105]-Marquardt[117] method

pk+1 = pk +(JT

k LTwLwJk + λI

)−1JT

k LTwLw (g −Z(pk)) , (4.43)

whereλ ≥ 0 is a control parameter. The convergence of the method depends on the parameterλ

and is in between the Gauss-Newton direction (whenλ = 0) and the steepest descent direction

(whenλ = ∞). The Levenberg-Marquardt method at each iteration minimizes the following

objective functional:

Φ(p) = ||Lw(g − (Z(pk) + Jk(p− pk)))||22 + λ||p− pk||22. (4.44)

4.5 Constrained optimization: The method of Lagrange

The majority of methods presented in the previous section§4.4.2 require a line search algorithm

for the calculation of the step length in the descent direction. Another approach is to define a

region where the linearized model is considered to be a good approximation of the objective

functional. An example of this methodology is the Levenberg-Marquardt method (eq. (4.43)),

which simultaneously evaluates both the step length and the direction. While minimizing the

linearized objective functionalΦ(p), the update∆p = pk+1 − pk is restricted to be within a

region of radiusδt, that is||∆p||22 ≤ δt. These methods are called trust-region or restricted step

methods. This is a constrained optimization problem and it can be expressed as:

minp

Φ(p) , subject to||∆p||22 ≤ δt. (4.45)

4.5. Constrained optimization: The method of Lagrange 61

In the standard Levenberg-Marquardt algorithm the trust region radius is indirectly controlled

with the use ofλ (eq. (4.43)). Modern trust region methods offer direct control ofδt. They do

not seek an exact solution to the above equation, but instead a near optimal one. More [125]

gives details of such methods. For further details on the Levenberg-Marquardt method refer to

Marquardt’s original paper [117] as well as [85] and [13]. Bazaraa et al in [11] discuss also the

necessary conditions for the existence of an optimal solution in nonlinear programming.

The methodology for constrained optimization problems was initially discovered by Leon-

hard Euler and Joseph Louis Lagrange in the midst of the 18th century. While Euler had orig-

inally worked out a geometrical proof of his method, Lagrange was able to prove the same

using analysis alone. Lagrange published some applications of his method in 1788 and further

generalized it in 1797. More details on the history of the calculus of variations can be found in

[57].

Consider the following optimization problem:

minp

Φ(p) , subject tofi(p) = 0 andcj(p) ≤ 0, (4.46)

wherep ∈ Rm, fi is an array of constraint functions withi ∈ [1, Ne] and similarlycj is an array

of constraint functions withj ∈ [1, Ni]. By the method of Lagrange the constrained problem in

m variables is transformed in to an unconstrained problem with additional variables. We form

the following objective functional:

Λ(p, λ, µ) = Φ(p) + λf(p) + µc(p), (4.47)

whereΛ(x, λ, µ) is the Lagrangian andλ ∈ RNe andµ ∈ RNi are the Lagrange multipliers.

The inequality constraints can be converted in to equality constraints with the introduction of

a slack variable,cj(x) ≤ 0 is equivalent tocj(x) + a2 = 0 [111]. An alternative to the

slack variable method was presented in [178]. The unconstrained problem will be a problem in

m + Ne + Ni variables with the additional ones being the Lagrange multipliers. As previously,

to minimize this we need to find the stationary points. We set the derivatives with respect tox

and to the Lagrange multipliersλ andµ equal to zero

∂Λ(x, λ, µ)∂x

= 0

∂Λ(x, λ, µ)∂λ

= 0

∂Λ(x, λ, µ)∂µ

= 0.


Solutions are obtained by solving the above system of equations subject to existence and opti-

mality conditions; typically the Karush-Kuhn-Tucker conditions are used [11], [45]. These sta-

tionary points of the Lagrangian are potential solutions of the constrained problem in eq. (4.46).

In general equality constraints are simpler to solve than inequality ones, both though do

not usually have a trivial solution. Constraints can be employed to restrict the solutions in to

some specific set, such as positive numbersR+ or more generally to impose prior knowledge

in to the solution set. More details on constrained optimization and Lagrangian methods can be

found in [183] and [45].

4.6 Tikhonov regularisation

Ill-posed problems suffer from instability in their solutions. The ill-conditioning nature of the

inverse operator can cause the solution to be far away from the real or sought solution. To

cure this, the ill-posed problem is replaced by a well-posed approximation, such that the new

approximate solution is stable and unique. Common regularization techniques include methods

based on the TSVD, as discussed in§4.4.1, and Tikhonov regularization. Tikhonov regulariza-

tion became widely known from the works of Andrey Nikolayevich Tikhonov in 1943[165] and

David Phillips in 1962[136]. Both applied the method to integral equations. Essentially, it is a

method of obtaining stable solutions in ill-posed problems by forming a constrained optimiza-

tion problem, the constraint being that the set of solutions is a compact one [164].

The generalized Tikhonov regularization assumes the following objective functional:

Φ(p) = ||Lw (g −Z(p)) ||22 + λP (p), (4.48)

whereLw is a weight matrix,λ > 0 is a regularization parameter andP is a penalty func-

tional. The penalty functional is used to penalize unwanted solutions, typically non-smooth

ones. The Tikhonov regularization method is essentially a solution to the following constrained

optimization problem [13]:

minp||Lw (g −Z(p)) ||22 , subject toP (p) ≤ ε, (4.49)

whereε ∈ R+ is a scalar governing the balance between a small residual and a penalized

solution. Both Tikhonov [164, p. 57] and Phillips [136, p. 86] used Lagrange multipliers to

derive the regularization method. An analysis of constraints and penalty functionals can be

found in [11].

Typical choices for the penalty functional are thel1 (see for e.g. [48]) andl2 norms,

introduced in§4.3, as well as the total variation (TV) functional

4.6. Tikhonov regularisation 63

TV (f) =∫

Ω

∣∣∣∣∂f

∂x

∣∣∣∣ dx =∫

Ω|∇f |dx, (4.50)

wherex ∈ Ω is anm-dimensional vector,Ω ⊆ Rm is a bounded open set and∂f∂x = ∇f is the

gradient off . The TV functional has been used in image processing optimization problems in

the works of Candes et al [23], Rudin et al [143] and Chan et al [25]. For the theory on TV

regularization refer to [174]. The discussion here will be restricted to thel2 penalty functional,

as in the original formulation by Tikhonov.

4.6.1 Linear case

The forward modelZ can be expressed as a linear mappingZ : P 7→ Y, whereP ⊂ Hm

andY ⊂ Hn are defined as usual. We set the penalty functional toP (p) = ||L(p − p∗)||22,

wherep∗ is ana priori parameter estimate andL is a regularization operator. The Tikhonov

regularization functional is formulated then as follows:

Φ(p) = ||Lw(g − Zp)||22 + λ||L(p− p∗)||22. (4.51)

Setting the derivative of the objective functionalΦ(p) equal to zero, we obtain the set of normal

equations

(ZT LT

wLwZ + λLT L)p = ZT LT

wLwg + λLT Lp∗. (4.52)

If the regularization operatorL is chosen so thatNull(Z) ∩ Null(L) = 0, which directly

implies thatZT LTwLwZ +λLT L is positive definite, then the solution is unique and defined as:

p =(ZT LT

wLwZ + λLT L)−1 (

ZT LTwLwg + λLT Lp∗

). (4.53)

The incorporation ofa priori information using the regularizing operator can be seen better in

the “stacked form”

Φ(p) =

∣∣∣∣∣∣

∣∣∣∣∣∣

Lwg√

λLp∗

−

LwZ√

λL

p

∣∣∣∣∣∣

∣∣∣∣∣∣

2

2

. (4.54)

Intuitively speaking, this can be interpreted as converting the (numerically) underdetermined

least squares problem to an overdetermined system of equations by adding thea priori infor-

mation encoded in the regularization operatorL.

As in the ordinary Tikhonov scheme we assume that the regularization operatorL and the

weighting matrixLw are equal to the identity matrixI and that we have no prior estimate ofp,

i.e. p∗ = 0. Performing these substitutions in eq. (4.53) we obtain the solution


p =(ZT Z + λI

)−1ZTg, (4.55)

which minimizes the following objective functional:

Φ(p) = ||g − Zp||22 + λ||p||22, (4.56)

with its equivalent “stacked form”:

Φ(p) =

∣∣∣∣∣∣

∣∣∣∣∣∣

g

0

−

Z√

λ

p

∣∣∣∣∣∣

∣∣∣∣∣∣

2

2

. (4.57)

This is known as damped least squares [171]. In the statistical literature it is referred to as ridge

regression [13].

The choice of the regularization parameter is an important research topic. The effort is to

find a method that automatically calculatesλ. Such methods include generalized cross valida-

tion, the discrepancy principle and the L-curve method [174],[68]. A discussion on the L-curve

method can be found in [173]. This method requires to solve the minimisation problem with

a variety ofλ parameters and then choose the optimal. This is prohibitive for many applica-

tions due to the computational cost. The literature on the topic of “automatic” regularisation

parameter choice is vast and a proper discussion on these methods exceeds the purposes of this

thesis.

4.6.2 Nonlinear case

For the nonlinear observation modelZ(p), the generalized Tikhonov regularization is expressed

as in eq. (4.48)

Φ(p) = ||Lw (g −Z(p)) ||22 + λP (p). (4.58)

The Newton-Rapshon method, based on successive linear approximations, is derived similar to

§4.4.2 by setting the derivative of the linearized objective functional equal to zero∂Φ∂p = 0. This

leads to the following formula:

pk+1 = pk −(

∂2Φ∂p2

(pk))−1

∂Φ∂p

(pk). (4.59)

For convenience we will use Newton’s notation and set∂P∂p (pk) = P ′(pk) and ∂2P

∂p2 (pk) =

P ′′(pk). The gradient of the objective functional∂Φ∂p ∈ Rm is then expanded to

4.7. Statistical estimation: Kalman filters 65

∂Φ∂p

(pk) = −2JTk LT

wLw(g −Z(pk)) + λP ′(pk). (4.60)

The Hessian matrix∂2Φ

∂p2 ∈ Rm×m is evaluated as

∂2Φ∂p2

(pk) = 2Kk + 2JTk LT

wLwJk + λP ′′(pk). (4.61)

Ignoring theKk term in the Hessian matrix, we obtain the Gauss-Newton regularized solution

with an iteration formula of the following kind:

pk+1 = pk + sk

(JT

k LTwLwJk +

12λP ′′(pk)

)−1 (JT

k LTwLw(g −Z(pk))− 1

2λP ′(pk)

).

(4.62)

Typically the 12 term can be absorbed by the regularization parameterλ to simplify the iteration

formula further. Setting the penalty functionalP (p) = λ||p||22, one arrives at the Levenberg-

Marquardt method described in§4.4.2.

Additional constraints, such as positivity of the solutionsp ∈ Rm+ can be incorporated

in a similar way as presented here. Up to this point all presented methodology is based on an

algebraic or deterministic approach to minimization. In the next section a statistical approach

to minimization will be discussed.

4.7 Statistical estimation: Kalman filters

An alternative to the deterministic approach to inverse problems is to formulate the problem

using tools from statistical analysis. More details on statistical approaches and Bayesian meth-

ods can be found in [161] and [86]. The focus on this section will be on the theory of Kalman

filters.

During World War II Norbert Wiener(1894-1964) and Andrey Kolmogorov(1903-1987)

set the foundations for the theory of filtering. Their efforts were to solve target tracking prob-

lems and they formulated independently a theory, now known as Wiener-Kolmogorov filtering

theory, in 1941 (Kolmogorov [98]) and in 1942 (Wiener [185]). While many people worked on

filtering theory, the best known results are the Kalman filters. Rudolf Kalman (1930-) published

in 1960 [87] his results on the Wiener-Kolmogorov problems. Kalman went further than Wiener

and Kolmogorov by allowing time variation in the system. For a further historical perspective

refer to [156].

Kalman filtering has found many applications in imaging and in engineering in general.

In Electrical Impedance Tomography (EIT) Kolehmainen et al in [97] presented a method for


Figure 4.6: From left to right. N. Wiener, A. Kolmogorov and R. Kalman.

estimating time varying boundaries of regions with known conductivity. In [93] Kim et al

developed a method for dynamic imaging in EIT. Baroudi et al [8] worked on the estimation of

gas temperature distribution in electric wire tomography. In [88] Kao et al presented a sinogram

restoration technique for image reconstruction in Positron Emission Tomography (PET). In

[123] Kalman filters were used in cardiac MRI to estimate myocardial motion using velocity

fields.

4.7.1 Linear case: Discrete Kalman filters

The discrete Kalman filter algorithm solves the estimation problem of the statep ∈ Rm of a

time controlled process given some measurementsg ∈ Rn. This process is governed by a linear

equation of the form:

pt = Spt−1 + wt−1. (4.63)

The process state is related to the measurement by

yt = Zpt + nt. (4.64)

S is the state transition model, it relates the states of time stept − 1 to next time step. It

represents our knowledge of the motion of the states. If the motion is unknown, the identity

matrix can be used asS to express a random-walk model.Z : P 7→ Y is assumed to be linear

and it represents the mapping from the state space to the measurement space. In the inverse

problem terminology, this is referred to as the forward model. The variableswt andnt rep-

resent the process and measurement white noise, they are assumed to have normal probability

distributions.

p(wt) ∼ N(0, Cw,t) (4.65)

p(vt) ∼ N(0, Cn,t), (4.66)


whereCw,t and Cn,t are the process and measurement noise covariance matrices at timet,

respectively.

We definept|t−1 to be thea priori state estimate representing our knowledge of the process

prior to the stept andpt|t to be thea posterioristate estimate at time stept. Thea priori and

thea posterioriestimate errors are

et|t−1 ≡ pt − pt|t−1 (4.67)

et|t ≡ pt − pt|t. (4.68)

Then thea priori anda posterioriestimate error covariances are

Ct|t−1 = E[et|t−1eTt|t−1] (4.69)

Ct|t = E[et|teTt|t], (4.70)

whereE[f(x)] is the expectation operator.

We seek to find an equation that computes ana posterioristate estimatept|t as linear com-

bination of thea priori state estimatept|t−1 and weighted difference between a measurement

gt and a predicted measurementZpt

pt|t = pt|t−1 + Gt(gt − Zpt|t−1). (4.71)

The matrixGt is called the Kalman gain. The aim is to findGt such that the mean-square

estimation error is minimized. The derivation presented in this thesis is along the general lines

of [16]. An alternative derivation from the viewpoint of regression analysis can be found in

[40]. To minimize the mean-square error, the trace of thea posteriorierror covariance matrix

has to be minimized. The trace ofCt|t represents the sum of the error expectations between

measurements and predictions. The argument is that the individual mean-square errors will be

minimized when the total is minimized [16, p. 215]. Continuing from eq. (4.71), we substitute

gt from eq. (4.64)

pt|t = pt|t−1 + Gt(Zpt + nt − Zpt|t−1).

Substituting this into eq. (4.68) and the result into eq. (4.70)


Ct|t = E[(pt − pt|t−1 + Gt(Zpt + nt − Zpt|t−1)

) ·(pt − pt|t−1 + Gt(Zpt + nt − Zpt|t−1)

)T ].

Calculating the expectations and assuming that thea priori error (pt − pt|t−1) is uncorrelated

with the measurement errornt, the following is obtained:

Ct|t = Ct|t−1 −GtZCt|t−1 − Ct|t−1ZT GT

t + Gt

(ZCt|t−1Z

T + Cn,t

)GT

t . (4.72)

Taking the derivative of the trace ofCt|t with respect toGt and noting thatTr[A] = Tr[AT ]

andCt|t−1 is a covariance matrix and therefore symmetric

dTr[Ct|t]dGt

= −2dTr[GtZCt|t−1]

dGt+

dTr[Gt

(ZCt|t−1Z

T + Cn,t

)GT

t ]dGt

. (4.73)

Using matrix derivative identities in eq. (4.73), we obtain

dTr[Ct|t]dGt

= −2(ZCt|t−1)T + Gt

((ZCt|t−1Z

T + Cn,t

)+

(ZCt|t−1Z

T + Cn,t

)T)

.

The matrixZCt|t−1ZT +Cn,t is symmetric sinceCt|t−1 andCn,t are both covariance matrices.

dTr[Ct|t]dGt

= −2Ct|t−1ZT + 2Gt

(ZCt|t−1Z

T + Cn,t

).

Setting this derivative equal to zero and solving forGt

Gt = Ct|t−1ZT

(ZCt|t−1Z

T + Cn,t

)−1. (4.74)

WhenCn,t approaches zero,Gt weights the residual more heavily

limCn,t→0

Gt = Ct|t−1ZT

(ZCt|t−1Z

T)−1

. (4.75)

SettingZ = ZC1/2t|t−1

limCn,t→0

Gt = C1/2t|t−1Z

T(ZZT

)−1. (4.76)


Noting thatZT(ZZT

)−1is the underdetermined version of the Moore-Penrose inverseZ†

limCn,t→0

Gt = C1/2t|t−1C

−1/2t|t−1Z†

limCn,t→0

Gt = Z†. (4.77)

In simple terms it means that as the measurement error covariance decreases the actual mea-

surements are trusted more. If thea priori error covarianceCt|t−1 approaches zero, the Kalman

gainGt will weight less the residual

limCt|t−1→0

Gt = 0. (4.78)

This means that as thea priori error covariance decreases, less trust is put on the actual mea-

surementsgt and more on the predicted measurementsyt.

Having defined the matrixGt, the full expression for thea posterioricovariance matrixCt|t can

be obtained. Substituting eq. (4.74) in to eq. (4.72) and settingD = ZCt|t−1ZT + Cn,t

Ct|t = Ct|t−1 − Ct|t−1ZT D−1ZCt|t−1 − Ct|t−1Z

T(Ct|t−1Z

T D−1)T

+ Ct|t−1ZT D−1D

(Ct|t−1Z

T D−1)T

.

Noting thatD andCt|t−1 are symmetric, we obtain

Ct|t = Ct|t−1 −GtZCt|t−1. (4.79)

The first step of the discrete Kalman filter algorithm is to project the state ahead

pt+1|t = Stpt|t. (4.80)

Then the error covariance

Ct+1|t = StCt|tSTt + Cw,t. (4.81)

The previous two equations constitute the prediction step of the algorithm. Next step is to

correct. First we compute the Kalman gainGt as in eq. (4.74)

Gt = Ct|t−1ZT (pt|t−1)

(Zpt|t−1Ct|t−1Z

Tpt|t−1 + Cn,t

)−1. (4.82)


Next we update the state estimate with respect to the measurementgt

pt|t = pt|t−1 + Gt(gt − Zpt|t−1). (4.83)

The final step is to update the error covariance

Ct|t = Ct|t−1 −GtZpt|t−1Ct|t−1. (4.84)

The previous equations are iterated until convergence and they are known as the discrete

Kalman filter algorithm. As mentioned previously this derivation assumes that the process is

linear, for nonlinear systems the extended Kalman filter is used. This is derived by linearization,

as presented previously§4.4.2, details will be given in the following section.

4.7.2 Nonlinear case: Extended Kalman filters

The parameters of the nonlinear forward model are mapped to the data spaceY using the fol-

lowing relation:

yt = Z(pt) + nt, (4.85)

whereZ(p) : Rm → Rn is a mapping from the parameters space to the data space andnt

is some noise process related to the forward mapping at timet. The forward modelZ(p) is

nonlinear and it will be linearized. The observation equation becomes

yt = Z(p∗,t) + Jp,t(p∗,t)(pt − p∗,t) + nt, (4.86)

wherep∗,t is the linearization point,Jp,t = ∂Z(pt)∂p , is the Jacobian matrix andnt is a zero-mean

Gaussian observation noise process with covariance matrixCn,t.

The linearized problem can be solved recursively using the extended Kalman filter algo-

rithm equations

Gt = Ct|t−1JTp,t(p∗,t)(Jp,t(p∗,t)Ct|t−1J

Tp,t(p∗,t) + Cn,t)−1 (4.87)

pt|t = pt|t−1 + Gt(gt −Z(p∗,t)− Jp,t(p∗,t)(pt − p∗,t)) (4.88)

Ct|t = Ct|t−1 −GtJp,t(p∗,t)Ct|t−1 (4.89)

pt+1|t = Stpt|t (4.90)

Ct+1|t = StCt|tSTt + Cw,t. (4.91)

4.8. Discussion 71

Gt is the Kalman gain and the matricesCt|t and Ct+1|t are the covariance matrices of the

Kalman filtered state vectorpt|t and the predictorpt+1|t respectively. If the observation model

is linearized at the current value of the predictor,p∗,t = pt|t−1, eq. (4.87) becomes

pt|t = pt|t−1 + Gt(gt −Z(pt|t−1)). (4.92)

Eq. (4.88) is correcting the parametersp of the model according to the measured datag.

Eq. (4.90) is projecting the parameters to the next time point usinga priori knowledge in-

corporated in the state transition matrixSt.

4.7.3 Fixed interval smoother

The extended Kalman filter algorithm can be used in real time processing as it only uses current

and past measurements. In the case where data does not have to be processed in real time, esti-

mates can be calculated from all measurements. The estimates from the Kalman filter algorithm

can be further processed using the fixed interval smoother algorithm [4, pp. 187-190]

Xt−1 = Ct−1|t−1STt−1C

−1t|t−1 (4.93)

pt−1|T = pt−1|t−1 + Xt−1(pt|T − pt|t−1). (4.94)

4.8 Discussion

In this chapter various approaches to minimization were presented. Optimization is still a mod-

ern topic of research and we have only covered the area that is relevant to the work presented

in this thesis. Least squares estimation gives an introduction on the principles of optimization

methods. Standard Newton-type algorithms typically include a step length parametersk, which

requires a line search at each iteration. In addition the regularization parameterλ needs to be

defined. Trust region approaches, e.g. the Levenberg-Marquardt method, are robust in most

cases and do not require the step length nor the regularization parameter to be defined.

Using the method of Tikhonov, constraints can be incorporated in to the problem with

the use of penalty functions. In the original formulation of Tikhonov the constraints aimed to

replace the inverse(ZT Z)−1, which is often singular, with a well-behaved bounded inverse

(ZT Z + λP )−1. These penalty functions are a representation of our prior knowledge of the

process governing a particular problem. The Tikhonov regularization can be seen from the

stochastic point of view as a maximuma posterioriestimate. Deterministic methods in general

make many assumptions, for example on the distribution of noise in a process. Statistical esti-

mation on the other hand offers the ability to express the problem with stochastic assumptions


about the distribution of data and noise. The focus in deterministic methods is to obtain as much

information as possible from measured data by building an exact forward model with some ba-

sic assumptions about the noise distribution. In stochastic methods the aim is to incorporate

accuratelya priori information about the sought solutionp and the noise statistics. The uncer-

tainty in the accuracy of the measurements and the forward model is expressed in probability

density functions.

Kalman filters have the cognitive machinery to estimate a temporal process with much

more control of the error distributions, when compared to standard Tikhonov regularization.

Kalman filters also have a built-in matrixSt for progressing the parameters of the model ahead

in time. The extended Kalman filter algorithm will be used in this thesis for the solution of

the dynamic shape estimation problem in§8. In the absence of noise both methods reduce

to the Moore-Penrose inverse, as it can be seen in§4.7.1 eq. (4.77). Setting the weighting

matrix Lw in the Tikhonov method equal to the estimate error covariance matrixC1/2t|t shows

the close relation between the two methods. In the case of TikhonovLw is a typically set as

a diagonal matrix, representing the variances between parameters of the model. The estimate

error covariance matrixCt|t has values off the diagonal representing the covariances between

different parameters of the model, while they change in time. In this sense, it can be said that

Kalman filters offer a formula for calculating the weighting (or covariance) matrix method in

time. A derivation of Newton-type methods from a Wiener-Kolmogorov filtering perspective

has been discussed in [51].

This chapter concludes the discussion on the background of this thesis. In the following

chapters we present methods based on numerical optimisation techniques for the reconstruction

of images and shapes. We begin with the image reconstruction method in the next chapter.

Chapter 5

Image reconstruction method

5.1 Introduction

Standard techniques for reconstructing images from tomographic data such as filtered back-

projection and gridding, fail in limited data cases as they assume that the data set is complete.

Recent results have been very promising in solving problems with limited data. In [151] and

[96] Kolehmainen et al used statistical analysis to reconstruct images in dental radiology. They

chose to minimize an approximation to the total variation functional, because of the difficulty to

calculate its derivatives. More examples of the Bayesian approach in optimisation can be found

in [146],[69]. Candes et al [22], [23] presented exact reconstructions of simulated phantoms

using very limited data (22 radial profiles). An application of the theories presented in [22],

[23] incorporating the k-t approach was presented in [112] for case of cardiac MRI. Candes

has also presented methods for image reconstruction using curvelets [21], [20]. Curvelets are a

particular application of wavelet analysis to the Radon transform. Ye et al [187] used a level set

methodology to reconstruct images from randomly undersampled Fourier data. The minimisa-

tion process was applied on a least-squares functional. Yu and Fessler [188] presented a level

set method for the reconstruction of images in PET.

In this chapter we present a method for the reconstruction of images from limited data sets,

which are typically encountered in dynamic imaging. These limited data sets have the benefit

that data for each time frame can be collected very fast. Thus, motion of the imaged object

during the collection of data is reduced significantly. The presented method does not make any

assumptions about the completeness of the data set. The aim is minimize the difference between

predicted data and measured data. To achieve this, the image reconstruction is formulated as a

least squares problem, as it will be explained in the following sections.

74 Chapter 5. Image reconstruction method

5.2 Forward problem

We approximate the imagef(r) with local basis function as in eq. (4.17)

f = Bpζ , (5.1)

wherer ∈ R2 is a vector of the following formr = rx, ry, Bk is a basis function,pζ ∈ P is

the parameter vector andNζ is the total number of basis functions used in a given grid.

The choice of basis functions was based on their performance in terms of image quality in

the results of [106] and [147]. The Kaiser-Bessel radially symmetric basis functions,or as they

are commonly referred to blobs, are of the following type:

b(r) =1

Im(α)(√

1− (d/a)2)mIm(α√

1− (d/a)2) , (5.2)

whereIm is the modified Bessel function of the first kind,m is the degree,a is the support,

d is the distance from the center of the basis function andα is a shape parameter. Optimal

combinations of the shape parameterα and the support can be found in [147] and [61].

1 2 3 4 5 6 7 8

20

40

60

80

100

120

140

160

180

Figure 5.1: Radon data. A sinogram with 8 projections each with 185 line integrals.

The projection datag = R(f) ∈ Rn (fig. 5.1) will then be equal to

g = RBpζ , (5.3)

wherepζ ∈ RNζ are the unknown blob coefficients andB ∈ RNr×Nζ is the basis functions

matrix, whereNr is the total number of pixels. Each column of matrixB is going to be the

image vector of each basis function. A discrete method for the calculation of the system matrix

J = RB (fig. 5.3) is to create the image of each blob and then take its Radon transform. The

5.3. Inverse problem: Direct solution 75

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 5.2: Radial profile of the Kaiser-Bessel blob in Fourier space (Left) and Radon space

(Right).

matrix can also be calculated analytically in both Fourier and Radon domains. The Kaiser-

Bessel blobs remain radially symmetric in both cases. The derivation of the Fourier transform

of the basis functions uses the Hankel transform, which is equivalent to the Fourier transform

for a rotationally symmetric function. The analytic Fourier formula (fig. 5.2 (Left)) is given in

[106], [61]

bF (k) =akαm(2π)k/2Jm+k/2(z)

Im(α)zm+k/2, with z =

√(2πa||k||)2 − α2, (5.4)

whereJm is them-th degree Bessel function of the first kind andk is the dimension of the

space.

The Radon transform of the basis functions (fig. 5.2 (Right)) is calculated with the use of the

central slice theorem and its formula is given in [106],[62]

bR(t) = aIm+1/2(α)

Im(α)

√2π

αbm+1/2(t). (5.5)

The linear system in eq. (5.3) is typically non-square and a unique solution is unobtainable. We

use the least squares functional to obtain a solution of minimum norm

pζ,min = arg minpζ

||g − Jpζ ||22. (5.6)

5.3 Inverse problem: Direct solution

5.3.1 Least squares estimation

A direct solution for the minimization problem of eq. (5.6) is to use the Moore-Penrose pseu-

doinverse, which gives the following solution:


500 1000 1500 2000 2500 3000 3500 4000

200

400

600

800

1000

1200

1400

Figure 5.3: The system matrixJ . Each column corresponds to the vectorised basis function in

the Radon space.

pζ = (JT J)−1JTg. (5.7)

We test this method with the Shepp-Logan phantom [150] in fig. 5.4. The resolution of

the image is128 × 128 pixels. Data is simulated by taking the Radon transform at 8 angles.

Images are reconstructed using filtered back-projection and the proposed method (fig. 5.5). To

compare images numerically, we use the relative mean square errorrms = ||fg − f ||22/||fg||22 ,

wherefg andf are the ground truth and predicted vectorised images.

Figure 5.4: Ground truth image. Shepp-Logan phantom.

5.3. Inverse problem: Direct solution 77

Figure 5.5: 8 projections. (Left) Filtered back-projectionrms = 1.2521. (Right) Least squares

reconstruction8× 8 grid rms = 0.73092.

5.3.2 Damped least squares estimation

The use of such a small number of basis functions results in very poor reconstructions. The

increase of this number has direct effect in the decrease of the conditioning of matrixJT J ,

making it practically singular. To stabilize this inversion we augment the effectively underde-

termined system in to an overdetermined one with the incorporation ofa priori information

J

λI

p =

g

0

. (5.8)

This corresponds to the ordinary Tikhonov regularization functional

Φ(p) = ||g − Jpζ ||22 + λ||p||22, (5.9)

which has the following minimizer

pζ = (JT J + λI)−1JTg. (5.10)

Using the damped least squares method, the grid resolution can be increased to64 × 64. As

seen in fig. 5.6 the reconstructed image is much more detailed than the least squares version

in fig. 5.5. The increase on the grid resolution produces angular artifacts, that are common

with filtered back-projection methods. In the next section we discuss iterative reconstruction

methods, which can significantly reduce these artifacts.


Figure 5.6: 8 projections. (Left) Filtered back-projectionrms = 1.2521. (Right) Damped least

squares reconstruction 64x64 gridrms = 0.61756.

5.4 Inverse problem: Iterative solution

Using the damped least squares as an initial estimate, we can further increase the quality of the

reconstructions by solving the following constrained optimization problem

pmin = arg minpζ

TV (p) subject to||g − Jpζ ||22 = σ2, (5.11)

whereTV is the total variation functional introduced in§4.6 andσ is a known noise level. We

solve the equivalent Tikhonov regularization problem by minimizing the following objective

functional

Φ(pζ) = ||g − Jpζ ||22 + λTV (pζ). (5.12)

The minimization of this can be thought as a penalty approach to the constrained problem [111].

Well-posedness and convergence of this problem have been proved in [1].

TheTV functional favors solutions with small total variation, yet allowing discontinuities

in the solution. It tends to preserve edge information, while favoring solutions with smaller

derivatives. A discussion in the edge-preserving properties of theTV functional in statistical

estimation can be found in [102]. The total variation functional was originally used for image

denoising tasks [143], [177], [27], [108] and [36]. The method has been applied to tomographic

reconstruction problems both with full and limited data. In [28] the total variation functional

was used for the reconstruction from full data. Their algorithm was named ARTUR. A gen-

eralization of ARTUR was discussed in [34], where the authors reconstructed a phantom from

limited data. From the Bayesian point of view theTV approach of Kolehmainen et al [96],

with its theoretical background given in [151], was applied to both limited sparse and limited

5.4. Inverse problem: Iterative solution 79

view data. They reconstructed 2D and 3D images from dental x-ray data. Persson et al [135]

described an expectation maximization (EM) algorithm. They applied their method for 3D

reconstructions of cardiac phantoms from limited view data.

The TV functional is defined as in eq. (4.50)

TV (pζ) =∫

Ω|∇pζ |dpζ . (5.13)

The presence of the absolute value function makes theTV functional nondifferentiable at the

origin. Therefore we approximate the absolute function, with a continuous functionψ(t) =√

t2 + β2, as seen in fig. 5.7. Other options for this function can be found in [174]. The

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

y

Figure 5.7: The solid line represents the absolute function|t| and the dashed line represents the

approximationψ(t) =√

t2 + β2 with β = 0.1.

approximatedTV functional is

TV (pζ) =∫

Ωψ(|∇pζ |)dpζ . (5.14)

Setting the gradient|∇pζ | =√

(Dxpζ)2 + (Dypζ)2, whereDx andDy arex andy differential

operators, we obtain the discrete version

TV (pζ) =Nζx∑

i=1

Nζy∑

j=1

√((Dx,ijpζ)2 + (Dy,ijpζ)2) + β2, (5.15)

whereNζx andNζy are the total numbers of basis functions in thex andy directions. A variety

of choices for the differential operators can be found in [83] and [137].

It is clear now that the objective functionalΦ in eq. (5.12) is nonlinear, since theTV

penalty term is nonlinear. The derivative of theTV penalty functional is


TV ′(pζ) = DTx diag(ψ′(∇pζ))Dx + DT

y diag(ψ′(∇pζ))Dy, (5.16)

whereψ′(∇pζ) denotes the derivative of the absolute value approximation. TheTV ′(pζ) ∈RNζ×Nζ matrix is block tridiagonal, with the blocks on the diagonal being tridiagonal matrices

and the off-diagonal blocks being diagonal matrices. This is a sparse matrix, as shown in

fig. 5.8.

10 20 30 40 50 60

10

20

30

40

50

60

Figure 5.8: TheTV ′ block tridiagonal matrix.

The next sections are presenting two methods for the solution of the nonlinear problem in

eq. (5.12).

5.4.1 Lagged diffusivity fixed point iteration

Using the Euler-Lagrange equation for the unconstrained optimization problem in eq. (5.12) we

obtain a nonlinear partial differential equation

g(pζ) = −λ∇(

∇pζ√|∇pζ |2 + β2

)+ Z∗(Zpζ − g) = 0, (5.17)

whereZ∗ denotes the adjoint operator ofZ. Rudin, Osher and Fatemi [143] solve this system

of nonlinear equations using a time marching method. It results in a gradient descent method,

or steepest descent with the addition of a line search algorithm for globalization. The slow

convergence of the steepest descent method is not ideal for many applications. In this section

we use the lagged diffusivity fixed point iteration of Vogel [176]. It is based on successive

linearisations of the objective functional using a quasi-Newton approach. It can be considered

as a special case of the ARTUR algorithm [28]. ARTUR has also been applied in the limited

data case in [34]. A complete derivation of the fixed point algorithm can be found in [176] and

[177].


From another point of view the constrained problem can be thought of as Tikhonov regular-

ization§4.6, with a nonlinear penalty functional. Unconstrained problems are in general easier

to handle from a computational standpoint. Both methods (constrained and unconstrained) pro-

duce the same results as long as the parameters are selected appropriately [176]. The iteration

formula resembles the nonlinear Tikhonov regularization iteration

pk+1 = pk + (ZT Z + λTV ′k)−1

(ZT (g − Zpk)− λTV ′

kpk

), (5.18)

whereTV ′k = TV ′(pk) denotes the derivative of the total variation functional with parameters

pk at thek-th iteration. Note that we have dropped the second derivative in the Hessian.(ZT Z+

λTV ′(pk)) of the objective functional, which results in a quasi-Newton approach.

Figure 5.9: 8 projections. (Left) Initial (damped least squares)rms = 0.61756. (Right) Fixed

point reconstructionrms = 0.5975.

0 5 10 15 20 250.596

0.598

0.6

0.602

0.604

0.606

0.608

0.61

0.612

0.614

k

rms

0 5 10 15 20 250

50

100

150

200

250

300

350

400

450

k

|gr ad|

Figure 5.10: 8 projections. (Left)rms error over iteration plot. (Right) Gradient norm plot.

The results of the fixed point method are presented in figs. 5.9 and 5.10. As seen in the right

plot of fig. 5.10, the norm of the gradient has dropped most of the way at about 5 iterations. The


decrease after that is very small, yet it is not insignificant. Examining therms plot (fig. 5.10

(Left)) the error on the image space reduces significantly up to about 15 iterations.

5.4.2 Primal-dual Newton method

Another option for the solution of the constrained minimization problem has been proposed

by Chan et al [27]. Their approach is based on a full-Newton scheme. The highly nonlinear

nature caused by theTV functional is linearized more efficiently with the introduction of a dual

variable. We introducea = ∇pζ√|∇pζ |2+β2

as the new variable. This transforms the problem of

eq. (5.17) in to a system of nonlinear equations

−λ∇a + Z∗(Zpζ − g) = 0

a√|∇pζ |2 + β2 −∇pζ = 0. (5.19)

For the 2D case we are examining in this chapter, we split the dual variable in to itsx andy

components,a = u,v. SettingB(∇pζ) = diag(ψ′(∇pζ)), the system of (5.19) is now

defined

M(u,v,pζ) =

B(∇pζ)u−Dxpζ

B(∇pζ)v −Dypζ

λDTx u + λDT

y v + ZT (Zpζ − g)

= 0 (5.20)

and its derivative

M ′(u,v,pζ) =

B(∇pζ) 0 B′(∇pζ)u−Dxpζ

0 B(∇pζ) B′(∇pζ)v −Dypζ

λDTx λDT

y ZT Z

. (5.21)

The solution of the system (5.20) by Newton’s method requires the iterative solution of

Bk 0 B′kuk −Dxpk

0 Bk B′kvk −Dypk

λDTx λDT

y ZT Z

∆u

∆v

∆p

= −

Mu

Mv

Mpk

, (5.22)

wherepk, uk, vk are the parameter and dual vectors at thek-th iteration,Bk = B(∇pk) and

Mu, Mv andMpkdenote theu, v andpζ components ofM at thek-th iteration. We convert

the system (5.22) to block upper triangular form by block row reduction


Bk 0 −E11Dx − E12Dy

0 Bk −E21Dx − E22Dy

0 0 ZT Z + λH

∆u

∆v

∆p

= −

Mu

Mv

gradk

, (5.23)

where

E11 = diag(w. ∗Dxpk. ∗ uk)

E12 = diag(w. ∗Dypk. ∗ uk)

E21 = diag(w. ∗Dxpk. ∗ vk)

E22 = diag(w. ∗Dypk. ∗ vk),

the discretised diffusion operator is

H = DTx BkE11Dx + DT

x BkE12Dy + DTy BkE21Dx + DT

y BkE22Dy (5.24)

andgradk = ZT (Zpk − g) + λTV ′kpk is the gradient of the objective functional at thek-th

iteration. From the system (5.23) we obtain first the update forpζ

∆p = −(ZT Z + λH)−1gradk (5.25)

and the updated parameter vector

pk+1 = pk + ∆p. (5.26)

The updates for the dual variables are

∆u = −uk + Bk(Dxpk + (E11Dx + E12Dy)∆p) (5.27)

∆v = −vk + Bk(Dxpk + (E21Dx + E22Dy)∆p). (5.28)

The dual variables have to be restricted to be within the conjugate setC∗ = y ∈ Rn| |y| ≤ 1,i.e. the unit ball, to ensure convergence. This can be done by a line search method as follows:

sk = max0 ≤ s ≤ 1|(uk + s∆u,vk + s∆v) ∈ C∗. (5.29)

With the step lengthsk calculated, we can update the dual variables


uk+1 = uk + sk∆u

vk+1 = vk + sk∆v.

For a more detailed discussion on the primal-dual method refer to [27] and [174]. The

reconstructed images are presented in fig. 5.11.

Figure 5.11: 8 projections. (Left) Initial (damped least squares)RMS = 0.61756. (Right)

Primal-dual reconstructionRMS = 0.5975.

0 5 10 15 20 250.596

0.598

0.6

0.602

0.604

0.606

0.608

0.61

0.612

0.614

k

rms

0 5 10 15 20 250

50

100

150

200

250

300

350

400

450

k

|gr ad|

Figure 5.12: 8 projections. (Left)RMS error over iteration plot. (Right) Gradient norm plot.

The primal-dual method produces the same reconstruction (fig. 5.11) with the fixed point

method (fig. 5.9). It is the convergence speed that is of interest. While the gradient norm plot

(figs. 5.10 and 5.12 (Right)) of both methods are very similar, therms image error of the

primal-dual method in fig. 5.12 converges in 5 iterations. In the next section the method is

further developed to incorporate constraints on the maximal and minimal intensity expected in

the reconstructed images.


5.4.3 Constrained optimisation

Further to the previous results, we can enforce constraints on the intensity values of the parame-

ters. We can incorporate thea priori knowledge that the image will not contain negative values.

The constrained optimization problem to be solved is

minpζ

Φ(pζ) subject toc(pζ) ≥ 0. (5.30)

For the solution of this problem we modify the primal-dual method with a projected method

[174],[111],[49]. We define an active set as follows:

A(p) = i|0 ≤ pi ≤ ε, (5.31)

whereε is a small number, typically reducing as the iteration progress. This can be based on

statistical analysis or assumptions about the noise in the system. The active parameters are

going to be on the boundaries of the feasible region. We reduce the Hessian as follows:

H(p) =

δij if i ∈ A(p) or ∈ A(p)∂2Φ

∂pi∂pjotherwise

. (5.32)

We obtain a solution from the linear system of equations and update the parameter vector

pk+1 = T pk + ∆p, (5.33)

whereT is a projection operator defined as:

T (p) =

pi if pi ≥ 0

0 if pi ≤ 0. (5.34)

An alternative to the projection methods are the interior and exterior point methods [45].

An interior point method has been used in [108]. Kolehmainen et al [96] used an exterior point

method to apply the positivity constraint.

Using the projected method, as described, we can also enforce an upper bound constraint

by replacing0 in the above equations with a maximal value and changing the order of the

inequalities in eq. (5.34). This maximal value can be obtained in cardiac MRI with fair accuracy,

by building a time averaged image, which is fully sampled.


Figure 5.13: 8 projections. (Left) Initial (damped least squares)rms = 0.61756. (Right)

Projected primal-dual reconstructionrms = 0.4833.

1 1.5 2 2.5 3 3.5 4 4.5 50.48

0.5

0.52

0.54

0.56

0.58

0.6

k

rms

1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

k

|gr ad|

Figure 5.14: 8 projections. (Left)rms error over iteration plot. (Right) Gradient norm plot.

The application of intensity inequality constraints combined with the total variation penalty

functional has a great effect in the reduction of artifacts, as seen in fig. 5.13. As the constraints

are enforced, the gradient norm (fig. 5.14 (Right)) increases as it climbs out of an infeasible

solution area of negative and very high intensity values. In the next section we apply the method

to simulated and measured cardiac data sets.

5.5 Results

In this section we apply the primal-dual method with both constraints enforced by an active set

method. The grid resolution as previously was fixed at64 × 64 and the reconstructed images

128 × 128. Results are obtained for various degrees of undersampling. For comparison we

reconstruct images using filtered back-projection and gridding, currently considered to be the

standard in image reconstruction from tomographic data.

5.5. Results 87

5.5.1 Simulated cardiac data

Data was simulated by taking the Radon transform at 4, 8, 13 and 16 equispaced angles from a

fully sampled cardiac image in fig. (5.15). This acts as a ground truth image for the numerical

comparison, withrms, between the filtered back-projection and the primal-dual method.

Figure 5.15: Ground truth image. Fully sampled cardiac image.

The reconstructed images for both methods are shown in fig. 5.16 with various degrees of

radial undersampling. Fig. 5.17 shows the correspondingrms errors. In the case of 4 profiles,

the filtered-backprojection produces an image dominated by streaks, making it practically im-

possible to distinguish the location and shape of the heart. In the case of the primal-dual method

some gross features of the anatomy are visible. As the number of profiles increase these features

become clearer in both images. The corruption of the filtered back-projection images by an-

gular artifacts deforms small anatomical features completely and in some locations introduces

new features which are not present in the ground truth image. In the primal-dual reconstruction

the streaky artifacts are missing. While general features are represented well, even some of the

finer ones are also visible. On this finer scale the primal-dual method does not introduce new

signals or deforms their apparent shape. Details that are not reconstructed are typically blurred.


4

8

13

16

Figure 5.16: Simulated data reconstructions. The numbers on the left column indicate the num-

ber of profiles. (Left) Filtered backprojection. (Right) Projected primal-dual reconstruction.

5.5.2 Measured data from MRI

ECG gated data was acquired from a healthy volunteer. A total of 25 phases each with 208

radial profiles were collected using a five-element array receive coil. The data used in this

5.5. Results 89

4 6 8 10 12 14 160.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Filtered backprojectionPrimal−dual

rms

number of profiles0 20 40 60 80 100 120

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Ground truthFiltered backprojectionPrimal−dual

x

intensity

Figure 5.17: (Left)Simulated cardiacrms plot over the number of profiles. The dashed line

represents the filtered backprojection method and the solid the primal-dual method. (Right)

Comparison of central lines of the ground truth and reconstructed images for the case of 8

radial profiles.

experiment was generated from the first phase of the acquired data by undersampling in various

degrees. This was done by using everyn-th profile according to the total number (Nunder) of

profiles in the undersampled set withn = 208/Nunder. Using 8 radial profiles results in a

26-fold acceleration compared to the original radial acquisition. For the case of real-time MRI,

a total of about 200 radial profiles can be collected within a single heart beat using a fast steady

state free precession sequence. To transform the data in the Radon space, we 1D inverse Fourier

transformed along each radial profile, according to the central slice theorem.

The quality of the reconstructions is compared using normalized images. First we will

investigate the single coil reconstructions. The fully sampled gridding reconstruction is shown

in fig. 5.19. This was used as ground truth data and undersampled reconstructions with the

gridding method from§2.3 are compared with primal-dual ones in fig. 5.18. In fig. 5.20 the

rms errors are presented for both methods using different degrees of undersampling.


Single coil reconstructions

4

8

13

16

Figure 5.18: Coil 1 reconstructions from measured data. The numbers on the left column

indicate the number of profiles. (Left) Gridding. (Right) Projected primal-dual reconstruction.

5.5. Results 91

Figure 5.19: Coil 1. Fully sampled gridding reconstruction used as ground truth image.

4 6 8 10 12 14 160.4

0.6

0.8

1

1.2

1.4

1.6

1.8GriddingPrimal−dual

number of profiles

rms

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Ground truthGriddingPrimal−dual

x

intensity

Figure 5.20: (Left) Coil 1rms plot over the number of profiles. The dashed line represents the

gridding method and the solid the primal-dual method. (Right) Comparison of central lines of

the ground truth and reconstructed images for the case of 8 radial profiles.


Multiple coil reconstructions

To reconstruct images from multiple receive coils using the proposed method, the operator

J was changed to a Radon transform followed with a multiplication with the coil sensitivity

matrix. As there was no body coil used for the collection of data, sensitivity matrices were

calculated by dividing a time average image of each coil with the square root of the sum of

squares of all the coil images, similarly to [139]. Time averaged images can be reconstructed

with a gridding method using data from all time points in a scanning sequence where profiles are

interleaved in order to span the 180 degrees. For comparison the gridding reconstructed single

coil images were combined in a least-squares sense using the sensitivity matrices by solving the

following linear system for each pixel

Cx,y = Sx,yf(x, y) (5.35)

f(x, y) = S†x,yCx,y,

whereSx,y is a vector containing all coil sensitivity values at pixel location(x, y), f is the image

andCx,y is the vector containing the intensity value of each coil image atx, y. This method

produces superior results when compared with the square root of the sums of squares of the coil

images. It is computationally very efficient since it can be solved per pixel. Normalised images

of the proposed method and the least squares gridding method were compared with the fully

sampled image (fig. 5.21), which was reconstructed using the least squares gridding approach

with the sensitivity matrices described previously. The results of the LS gridding and projected

primal-dual reconstructions are displayed in fig. 5.22 and the correspondingrms errors in fig.

5.23.

Figure 5.21: Multiple coil. Fully sampled LS gridding reconstruction used as ground truth

image.

5.6. Discussion 93

4

8

13

16

Figure 5.22: Multiple coil reconstructions from measured data. The numbers on the left column

indicate the number of profiles. (Left) LS gridding. (Right) Projected primal-dual reconstruc-

tion.

5.6 Discussion

In this chapter we have presented direct and iterative methods for the reconstruction of images

using an inverse problem approach. The total variation functional in combination with the ap-


4 6 8 10 12 14 160.5

0.6

0.7

0.8

0.9

1

1.1

1.2LS griddingPrimal−dual

number of profiles

rms

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7Ground truthLS griddingPrimal−dual

x

intensity

Figure 5.23: (Left) Multiple coilrms plot over the number of profiles. The dashed line rep-

resents the LS gridding method and the solid the primal-dual method. (Right) Comparison of

central lines of the ground truth and reconstructed images for the case of 8 radial profiles.

plication of constraints improves on the quality of the reconstructed images and greatly reduces

angular artifacts. The approaches discussed were based on direct inversion of a large sparse ma-

trix. This is a computationally and memory demanding task. Apart from the typical argument

of increasing memory capacity and processing power in computers, the direct inversion can be

replaced with a more efficient iterative linear system solver, for example a conjugate gradient

method [174]. Conjugate gradient methods do not necessitate the complete matrix to be stored,

thus allowing higher grid resolutions. Further improvements to the computational aspect of the

minimization problem might be achieved using a multigrid scheme [175].

While the damped least squares method can be solved efficiently using techniques for

sparse matrix inversion, the nonlinear methods require a lot more inversions due to their iterative

nature. The primal-dual method was the most computationally demanding method per iteration.

When compared though to the fixed point method in terms of convergence speed, its benefits

become clear, as it reaches the desired solution in approximately a third of the iterations needed

by the fixed point method. The difference in convergence speed is due to the use of second

order derivatives and the improved linearisation in the primal-dual method. The novel methods

introduced in this chapter were compared to the standard filtered back-projection and gridding

approaches. It has to be noted that both of these approaches produce very similar results in

limited data cases. These methods were originally designed to deal with complete data sets and

propagate high frequency information very well in that case. In the case of undersampled radial

MRI high frequency information is very sparsely sampled, while sampling near the center is

dense. Standard approaches can be smoothed to overcome these problems, but they tend to blur

a lot of useful information as well. The iterative approach discussed in this chapter does not

throw away high frequency information, but attempts to reconstruct it, with the only restriction

5.6. Discussion 95

being the grid resolution, some times successfully, while other times it produces smooth results.

In the majority of cases it does not produce high frequency artifacts, like introduction and

deformation of small structures, while greatly reducing the angular artifacts.

It was shown that the inverse problem approach produces superior results both visually and

numerically to standard methods in both simulated and measured data studies. For the measured

data case we applied the method to both single and multiple receive coil data. It is not the aim of

our method to replace existing parallel imaging technologies but to demonstrate the feasibility

of its application with multiple receive coil data. We believe that our method could potentially

be combined with SENSE (§2.4.2). It is also worth noting that while the results of this chapter

were based on angular samples spanning the 180 degrees, the method can be also be applied

to limited view problems, where the total imaging angle is less than 180, and generally to any

Fourier sampling pattern.

In the next chapter we leave temporarily the problem of image reconstruction aside and

focus on the direct reconstruction of shapes from measured data.


Chapter 6

Shape reconstruction method

In this chapter we present a method for the reconstruction of shapes directly from measured

data. Our model-based approach is stated as a minimisation problem between the predictions

of our model and the data. It has the benefit that images can be segmented from very limited

data by working directly in the data space. Feng et al [65] have presented a level set method,

which segments objects of constant interior in simple backgrounds. A level set method for

the reconstruction of surfaces was discussed in [182] and [42]. The surfaces contained volumes

with homogeneous intensity in an homogenous but separable background. In [10] and [9] Battle

et al reconstructed surfaces using a Bayesian approach.

We assume in this chapter that the background and constant interior intensity of the object

are known. Our approach is based on least squares minimisation using global basis functions for

the boundary of the shape. Using this direct approach shapes can be reconstructed from very

few samples, which would not be adequate to form an image and segment it with traditional

‘snake’ approaches, presented in§3.1.

6.1 Forward problem

A planar curve can be modelled with the use of basis functions

C(s) =

x(s)

y(s)

=

Nγ∑

n=1

γx

nθn(s)

γynθn(s)

, s ∈ [0, 1], (6.1)

whereθn are periodic and differentiable basis functions,Nγ is the number of harmonics and

γxn, γy

n ∈ R are the weights ofθn. The basis functions used are trigonometric functions [157],

as described in§4.2.2.

A global representation of the form (6.1) does not have such limitations as convexity for

admissible domains, but it does have the drawback that it is difficult to set constraints such as

non-self-intersection. Self-intersection occurs when the contour intercepts itself, that isC(s1) =

98 Chapter 6. Shape reconstruction method

C(s2) for two parametric pointss1 ands2. This clearly is a problem as it does not represent

realistic boundaries.

So far there is no analytic solution to the problem of defining a relation in the parameters

γ, when a self-intersection occurs. We solve this problem by detecting the self-intersection

with an exhaustive search. This is feasible due to the small size of the search space, that is the

number of pixels belonging to the curve. This can be reduced further by taking in to account

that neighboring pixels by definition cannot intersect each other.

Given a point of self-intersectionse, the pixels belonging to the smallest loop are removed

from the curve, as seen in fig. 6.1. The remaining curve, free of self-intersections, requires

20 40 60 80 100 120

20

40

60

80

100

120

se

C( s)

rx

ry

20 40 60 80 100 120

20

40

60

80

100

120

rx

ry

C (s)

Figure 6.1: (Right) Contour with self-intersection at parametric pointse. (Left) Corrected

contour with the small loop removed.

reparameterisation, as thes points will not be spread evenly in the unit interval[0, 1). There

will be a clear gap where the self-intersection was removed. The parametric length of the curve

will be reduced. This reduction can be calculated by taking the parametric difference between

the exit and the entry sample of the self-intersection. Due to this reductionsr in parametric

length it is required to scale the corrected contour to have again length equal to 1. The scale

factor is calculated as follows:

sc =1

1− sr.

The scaling of the new list for the non-self-intersecting contour is preformed by multiplying

each parametric difference between two consecutive samples withsc. At the point where the

self-intersection occurred, the new parametric distance is unknown. This is approximated from

the mean of all parametric distances in the corrected contour. The new list in the correct scale

will be used to calculate the boundary parametersγ in a similar manner to [5].

The parametarisation formula based on the fact that the basis functionsθn resemble a

6.1. Forward problem 99

truncated Fourier series, is the following:

If k is odd

γxk = 2

∫ 1

0Cx(s) cos((k − 1)πs) ds

γyk = 2

∫ 1

0Cy(s) cos((k − 1)πs) ds.

If k is even

γxk = 2

∫ 1

0Cx(s) sin(kπs) ds

γyk = 2

∫ 1

0Cy(s) sin(kπs) ds.

Using the trapezoidal rule for numerical integration, the discretised form of the previous equa-

tions becomes:

if k is odd

γxk = 2

Ns∑

n=1

((Cx(n) cos((k − 1)πsn)) + (Cx(n + 1) cos((k − 1)πsn+1)))∆sn

2

γyk = 2

Ns∑

n=1

((Cy(n) cos((k − 1)πsn)) + (Cy(n + 1) cos((k − 1)πsn+1)))∆sn

2,

if k is even

γxk = 2

Ns∑

n=1

((Cx(n) sin(kπsn)) + (Cx(n + 1) sin(kπsn+1)))∆sn

2

γyk = 2

Ns∑

n=1

((Cy(n) sin(kπsn)) + (Cy(n + 1) sin(kπsn+1)))∆sn

2,

where∆sn is then-th parametric difference between then-th sample and the(n+1)-th sample

andNs is the total number of samples in the contour. Keep in mind that the DC terms in the

parameters of contours are equal to12γx

1 and 12γy

1 . Results for the re-parameterisation of curves

with Nγ = 7 are presented in table 6.1. The parametersγ are estimated from pixel locations

and parametric differences of each sample of the contour using the above equations. The small

deviations from the original parametersγ used to produce the two contours are in the sub pixel

level.


γx 32 10 0 0 0 0 0

γx 32 10.0049 -0.0314 0.0133 0 0.0075 0

γy 27 0 12 0 0 0 0

γy 26.9880 0.0377 11.9939 0.0001 0.0080 0 0.0030

γx 32 10 0 4 0 0 0

γx 32 9.9996 -0.0313 4.0152 -0.0251 0.0171 0

γy 27 1 11 0 0 0 0

γy 26.9880 1.0356 10.9881 0.0027 0.0073 0.0015 0.0027

Table 6.1: Results for 2 re-parameterised contours.γx andγy are the original coefficients of the

basis functions andγx,γy are the reconstructed coefficients.

Pixel intensities are calculated depending on whether they belong in the interior, boundary

or background. This constitutes the mapping of the model to the image spaceG(p) : P 7→ X,

whereP is the parameter space andX is the pixel space. The forward problem is completed

by the mapping of the predicted image to the data spaceR : X 7→ Y, whereR is the Radon

transform.

The combined mappingZ = R(G(γ)) is nonlinear and ill-posed. We seek a solution to

the following Tikhonov regularization problem

γmin = arg minγ||g −Z(γ)||22 + λ||γ||22. (6.2)

6.2 Inverse problem

The nonlinear problem of eq. (6.2) can be solved by the method of Levenberg [105] and Mar-

quardt [117]. This method solves a linear approximation at each iteration. The update is given

by

γk+1 = γk + (JT J + λI)−1JT (g − Jγk), (6.3)

whereJ = ∂(RG)(γ)∂γ is a Jacobian matrix andλ is a regularization parameter controlled by the

optimization method dependant on the objective error. The Jacobian of the operatorG has been

analytically calculated in [94]

JG =

∫ s2

s1ny(s)θn(s) ds n ∈ γx

− ∫ s2

s1nx(s)θn(s) ds n ∈ γy

, (6.4)

6.3. Results 101

wheres1 ands2 are the parametric intersection points of the curve with a boundary pixel (fig.

6.2) andny = γyn

∂θyn

∂s andnx = −γxn

∂θxn

∂s are thex andy directions of the normal to curve. The

derivatives of the basis functionsθn are as follows:

∂θ1∂s = 0∂θn∂s = 2π n

2 cos(2π n2 ), if n is even

∂θn∂s = −2π n−1

2 sin(2π n−12 ), if n is odd

.

Figure 6.2: Exact parametric pointss1 ands2 of the intersection of the curve with a pixel.

Due to the very small size of a pixel it can be assumed thatθn andnx(s) are constant over the

pixel, eq. (6.4) becomes forx∫ s2

s1

ny(s)θxn ds ≈ (s2 − s1)ny(

s1 + s2

2)θx

n(s1 + s2

2), (6.5)

wheres1 ands2 are the parametric points of intersection. Similarly eq. (6.4) fory is

∫ s2

s1

nx(s)θxn ds ≈ −(s2 − s1)nx(

s1 + s2

2)θy

n(s1 + s2

2)

∫ s2

s1

nx(s)θxn ds ≈ (s1 − s2)nx(

s1 + s2

2)θy

n(s1 + s2

2). (6.6)

6.3 Results

To test the shape reconstruction method we examine two cases: the background being equal to

zero and the more general one where the background is non-zero but it is known. For both of

these cases we assume that the interior of the shape is known and constant. Data was simulated

by taking the Radon transform at 8 angles that span the 180 degrees. For all the experiments we

have used a total of 14 basis functions coefficients, 7 for each dimension, for the description of

the contours.


6.3.1 Simulated data

The first case where background contains no signal is demonstrated with a cartoon heart of

constant interior (fig. 6.3). The boundary of the object has high curvature at some locations,

yet the method is capable of the recovering the shape almost perfectly (fig. 6.4). The norm

of the gradient of the objective functional is reducing till it converges approximately after 7

iterations, as seen in fig. 6.5. The shape is also reconstructed accurately even if the data has

been corrupted with noise. Reconstructed data with 15% added Gaussian noise is shown in

fig. 6.6. The gradient norm over iteration plot can be seen in fig. 6.7. The shape reconstruction

approach can be extended for multiple objects. In multiple shape reconstruction it would be

desirable to impose constraints such as spring models between the contours. The trigonometric

approach makes it difficult to apply these local methods as there are no control points to where

the springs can be attached and the forces calculated. To overcome the problem of the shapes

intersecting each other, we fill the intersected area with the sum of both constant interiors. This

imposes a heavy penalty on intersecting areas and if the model is accurate enough it tends to

overcome this problem. We demonstrate the case of two objects in fig. 6.9. Both of the contours

are reconstructed with accuracy (fig. 6.9 and fig. 6.10). In figs. 6.12-6.13 results are shown with

simulated data from a cardiac image. The cardiac image (fig. 6.11) was created from a fully

sampled data set with a filtered back-projection method. The reconstructed shape approximates

the truth well. There are two sources of error: the approximation of the interior of the shape

with a constant value and the papillary muscle which has very low intensity values compared

with the rest of the interior of the left ventricle.

Figure 6.3: Ground truth image. Cartoon heart.

6.3. Results 103

Figure 6.4: Simulated data with no background. (Top Left) Initial superimposed to ground truth

image. (Top Right) Initial predicted image. (Bottom Left) Final superimposed to ground truth

image. (Bottom Right) Final predicted image.

0 2 4 6 8 10 12 14 160

50

100

150

200

250

300

k

|gr ad|

Figure 6.5: Simulated data with no background. Gradient norm plot over iteration.


Figure 6.6: Simulated data with no background and 15% added Gaussian noise. (Top Left)

Initial superimposed to ground truth image. (Top Right) Initial predicted image. (Bottom Left)

Final superimposed to ground truth image. (Bottom Right) Final predicted image.

0 2 4 6 8 10 12 14 160

20

40

60

80

100

120

140

k

|gr ad|

Figure 6.7: Simulated data with no background and 15% added Gaussian noise. Gradient norm

plot over iteration.

6.3. Results 105

Figure 6.8: Ground truth image with multiple shapes.

Figure 6.9: Simulated data with no background. (Top Left) Initial superimposed to ground truth

image. (Top Right) Initial predicted image. (Bottom Left) Final superimposed to ground truth

image. (Bottom Right) Final predicted image.


0 5 10 150

20

40

60

80

100

120

140

k

|gr ad|

Figure 6.10: Simulated data with no background. Gradient norm plot over iteration.

Figure 6.11: Ground truth image. Simulated cardiac phantom.

6.3. Results 107

Figure 6.12: Simulated data with known background. (Top Left) Initial superimposed to ground

truth image. (Top Right) Initial predicted image. (Bottom Left) Final superimposed to ground

truth image. (Bottom Right) Final predicted image.

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

35

40

45

k

|gr ad|

Figure 6.13: Simulated data with known background. Gradient norm plot over iteration.


6.3.2 Measured data from MRI

Data was obtained directly from an MRI scanner as described in§5.5.2. As seen, in fig. 6.14,

from the fully sampled reconstruction the signal contains significant noise, which appears as

freckles in the fully reconstructed image. The shape reconstruction method was able to approx-

imate the true boundary closely (fig. 6.15). The gradient norm plot is shown in fig. 6.16. In

fig. 6.18 the shape was reconstructed from multiple coil data. The fully sampled reconstruction

using the LS gridding method, presented in§5.5.2, is the ground truth image fig. 6.17. The

sensitivity matrices for the multiple coil reconstruction were calculated as described previously

in §5.5.2.


Figure 6.14: Ground truth image calculated from a fully sampled single coil data set.

6.3. Results 109

Figure 6.15: Measured single coil data with known background. (Top Left) Initial superimposed

to ground truth image. (Top Right) Initial predicted image. (Bottom Left) Final superimposed

to ground truth image. (Bottom Right) Final predicted image.

1 2 3 4 5 6 7 813.2

13.4

13.6

13.8

14

14.2

14.4

14.6

14.8

k

|gr ad|

Figure 6.16: Measured single coil data with known background. Gradient norm plot over itera-

tion.



Figure 6.17: Ground truth image calculated from a fully sampled multiple coil data set.

Figure 6.18: Measured multiple coil data with known background. (Top Left) Initial superim-

posed to ground truth image. (Top Right) Initial predicted image. (Bottom Left) Final superim-

posed to ground truth image. (Bottom Right) Final predicted image.

6.4. Discussion 111

1 2 3 4 5 6 7 81.76

1.78

1.8

1.82

1.84

1.86

1.88

1.9

1.92

k

|gr ad|

Figure 6.19: Measured multiple coil data with known background. Gradient norm plot over

iteration.

6.4 Discussion

The shape reconstruction method has been demonstrated to work with both simulated and mea-

sured data. Results have been demonstrated for the case of 8 radial profiles. The method though

can be applied to other degrees of angular sampling. The number of objects has to be known

in advance. This might be considered as a disadvantage when compared to level set methods

which are topologically adaptive. It is often the case though that the number of shapes is known

in many applications and additional constraints have to be applied to level set methods to re-

strict the topology. The quality of the reconstruction is dependant on the knowledge of the

interior of the shape. If the model of the shape is accurate, then the shape reconstruction can

be expected to be robust and precise. Nevertheless the shape reconstruction method gives good

approximations even when this knowledge is not available. In the case of noisy data combined

with an inaccurate interior intensity model the quality of the reconstructed shapes decreases as

seen in figs. 6.15 and 6.18.

As it is typical with many ‘snake’ approaches, a close initialisation will result in a good

approximation of the boundary. Our inverse problem approach is not based on derivative in-

formation, but on a accurate model of the object. As it was demonstrated in the results of the

previous section, when the model is accurate the segmentation is robust and the initial posi-

tion does not have to be very close to the real boundary. In this case, our shape reconstruction

method will produce accurate results even from less than 8 angular samples. When the model

is precise then the solution can be thought of as a global minimum for the objective func-

tions. On the other hand, when the model is not a good approximation of the true object then

g /∈ Range(Z(γ)), which typically implies that there are many local minima, the true solution


is hard to find. The problem of reconstruction from incomplete Radon data also depends on the

background structure. In a simple, for example a zero intensity, background, the solutions are

easy to find as they lie on a deep valley of the objective function. In a complex background

with a variety structures, many solutions exist that minimize the objective function locally. The

valleys are smaller, making the true solution much harder to be tracked. It is then required

to incorporatea priori information that will restrict the solution space to a well-defined set of

meaningful results.

In the next chapter we discuss the problem of reconstructing a shape of unknown, but

smooth, interior intensity in an unknown background.

Chapter 7

Combined reconstruction method

Total-variation-based methods in image reconstruction from tomographic data have been

demonstrated in the works of [28], [96] as discussed in§5.4. The approach discussed in this

chapter differs significantly from these methods in the fact that it combines the image recon-

struction with the shape reconstruction. Ye et al [187] have presented a method, which alternates

the minimization process between the reconstruction of the image and the shape. Our approach

is based along the same general lines of an alternating minimisation approach. It uses the re-

constructed image for the estimation of the background and interior of the shape and then takes

advantage of the shape information in the image reconstruction.TV based methods have the

property of enhancing edges when compared to standard Tikhonov regularisation which tend to

produce smooth solutions. This edge enhancing property applies globally to all locations of the

image. The combined shape and image reconstruction method enjoys this global edge enhanc-

ing property while giving the ability to enhance edges further on the boundary of the estimated

shape.

Our method is based on the ideas presented in chapters§5 and§6. It is an initial solution to

the problem of estimating shapes with an unknown intensity in an unknown background from

limited data.

7.1 Forward and inverse problem

The first part of the problem is to estimate the boundary of the shape. The aim is to obtain the

solution to the nonlinear minimisation problem

γmin = ||g −Z(γ)||22 + λγ ||γ||22, (7.1)

where the operatorZ(γ) = (RG)(γ) is the forward operator,g is the measured data,γ is the

shape parameter vector andλγ is a regularization parameter.

The operatorG : P 7→ X maps the parameters of the shape in to the image space. Pixels

114 Chapter 7. Combined reconstruction method

are classified as belonging to the interior, exterior or background. The interior of the object is

modelled as a smooth varying distribution of intensities. These intensities are calculated from

the reconstructed image and they are restricted to be within a small percentage of the maximal

intensity in the interior of the object. The background intensities are estimated directly from

the reconstructed image. Boundary pixels are calculated relative to their area belonging in the

interior and the background. Surrounding these pixels, we assume that there is a narrow band of

low intensities. This is modelled from the reconstructed image in a similar way to the interior

intensities by selecting the minimum value within that band and restricting the corresponding

coefficients as before. The operatorR : X 7→ Y is the Radon transform, which maps the

predicted image to the data space. The shape reconstruction problem in eq. (7.1) is solved with

the Levenberg-Marquardt method, which gives the following update:

γk+1 = γk + (JT J + λI)−1JT (g − Jγk). (7.2)

In the image reconstruction part of the problem we solve

minpζ

= arg minpζ

||g − Jppζ ||22 + λTVβ(pζ) subject toc(pζ) ≥ 0, (7.3)

wherec are the constraint functions, as described previously in§5.4.3,pζ is the blob parameter

vector and the total variation functional

TVβ(pζ) =∫

Ωψβ(|∇pζ |)dpζ (7.4)

is modified to each specific shape by altering the functionψ according to the estimated shape.

Douiri et al [39] proposed the use of the Huber function [80] for the approximation of the

absolute value function. The Huber function is onlyC1 continuous, making it unsuitable for

the primal-dual method which requires the calculation of the second derivative. The proposed

ψ =√

t2 + β2 does not show severe signs of the staircase effect1 [26]. By increasingβ,

the derivative of theψ function is becoming more isotropic, approximating more a Tikhonov

solution. The reduction ofβ makes the function more anisotropic. All these changes inβ

mainly effect the area where the gradient is small (fig. 7.1). In the intervals where the gradient

is large the function is approximately the same.

Our image reconstruction approach is based on the estimation of the shape, which enhances

our belief that an edge exists in a particular location. Interpreting the behavior of theψ′ in

relation to our shape estimation method, leads us to the following conclusions. If a location

1Solutions that are piecewise constant

7.2. Results 115

−1 −0.5 0 0.5 10

2

4

6

8

10

12

14

16

18

20background β = 0.1interior β = 0.15boundary β = 0.05

t

ψ′(t)

Figure 7.1: Plot of the derivative ofψ(t) for different values ofβ. These values are assigned ac-

cording to the classification of intensity coefficients as background (solid line), interior (dotted

line) and boundary (dashed line).

is estimated as a boundary coefficient we will penalise even the small gradients. If a location

is estimated to be in the interior, then small gradients will be penalised less as we assume that

the interior intensities are smoothly distributed. On the background region, we seek to enhance

edges but we do not know their exact locations and for that reason we use a value that lies in

between the interior and the boundaryβ values. The minimisation problem in eq. (7.4) is solved

with the projected primal-dual method presented in sections§5.4.2 and§5.4.3.

7.2 Results

The intensity coefficients are initialised using the damped least squares method, presented in

§5.3.2. Given this reconstructed image the shape parameters are initialised near the object of in-

terest. We begin the iteration by solving first the shape estimation problem with the Levenberg-

Marquardt approach. After each iteration of the shape estimation problem we re-solve the im-

age reconstruction problem using a few iterations of the projected primal-dual method with the

shape specificβ values. In this approach we use the following beta valuesβi = 0.11, βe = 0.1

andβb = 0.09 for the interior, background and boundary coefficients. A more sophisticated

approach can chooseβ values from the statistical properties of the expected edges, interior of

the shape and background intensities. This alternating minimisation process is repeated until

the convergence criteria are met.

We demonstrate our approach using 5 image iterations for every shape iteration. We begin

with simulated cardiac data, produced by taking the Radon transform of a fully reconstructed

cardiac image (fig. 7.2) at 8 angles. The reconstructed shapes and images are shown in fig. 7.3.


An enhanced image has been created using the estimation of the boundary of the shape and the

reconstruction of the image ((Left) fig. 7.4). Further to that we apply our method to measured

MRI data with 8 radial profiles, as described in the previous chapters, from single and multiple

receiver coils. The single coil reconstructions can be seen in fig. 7.6. On the left of fig. 7.6

the initial and final estimated shapes are superimposed on the ground truth image (fig. 7.5).

In the left part of fig. 7.7 we display the enhanced image. The right plot of fig. 7.7 shows

the gradient norm over the iteration as it climbs out of an infeasible solution, which contains

intensities below zero and/or above the maximal value. The results from multiple coil data can

be seen in figs. 7.8 - 7.10.

Simulated data reconstructions

Figure 7.2: Ground truth image for the simulated experiments.

7.2. Results 117

Figure 7.3: Simulated data with unknown background. (Top Left) Initial superimposed to

ground truth image. (Top Right) Initial predicted image. (Bottom Left) Final superimposed

to ground truth image. (Bottom Right) Final predicted image. The error for the reconstructed

image isrms = 0.40217.

1 2 3 4 5 6 7 80.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

k

|gr ad|

Figure 7.4: Simulated data with unknown background. (Left) Enhanced reconstructed image.

(Right) Plot of the gradient norm of the shape reconstruction over iteration.



Figure 7.5: Ground truth image from fully sampled single coil data.

Figure 7.6: Measured data with unknown background. Coil 5. (Top Left) Initial superimposed

to ground truth image. (Top Right) Initial predicted image. (Bottom Left) Final superimposed

to ground truth image. (Bottom Right) Final predicted image. The error for the reconstructed

image isrms = 0.6509.

7.2. Results 119

1 2 3 4 5 6 7 81

1.5

2

2.5

3

3.5

4

4.5

5

5.5

k

|gr ad|

Figure 7.7: Measured data with unknown background. Coil 5. (Left) Enhanced reconstructed

image. (Right) Plot of the gradient norm of the shape reconstruction over iteration.



Figure 7.8: Ground truth image from fully sampled multiple coil data.

Figure 7.9: Measured data with unknown background. Multiple coils. (Top Left) Initial su-

perimposed to ground truth image. (Top Right) Initial predicted image. (Bottom Left) Final

superimposed to ground truth image. (Bottom Right) Final predicted image. The error for the

reconstructed image isrms = 0.56808.

7.2. Results 121

1 2 3 4 5 6 7 80.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

k

|gr ad|

Figure 7.10: Measured data with unknown background. Multiple coils. (Left) Enhanced re-

constructed image. (Right) Plot of the gradient norm of the shape reconstruction over iteration.


7.3 Discussion

By combining the image reconstruction method with the shape estimation, we have presented

a method which is capable of reconstructing objects with smoothly varying intensities in an

unknown background. TheTV -based image reconstruction method enjoys the benefit of global

edge-enhancement. Using the estimated shape we can further improve on the reconstruction of

edges at the boundary of the object. The smooth varying interior model used for the intensities

inside the object is an improvement on the constant interior model. Still the exactness of our

model suffers from the existence of very low intensities within the left ventricle. An improved

cardiac model should allow for outliers within the interior, but these outliers have to be restricted

according to somea priori knowledge. The problem of non-filled interiors is that the shape can

continue to grow even when it has passed through the correct boundary. This happens because

we are trying to include low intensities within the interior of the shape without knowing their

location in advance. The result is that the interior model ends up including low intensity values

belonging outside of the object and progresses even further.

A model for the reconstruction of shapes has to describe the interior and the background in

a distinct manner. In our approach the aim was to find an object which has a smooth distribution

of high intensity values. Given an exact model of the heart the problem is then to grow or shrink

it according to the estimated shape. This could potentially be achieved with a model of the heart

based on an anatomical atlas or on a learned model from a large data set. Then the problem

of shape reconstruction would be to fit the best, according to some metric, registered model to

the data. While this is an important subject and it raises interesting questions, it exceeds the

purposes of this thesis.

Results for our method have been presented in simulated and measured data studies. The

presented method can be used to reconstruct real-time free-breathing cardiac MR data by solv-

ing a minimisation problem for each time step. Free-breathing imaging has the potential of

being clinically more useful than breathold imaging due to the poorly understood changes in

blood flow and pressure in the region of the heart [122] during the extended breathholds re-

quired for the collection of data in traditional methods. The motion of the object is very small

during the collection of the limited amount of data (8 radial profiles) that we have used for our

experiments. This makes the method ideal for dynamic imaging as there will be very little cor-

ruption due to motion artifacts. The method can be applied in other dynamic imaging modalities

where the collected data is limited. As the method does not make any assumptions about the

motion of the object it removes the restriction of periodicity, which is a limiting factor in gated

studies. This offers the possibility to apply the method to patients suffering from arrythmia.

7.3. Discussion 123

In the next chapter we present a method which assumes that shapes are correlated in time.

The problem is solved in the temporal dimension as a state estimation problem using Kalman

filters.


Chapter 8

Temporally correlated combined

reconstruction method

In the case of cardiac MRI, data is temporally correlated. The location of the cardiac boundaries

at a particular time point is dependant on their previous location. We consider this to be a

Markov process, where the boundary is determined only by its previous position. The problem

of reconstructing the shape is expressed as a state estimation problem, where the states are the

parameters of the shape. We solve this stochastic problem using Kalman filters. A similar

approach for the calculation of diffusion and absorption coefficients in optical tomography has

been presented in [95].

While the temporally correlated problem can be solved with the combined method pre-

sented in chapter§7 as a sequence of minimisation problems, it would not be trivial to incorpo-

rate statistics that change in time. Kalman filters have the temporal estimation of statistics build

in. The formulation of the problem in this temporal case is also simpler using the Kalman filter

approach.

8.1 Forward and inverse problem

In the Kalman filters algorithm the aim is to minimise thea posteriorierror covariance

Ct|t = E[et|teTt|t], (8.1)

whereet|t = gt − yt|t is thea posterioriestimate error,gt is the measured data at timet and

yt|t = Z(γ) is the predicted data at timet given the data at timet. Choosing the linearisation

point to be the current estimate of the predictorγ∗,t = γt|t−1 simplifies the nonlinear Kalman

filter equations (4.87)-(4.91) to

126 Chapter 8. Temporally correlated combined reconstruction method

Gt = Ct|t−1JTγ,t(γt|t−1)(Jγ,t(γt|t−1)Ct|t−1J

Tγ,t(γt|t−1) + Cn,t)−1 (8.2)

γt|t = γt|t−1 + Gt(gt −Z(γt|t−1)) (8.3)

Ct|t = Ct|t−1 −GtJγ,t(γt|t−1)Ct|t−1 (8.4)

γt+1|t = Stγt|t (8.5)

Ct+1|t = StCt|tSTt + Cw,t, (8.6)

whereJγ,t is the shape Jacobian matrix, described in eq. (6.4). Eq. (8.3) updates the estimate

according to the measured data and eq. (8.5) according to the model of the state process. The

matrix St represents our knowledge about the change of the states from one time point to the

next. SettingSt = I we are representing random motion.

To solve the combined image and shape reconstruction we use an alternating minimisation

approach, similar to chapter§7. Intensity parameters are initialized using the projected primal-

dual method, presented in§5. Shape parameters are initialized near the object of interest. We

estimate the location and shape of the left ventricle with the extended Kalman filters using a

single radial profile as our datagt at time pointt. After a number of radial profiles has been

used for the shape estimation, we switch to the reconstruction of the intensities with the shape

specificTVβ projected primal-dual method (§7). As our data for the intensity reconstruction

we use all the radial profiles from the time point of the previous switch till the current timet.

The data vector used isg = gt−n, ...,gt, wheren is the number of profiles we have used for

the estimation of the shape before switching to the image reconstruction method. The intensity

reconstruction is iterated until sufficient convergence is achieved. The method is then switched

again to the estimation of the shape using the intensity parameters calculated previously. The

combined estimation of the shape and intensity parameters progresses by alternating everyn

time points until all radial profiles have been processed.

8.2 Results

We choose to solve the image reconstruction problem after 8 iterations of the shape estimation

method. The shapes are estimated with the Kalman filters for each profile. In our numerical

experiments we have found that the heart is not exhibiting much motion during the collection

of 8 radial profiles, and we take advantage of this to reduce the computational cost without any

significant loss on the quality of the reconstructions. The acquisition of data is separated in

groups of 8 profiles. In each group, these 8 profiles are chosen so they have the largest angular

distance between them in order to span the 180 degrees in a near optimal way. All groups are

8.2. Results 127

interleaved with each other to span the 180 degrees perfectly in time (fig. 8.1).

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

ky

kx

time point 1

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

kx

ky

time point 2

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

kx

ky

time point 3

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

kx

ky

all time points

Figure 8.1: Interleaved sampling pattern.

The interleaved pattern (fig. 8.1) allows to construct sensitivity matrices from the time averaged

images without using a body coil. The fully sampled reconstructed data was segmented manu-

ally for comparison with the automatic segmentations. For this task we use the Dice similarity

coefficient [35]

dsc(C, Cg) =2 N(C ∩ Cg)

N(C) + N(Cg)), (8.7)

whereC andCg are the areas of the predicted and ground truth contours, respectively.N(C)

is the number of pixels within an area. Further to that we show the predicted and ground truth

areas over the cardiac phase in a graph.

Simulated data was produced by reconstructing a fully sampled data set and then taking

the Radon transform at 8 angles per cardiac phase. The full data set consisted out of 64 cardiac

phases, which we have undersampled to 16 by taking every fourth phase. The resulting under-

sampled data set consists out of 128 radial profiles. The reconstructed shapes superimposed on

the ground truth images can be seen in the left column of fig. 8.2 next to the predicted images

with the smoothly varying constant interior. In fig. 8.3 we display the filtered back-projection

reconstructions next to the projected primal-dual ones. The similarity coefficient, therms im-

age errors and the areas over time are presented in fig. 8.4. In fig. 8.5 we show one line passing

through the center of the image over time for the ground truth, filtered back-projection and the

combined reconstructions.


Simulated data reconstructions

1

4

7

16

Figure 8.2: Reconstructions from simulated data. The numbers on the left column indicate the

time point in the sequence. (Left) Reconstructed shapes superimposed on ground truth images.

(Right) Reconstructed images with restricted interior intensities.

8.2. Results 129

16

7

4

1

Figure 8.3: Reconstructions from simulated data. The numbers on the left column indicate the

time point in the sequence. (Left) Filtered back-projection. (Right) Reconstructed images using

shape specificTVβ approach.


0 2 4 6 8 10 12 14 160.65

0.7

0.75

0.8

0.85

0.9

t

dsc

0 2 4 6 8 10 12 14 16

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Filtered backprojection

Time correlated combined approach

t

rms

0 2 4 6 8 10 12 14 16100

150

200

250

300

350

400

Ground truth

Predicted area

area

t

Figure 8.4: Error plots from simulated data reconstructions. (Left) Plot of the Dice similarity

coefficient over time (Middle) Plot ofrms over time. Filtered backprojection (solid line) and

temporally correlated combined approach (dotted line). (Right) Predicted and ground truth

areas over time.

t

r x

ground F BP TVβ TVβ+

left ventricle

right ventricle

C

Figure 8.5: x-t plots of the centralrx line in the image over time. The thick arrows point to the

papillary muscle. (Left) Ground truth. (Middle Left) Filtered backprojection. (Middle Right)

Shape specific total variation method. (Right) Combined shape and image method.

8.2. Results 131

Measured data

ECG gated data was acquired from a healthy volunteer. A total of 25 phases each with 208

radial profiles were collected using a five-element array receive coil, as described in§5.5.2. The

data used in this experiment was generated by undersampling each phase to 8 profiles. This was

done by using every8-th profile of the fully sampled data set. Using this very undersampled

data set results in a 26-fold acceleration compared to the original radial acquisition. For the case

of real-time MRI, a total of about 200 radial profiles can be collected within a single heart beat

using a fast steady state free precession sequence. To transform the data in the Radon space, we

1D inverse Fourier transformed along each radial profile, according to the central slice theorem.

The results for the single and multiple coil reconstructions are presented in figs. 8.6 - 8.13, as

previously.



1

4

7

16

Figure 8.6: Reconstructions from measured single coil data. The numbers on the left column

indicate the time point in the sequence. (Left) Reconstructed shapes superimposed on ground

truth images. (Right) Reconstructed images with restricted interior intensities.

8.2. Results 133

16

7

4

1

Figure 8.7: Reconstructions from measured single coil data. The numbers on the left column

indicate the time point in the sequence. (Left) Gridding. (Right) Reconstructed images using



0 5 10 15 20 250.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

dsc

t0 5 10 15 20 25

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Gridding

Time correlated combined approach

rms

t0 5 10 15 20 25

60

80

100

120

140

160

180

200

Ground truthPredicted area

area

t

Figure 8.8: Error plots from measured single coil data reconstructions. (Left) Plot of the Dice

similarity coefficient over time (Middle) Plot ofrms over time. Gridding (solid line) and tem-

porally correlated combined approach (dotted line). (Right) Predicted and ground truth areas

over time.

rx

t

ground T V β T V β+C

right ventricle

left ventricle

GRID


papillary muscle. (Left) Ground truth. (Middle Left) Gridding reconstruction. (Middle Right)


8.2. Results 135

Multiple coils reconstructions

1

4

7

16

Figure 8.10: Reconstructions from measured multiple coil data. The numbers on the left column

indicate the time point in the sequence. (Left) Reconstructed shapes superimposed on ground

truth images. (Right) Reconstructed images with restricted interior intensities.


1

4

7

16

Figure 8.11: Reconstructions from measured multiple coil data. The numbers on the left column

indicate the time point in the sequence. (Left) Gridding. (Right) Reconstructed images using


8.2. Results 137

0 5 10 15 20 250.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

t

dsc

0 5 10 15 20 250.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

GriddingTime correlated combined approach

t

rms

0 5 10 15 20 2540

60

80

100

120

140

160

180

Ground truthPredicted area

t

area

Figure 8.12: Error plots from measured multiple coil data reconstructions. (Left) Plot of the

Dice similarity coefficient over time (Middle) Plot ofrms over time. Gridding (solid line)

and temporally correlated combined approach (dotted line). (Right) Predicted and ground truth

areas over time.

ground GRID TVβ TVβ+ C

rx

t

left ventricle

right ventricle


papillary muscle. (Left) Ground truth. (Middle Left) Gridding reconstruction. (Middle Right)



8.3 Discussion

In this chapter a method for the estimation of both shape and intensity parameters has been

presented. The time correlated combined reconstruction does not make any assumptions about

periodic motion. This makes it applicable in many dynamic imaging problems, where gated

methods are simply infeasible. If the background is assumed to be completely stationary then

another interesting approach is to reconstruct the difference between two sequential time points.

With a stationary background the only signal left if we subtract data from two different time

points will be the motion of the object of interest. This approach does not require the knowledge

of the background structures, which will be completely removed by the subtraction. More

details on the difference imaging method are given in the appendix.

The benefits of time correlated combined approach can be seen in the reconstructed images

and especially in the x-t plots (figs. 8.5, 8.9 and 8.13). While standard reconstruction methods,

such as filtered back-projection and gridding, produce images which are highly corrupted by

noise, our reconstruction method shows results where the cardiac ventricles can be clearly de-

lineated and the presence of noise is limited. The restricted intensity model for the interior of

the shape causes underestimation of the area, as seen in figs. 8.4 and 8.12, of the cardiac bound-

ary. This mismatch is caused due to our model not representing the interior intensities correctly.

It is also clear that the low intensities that represent the papillary muscle are a source of error

for our model, as seen in fig. 8.5. A more sophisticated model should be able to capture the

expected intensity distribution within the heart with precision. This would result in improved

estimation of the boundary, and therefore the predicted area would approximate the truth closer.

While we would expect the quality of the multiple receive coils reconstructions to be su-

perior to the single coil case, this is not clear from the obtained results. A reason for this is our

approximation of the sensitivity matrices. As mentioned in§5.5.2 we have divided each time

averaged fully sampled single coil image with the root of the sum of squares of all coil images.

Sensitivity matrices could also be obtained by dividing the single coil images with an image

obtained from a body coil, which covers all the k-space sampling positions, instead of the sum

of squares that we have used. In our case a body coil signal was not available. To eliminate

noise we have low-pass filtered the sensitivity matrices. This results in errors at object edges.

An alternative approach for smoothing is to perform a polynomial fit for each pixel to the noisy

images [139]. The exact calculation of sensitivity matrices exceeds the purposes of this thesis.

Further improvement in the case of cardiac MRI is to enforce constraints on the center of

the estimated shape, limiting its motion to a bare minimum. It is the boundary of the object that

is moving and not its center. High frequency parameters should be changing faster than lower

8.3. Discussion 139

frequency ones, since they represent details on the boundary and these are changing faster

from one time point to the next. While we can solve the shape reconstruction problem in time

using a least squares approach such as the Levenberg-Marquardt method, it would be much

harder to incorporate statistics that change in time. Kalman filters offers the tools for this task.

Using the Kalman filter approach, we can solve the problem considering each radial profile to

belong to a different time point. In our experiments, this robustness was not evident with the

Levenberg-Marquardt method, which requires a number of radial profiles, for the estimation

of the shape at each time point. Thus, Kalman filters provide shape estimates for each radial

profile reaching the physical limit of temporal resolution in MRI, as it is the change of gradients

that takes considerably more time than to read data out on a particular profile. Another benefit

of the stochastic approach is that the state transition matrix can vary in time, representing the

movement from one state to the next. To calculate its elements we can incorporate them in the

minimisation problem as parameters of the objective function. If the data can be reconstructed

off-line we can use the fixed-interval smoother§4.7.3, which calculates the estimates from past

and future measurements. Further to that we can consider the non-trivial problem of a non-

Markov process where the parameters depend on a longer history of the motion of the states.


Chapter 9

Conclusions and future directions

Novel approaches for the reconstruction of images and shapes in cardiac MRI have been pre-

sented in this thesis. The presented methods have been applied to the problem of cardiac MRI.

Yet the methodology is general enough to be applicable to many limited data problems. It also

offers the ability to be combined with other MRI methods, such as k-t and SENSE approaches

(§2.4).

In chapter§2 and§3 we presented the foundations of dynamic imaging in MR and dis-

cussed current approaches in shape reconstruction. The basis of our methods is inverse problem

theory. This we have introduced in chapter§4.

The image reconstruction method in§5 produces superior results to standard methodol-

ogy, such as filtered back-projection and gridding algorithms. Our method reduces the severe

streaky artifacts, which dominate standard methods, without oversmoothing edge information.

This reduction of angular artifacts in the reconstructed images can be of importance in k-t ap-

plications where training about the motion in the images is required. These methods typically

use interleaved data acquisition patterns. This implies that the angular artifacts will be rotating

depending on the choice of angles. These moving artifacts will be detected as motion which is

clearly unwanted. The quality and usability of the reconstructed images can only be assessed

by clinicians, which we plan to include in our future work. Apart from the standard argument

of increasing computer capabilities, higher resolution images using finer basis functions grids

can be obtained by replacing direct matrix inversions with iterative linear system solvers, for

example preconditioned conjugate gradient methods. The application of these methods also

offers the extension of the presented methodology to three dimensions, where the blob basis

functions are naturally extended due to their symmetric nature. 3D dynamic imaging requires

the collection of large data sets. Using our approach these requirements can be significantly

decreased, which will reduce motion artifacts in the reconstructed images.

In chapter§6 we have presented a shape reconstruction method in the case of simple or

142 Chapter 9. Conclusions and future directions

known background. Shapes in our method are not estimated from edge information, but directly

from the measured data using a model-based approach. It is our model that defines the shape

we wish to reconstruct. In this chapter we have presented a basic approach, where the interior

of the shape is considered to be of constant intensity.

In chapter§7 we combined the image and shape reconstruction methods. The background

and interior intensities are estimated using the image reconstruction method. The shape recon-

struction assists the estimation of images by providing local information about the expected

intensity distribution at the edges and interior of the boundary. Our model for the interior inten-

sities shows its limitations in the estimation of real cardiac shapes. The definition of a more so-

phisticated cardiac model based on anatomical knowledge would make the shape reconstruction

method more exact and robust. The combination of our methods with registration techniques

could be a possible direction for the exact segmentation of cardiac images. The extension from

planar contours to surfaces will be another exciting future direction. The trigonometric func-

tions used in this thesis can be replaced by spherical harmonics for the description of surfaces.

The reconstruction of shapes offers quantitative analysis of the cardiac motion. Solving

this as a state estimation problem in chapter§8 offers a compact method for dynamic shape

detection directly from MR samples. As a future direction, we believe that estimating the

parameters of the motion, that is the state transition from one time point to the next, will improve

the reconstruction results.

The limited data used in our experiments has the potential to reduce scanning time and

make high temporal resolution real-time cardiac imaging a clinical possibility. The benefit

of reducing scanning time and breath hold requirements is clear in the case of many patients

who find MRI scanners claustrophobic. Some patients find it difficult to maintain even a short

breath hold. It would also simplify examinations under stress. Real-time imaging escapes

the problem of normalising, shrinking or stretching, monitored cardiac cycles to fit an average

cycle. As we have not assumed the periodic nature of cardiac motion, our novel approach

is applicable under conditions where gated methods are infeasible. It escapes the temporally

averaging nature of gated imaging methods and has the potential of reducing motion artifacts.

Applications of interest for the presented method in cardiac MRI are patients with arrythmia

and free-breathing imaging. Free-breathing imaging can be potentially more clinically relevant

due to the poorly understood changes in blood flow and pressure within the cardiac region

during extended breath holds [122]. On top of that, data acquisition time is no longer limited

by the ability of patients to hold their breath and can be extended in order to obtain images

from more slices or even complete volumes of the heart. Other potential applications include

143

the imaging during pregnancy, where it would be very hard to maintain the fetus static. Imaging

of infants in situations where it is hard to keep them still is also another possibility. Another

interesting application is 3D mammography where the 2D images are of high resolution, but the

limited number of views makes the use of standard tomosynthesis algorithms ([60],[179]) not

ideal for the reconstruction of 3D images. Generally our novel approach can be used in many

tomographic and Fourier imaging problems.

Our proposed method is applicable to any choice of angles, even in limited view problems,

where data cannot be collected from the whole 180 degrees. An interesting choice of k-space

positions is random sampling, which has been explored recently for its possibility of producing

higher quality reconstructions [23],[22] and [112]. In radial data sampling the choice of angles

can be exploited by an intelligent acquisition scheme, where the angles are chosen according

to the cardiac motion. In simple terms, at the end diastolic phase the heart is moving slower

and more data can be collected without fear of corruption by motion artifacts. The detection

of the shape can also be employed for the choice of scanning angles. Assuming that there is

more interest in reconstructing the cardiac contours precisely than reconstructing the surround-

ing structure, we can collect data at angles tangent to the reconstructed shape. If the object

of interest is not perfectly round, then more views where the curvature of the shape is high

contribute more in the correct reconstruction, than those that have a low curvature.

In this thesis we have presented methods for the reconstruction of images and shapes in

a dynamic problem. We have applied our methods to radially sampled cardiac MRI, and seen

the improvements and limitations of our model-based approaches. Reconstructed images are

visually and numerically superior to standard methods. Further evaluation of the image recon-

struction method with multiple data sets will be required to access its clinical feasibility. The

detection of boundaries of objects with unknown interior intensities in an unknown background

is a difficult problem. A robust shape reconstruction method will require further development,

especially of the interior intensity model. We believe that we have made a significant first step

in the solution of these limited data dynamic problems with the introduction of model-based

techniques, which aim to approximate data as close as possible without making assumptions

about completeness of the measurement set.

144 Chapter 9. Conclusions and future directions

Appendix A

Acronyms

MR Magnetic resonance

MRI Magnetic resonance imaging

CT Computed tomography

EIT Electrical impedance tomography

PET Positron emission tomography

ECG Electrocardiogram

FOV Field of view

SNR Signal to noise ratio

FT Fourier transform

LS Least squares

EM Expectation maximisation

SVD Singular value decomposition

TSVD Truncated singular value decomposition

TV Total variation

146 Appendix A. Acronyms

Appendix B

Table of notation

a Scalar

a Column vector

ai i-th element of vectora unless otherwise defined within the context

A Matrix or linear operator expressed as matrix

diag(a) Diagonal matrix with the elements ofa on its diagonal

I Identity matrix

Range(A) Range ofA

Null(A) Nullspace ofA

dim(A) Dimensions ofA

7→ Maps to

An n-dimensional space namedA

Z Nonlinear operator

T Transpose

H Conjugate transpose

† Pseudoinverse

∗ Adjoint

||a||p lp norm ofa

E[] Expectation operator

Φ Objective functional

Λ Lagrangian

∪ Set union

∩ Set intersection

⊂ Subset of

∈ Belongs to

148 Appendix B. Table of notation

∇ xy gradient

Cn Continuity of a function forn derivatives

III(x) Comb function

× Multiplication

rms Relative mean square error

dsc Dice similarity coefficient

∗ Convolution

.∗ Element-wise multiplication

./ Element-wise division

grad Gradient of the objective functional

Appendix C

Difference imaging

Another approach for the dynamic imaging problem is to calculate the difference data between

two time points by subtracting k-space profiles (5 in this experiment) at the same positions

belonging to different time points. Assuming that the background structures remain stationary,

then the difference data will show the only areas where an object moved or changed shape.

τ

θ θ

τ

θ

τ

Figure C.1: Difference imaging approach with stationary background. (Top Left) Phantom im-

age at time point 1. (Top Middle) Phantom image at time point 8. (Top Right) Image difference

between time point 1 and 8. (Bottom Left) Phantom sinogram data at time point 1. (Bottom

Middle) Phantom sinogram data at time point 8. (Bottom Right) Sinogram difference between

time point 1 and 8.

This requires that the collection of data at each time point is done at the same locations in

k-space instead of the interleaved sampling pattern. In a multiple coil experiment this will

complicate the construction of sensitivity matrices from time averaged images and will require

the use of a body coil for this task. The benefit of such an approach is that in single-breathhold

150 Appendix C. Difference imaging

cardiac MRI most of the motion in the image is from the heart and especially the left ventricle.

The background structures are practically stationary. We present results on simulated data with

15% added, to the Radon data, Gaussian noise (fig. C.1) to demonstrate the power of this

approach. In fig. C.2 the ground truth images and the reconstructed shapes are shown. In fig. C.3

we compare the reconstructions with manually segmented shapes. Data was simulated from a

dynamic phantom at 12 time points. Note that the difference imaging method does not require

knowledge of the stationary background structure. We initialize the shape parameters exactly

at the boundary of the object of interest. For each subsequent time point, we calculate the

difference data between the current time point and the previous one and estimate the shape using

the Kalman filter algorithm. The Kalman filter algorithm is essentially estimating motion by

comparing the predicted and measured difference data. The predicted motion is the difference

between the predictions in the Radon space at time pointt andt− 1. The measured motion, as

described before, is the difference between the data at two sequential time points. The interior

of the object is filled with a known constant value.

151

1

4

7

12

Figure C.2: Difference imaging reconstructions. The numbers on the left column indicate the

time point in the sequence. (Left) Ground truth images. (Right) Reconstructed shapes superim-

posed on groundtruth.


0 2 4 6 8 10 120.89

0.895

0.9

0.905

0.91

0.915

0.92

0.925

t

dsc

0 2 4 6 8 10 121050

1100

1150

1200

1250

1300

1350

1400

1450

Ground truth

Predicted area

t

area

Figure C.3: (Left) Plot of the Dice similarity coefficient over time (Right) Predicted and ground

truth areas over time.

The immediate problem with the difference imaging approach is that unless we can guar-

antee convergence of the estimated shapes to the true ones, errors will be propagated further

in time. If the shape does not approximate the true boundaries precisely, then on the next time

step we will be modelling different motion than what is happening in the data, as our initial

estimated position is wrong. This directly implies that the initialization of the shape parameters

has to be exact. In the case of single-breathhold cardiac MRI, there is more than one shape

in motion. Even though most of the background is removed there is still some differences at

locations other than the heart (fig. C.4). Using a simple model, which estimates only shapes of

the left and right ventricles, will be a cause of error. This error will be amplified as the method

propagates in time.

If the model is very precise and convergence can be guaranteed, then the difference imag-

ing approach could give exciting results. Assuming that such a precise model exists, then the

need for using the same angles in the sampling sequence can be removed, by predicting the data

at any angle from the image space model.

153

θ

τ τ

θ θ

τ

Figure C.4: Difference imaging approach with stationary background. (Top Left) Phantom im-

age at time point 1. (Top Middle) Phantom image at time point 8. (Top Right) Image difference

between time point 1 and 8. (Bottom Left) Phantom sinogram data at time point 1. (Bottom

Middle) Phantom sinogram data at time point 8. (Bottom Right) Sinogram difference between

time point 1 and 8.


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