Dynamic Image and Shape Reconstruction in Under Sampled MRI

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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


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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 employedin the Center of Medical Image Computing, UCL.


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4 Statement of intellectual contribution


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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 singleimage 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.


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6 Abstract


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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.


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8 Acknowledgements


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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


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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


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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


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

transform of the image at = 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. The t-axis represents time and the ky-

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

a complete kx line in the read out direction. . . . . . . . . . . . . . . . . . . . 37

2.5 Plot of an aliased function. The q-axis is the temporal frequency and the F-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 Regular 3 3 grid. The x and y axes represent the spatial location in R2 and

the z axis represents the intensity. . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Surface plot of the Kaiser-Bessel blob basis in 2D with support radius 1.45 and

= 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 with N = 7. Dashed curves are the cos (even)

terms and solid curves are the sin (odd) terms. . . . . . . . . . . . . . . . . . 53

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

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


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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 matrix J. Each column corresponds to the vectorised basis functionin the Radon space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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

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

squares reconstruction 8 8 grid rms = 0.73092. . . . . . . . . . . . . . . . 775.6 8 projections. (Left) Filtered back-projection rms = 1.2521. (Right) Damped

least squares reconstruction 64x64 grid rms = 0.61756. . . . . . . . . . . . . 78

5.7 The solid line represents the absolute function|t|

and the dashed line represents

the approximation (t) =

t2 + 2 with = 0.1. . . . . . . . . . . . . . . . 79

5.8 The T V block tridiagonal matrix. . . . . . . . . . . . . . . . . . . . . . . . . 80

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

Fixed point reconstruction rms = 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) RM S = 0.61756. (Right)

Primal-dual reconstruction RM S = 0.5975. . . . . . . . . . . . . . . . . . . 84

5.12 8 projections. (Left) RM S 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 reconstruction rms = 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 cardiac rms 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


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List of Figures 15

5.20 (Left) Coil 1 rms 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 coil rms 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 point se. (Left) Corrected

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

6.2 Exact parametric points s1 and s2 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


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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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.17 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 of (t) for different values of . 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 is rms = 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 is rms = 0.6509. . . . . . . . . 118


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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 is rms = 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 specific T V approach. . . . . . . . . . . . 129

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

larity coefficient over time (Middle) Plot of rms 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 central rx 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


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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 specific T V 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 specific T V 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


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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 andground 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


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20 List of Figures


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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 and I. 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


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22 Publications


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Chapter 1

Prologue

1.1 Introduction

As the World Health Organization states on their web site 1 : 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 is1www.who.int


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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 MRIthe 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-


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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 thatwill 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 ideasin 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. Inchapter 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 collectionis 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 imagereconstruction 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 ofthe 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.


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26 Chapter 1. Prologue


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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 hydrogen H. Hydrogen has odd atomic number and weight, a half-integral valued

spin, and is found in water molecules H2O. 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


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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 88

6

4

2

0

2

4

6

8

ky

kx8 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 function f(r) of the vector variable r Rn,


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2.3. Image reconstruction 29

now known as the Radon transform

g(, ) = (Rf) (, ) =

f(u + sv)ds, (2.1)

where [0, 2) is the slope of a line, R

is its intercept, u is the vector defining thedirection of the line and v is its normal. In the 2D case (n = 2) u = (cos , sin ) and

v = ( sin , cos ). The Radon transform R maps a function f Rn into the set of itsintegrals over the hyperplanes ofRn. In the case where f R2, then f will be mapped into theset of its line integrals at angle . 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) =1

2

20

Rf(, r, u + t)d, (2.2)

where r, u is the inner product. In the 2D case the inverse transform is

f(r) = 1

0

dFr(t)

t. (2.3)

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

uses the Hilbert transform, which is defined as follows:

fH(y) = H[f(x)] = 1

f(x)

x

y

dx. (2.4)

This is essentially a convolution operator fH(y) = (h f)(y) where the convolution kernelh(x) = 1/x. The equivalent Radon inversion formula is

f(r) =1

2

gH(, ry rx)ry

d, (2.5)

where r = {rx, ry}. The singularity is still present in the above integral, but it can be handledas 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(, ) = F2f(k), (2.6)

where the n-dimensional Fourier transform Fn and inverse Fourier transform Fn for a func-tion f(r), r Rn are


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rx

ry

10 20 30 40 50 60

10

20

30

40

50

60

s

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)( , ) =s

fs

(Rf)(, ) =

f(s)ds

= 45o

= 32

0 10 20 30 40 50 60 700

2

4

6

8

10

12

14

16

18

20

Rf

=45o

0

20

40

60

80

0

50

100

150

0

5

10

15

20

25

=0 o

=45o =90o

=135o

Rf

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

at = 45

o

. (Top right) A line integral for = 32. (Bottom left) The Radon transform ofthe image at = 45o. (Bottom right) The Radon transform at four angles. The purple circle

indicates the location of the line integral.

F(k) = (2)n/2Rn

f(r)eirkdr (2.7)

f(r) = (2)n/2 Rn F(k)eikrdk. (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 1970s, [59], [58] and

[79]. It is the application of Kaczmarzs method to Radons integral equations [126]. The main

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

g =R

f. (2.9)

ART approximates Rf cUf and the previous equation becomes


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2.3. Image reconstruction 31

g = cUf, (2.10)

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

f(r) and c is an approximate correction factor. The predicted data gtj for the j-th line integral

is calculated as:

gtj = cjUj ft, (2.11)

where ft is the t-th estimated image vector, Uj is a matrix (with a single row) with the i locations

corresponding to the j-th line integral equal to 1 and cj 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 vector ft+1 is given by:

ft+1 = max

0, ft +

gj

cj g

tj

cj

/Nj

, (2.12)

where Nj is the total number of intercepts of the j-th line integral with f(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 matrix U 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) =

N

Ni=1

Qi(r ui), (2.13)

where N is the number of projections, ui = (cos i, sin i) and Qi is the filtered data at angle

i

Qi(r ui) = gi h, (2.14)


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where gi is the projection at angle i, h is a high pass filter and denotes convolution. The highpass 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

Qi(r ui) = F1F1(gi) 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 Kumars

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 a sinc function, whichis the ideal interpolation function. The sinc 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. OSullivan

[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

gf r(k) =F2f(r) Ar(k), (2.16)


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2.4. Dynamic imaging 33

where Ar is a sampling function

Ar(k) =N

i=1

(k ki), (2.17)

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

the signal gf r as follows:

gfi(k) = gf r(k) h(k), (2.18)

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

weighting function w(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

gf wc(k) = gf wi(k) Ac(k), (2.20)

where Ac(k) =i=1

j=1

(kx i, ky j) is a comb function III(k). Combining eqs. (2.18),

(2.19) and (2.20), we obtain

gf wc(k) = gf r(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


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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


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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 gatedcardiac 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 undersamplingcan 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)where f(ry) is the signal and Sj (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) eimkyry , (2.23)

where Nc is the number of receiver coils, ky = 2/FOV, F OV is a scalar representing

the field of view and m Z

is the order of the spatial harmonics. This can be solved for theweights wmj by fitting the coil sensitivities Sj to the spatial harmonics e

imkyry . Using eqs.

(2.22) and (2.23), an expression for the calculation of shifted k-space lines g(ky +mky) using

the measured sensitivity matrices Sj can be derived

Ncj=1

wmj gj(ky) g(ky + mky). (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


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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,kRj,kgj,k, (2.25)

where fn 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 the n superimposed image positions and it is

calculated as follows:

R =

SHC1S1

SHC1, (2.26)

where S is the Nc Ns coil sensitivity matrix with Nc being the total number of coils andNs the total number of samples, C is the Nc

Nc receiver noise matrix and the superscript

H

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 matrices Sj as linear combinations of the

spatial harmonics

Sj (r) p

m=q

wmj eimkyry , (2.27)

where q, p Z are integers defining the number of Fourier coefficients wmj for thejth coil. Thisallowed the construction of a linear system not as restrictive as the original SMASH formula-

tion. Sodickson et al [153] included an extra term S0 in eq. (2.23) to account for sensitivity

variations in the phase encode direction

Nc

j=1

wm

jSj(r)

S0e

imkyry . (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].


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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. The t-axis represents time and the ky-axis

the sampled locations in the phase encoding direction. Each point denotes a complete kx 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


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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. The q-axis is the temporal frequency and the F-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) =

SHC1n|k,tS+ C1s|r,q

1SHC1n|k,tgk,t, (2.29)

where S is the Fourier transform, from x-f to k-t space Frqkt, of the sensitivity encodingmatrix S, Cn is the noise covariance matrix and Cs 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-


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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


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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.


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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 Elschlagers spring loaded templates [46] and

Widrows 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

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

Esnake(v) =

10

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

where v(s) =

x(s)y(s)

is a parametric contour with s [0, 1) with x(s) and y(s) definingthe 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. Finally Eimage

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.


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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 1970s, 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 = j pj, (3.2)

where CC is the covariance matrix of the aligned shapes, j is the j-th eigenvalue and pj is the


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3.1. Snake methods 43

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

vary

C=

C+ Pw, (3.3)

where C is the mean of the aligned shapes, P is the matrix of the first n eigenvectors and w isa 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


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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 ofsnakes 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 in R2 the level set function will


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3.2. Level set methods 45

be a surface in R3. Next we give a brief introduction to the level set approach along the lines of

[145]. The boundary of the region of interest Rn is described by a function (r)

= {r : (r) = 0}. (3.4)The level set function is build as a sequence of functions t(r) which approach the real region

as t increases t with t = {r : t(r) = 0}. Assuming that the image f(r) withr Rn can be modelled as

f(r) =

fint(r) ifr fext(r) ifr /

, (3.5)

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

f(r) =

fint(r) if(r) < 0fext(r) if(r) > 0 . (3.6)

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

such as splitting and merging, are rather difficult to deal in R2, the level set function, a surface

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

3.2). The same is true for any dimension Rn.

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


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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.


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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 wo

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Dynamic Image and Shape Reconstruction in Under Sampled MRI