Image Reconstruction from Projections

Antti Tuomas Jalava

Jaime Garrido Ceca

Overview

Reconstruction methods Fourier slice theorem & Fourier

method Backprojection Filtered backprojection Algebraic reconstruction

Diffractive tomography Display of CT images Tissue characterization with CT

Projection Geometry

Problem: Reconstructing 2D Image. Given parallel-ray projections.

1D projection (Radon transform). Density distribution Ray AB

Integral evaluated for different values of the ray offset t1. 1D projection or Radon transform.

dxdytyxyxfdsyxftpAB

11 sincos,,

1sincos tyx yxf ,

tp

The Fourier Slice Theorem

1D Fourier Transform of 1D projection of 2D image

is equal to the radial section (slice or profile) of the 2D Fourier Transform of the 2D image at the angle of the projection.

dxdyvyuxjyxfvuF 2exp,,

dtwtjtpwP 2exp

,, wFvuFwP

sin

cos

wv

wu

The Fourier Slice Theorem

How to obtain f(x,y) applying Fourier Slice Theorem: Assumption: we have projections available at all angles from 0º to 180º.

1. From projections, we take their 1D Fourier transform.

2. Fill the 2D Fourier Space with the corresponding radial sections.

3. Take an inverse 2D Fourier transform to obtain

Problem: finite number of projections available

Solution: Interpolation is needed in 2D Fourier space.

yxf ,

Backprojection

Simplest reconstruction procedure Assumptions:

Rays: Ideal straight lines. Image: dimensionless points.

Procedure Estimate of the density at a point by simply summing (integrating )

all the rays that pass through it at various angles.

sincos

,0

yxt

where

dtpyxf

Problem:

•Finite number of rays per projection

•Finite number of projections

Interpolation is required.

Backprojection

BP produces a spoke-line pattern blurring details. Finite number of projections produces streaking artifacts.

Reconstructed image modeled by convolution between PSF (impulse response) and the original image.

Solution: Applying deconvolution filters to the reconstructed image.

Filtered BP technique.

Point density function

Filtered Backprojection

After some manipulations, we get:

where

In practice, smoothing window should be applied to reduce the amplification of high-frequency noise.

0

, dtqyxf

dwwtjwwPtq 2exp

Filter is represented by this function:

Ramp filter

Discrete Filtered Backprojection

Projection in frequency domain is manipulated:

2/

2/

2exp

22

12 N

Nm

mkN

jW

mp

WN

WkP

Frequency axis discretized

Finite number of samples

Samples at the sampling rate 2W

tp

Discrete Filtered Backprojection

The filtered projection may then be obtained as:

2/

2/

22exp

2222exp

N

N

tN

Wkj

N

Wk

N

WkP

N

WdwwtjwwPtq

2/

2/

2exp

222

2

N

N

mkN

jN

Wk

N

WkP

N

W

W

mq

• Problem: control noise enhancement

• Solution we apply hamming window:

2/

2/

2exp

2222

2

N

N

mkN

jN

WkG

N

Wk

N

WkP

N

W

W

mq

WN

Wk

N

WkG

2cos04654,0

2

tq

Discrete Filtered Backprojection

Finally, we get this expression :

Algorithmic:1. Measure projection.

2. Compute filtered projection.

3. Backproject the filtered.

4. Repeat 1-3 all projection angles

L

lll yxq

Lyxf

l1

sincos,~

Original

Back

projection

1°

Filtered

back

projection

1°

Filtered

back

projection

10°

Algebraic Reconstruction Techniques

Projections seen as set of simultaneous equations. Kaczmarz method

Iterative method. Implemented easily.

Assumptions: Discrete pixels. Image density is constant within each cell.

Equations

MMMNMM

NN

NN

pfwfwfw

pfwfwfw

pfwfwfw

2211

22222121

11212111

Contribution factor of the nth image element to the mth ray sum.

Algebraic Reconstruction Techniques

Karzmarz method take the approach of successively and iteratively projecting an initial guess and its successors from one hyperplane to the next.

In general, the mth estimate is obtained from the (m-1)th estimate as:

Because the image is updated by altering the pixels along each individual ray sum, the index of the updated estimate or of the iteration is equal to the index of the latest ray sum used.

mmm

mmm

mm www

pwfff

11

Algebraic Reconstruction Techniques

Characteristics worth:

ART proceed ray by ray and it is iterative Small angles between hyperplanes

Large number or iterations It should be reduced by using optimized ray-access schemes.

M>N noisy measurements oscillate in the neighborhood of the intersections of the hyperplanes.

M<N under-determined. Any a priori information about image is easily

introduced into the iterative procedure.

Approximations to the Kaczmarz method

We could rewrite reconstruction step at the nth pixel level as:

Corrections could also be multiplicative:

m

mmmn

mn N

qpff 1,0max

Number of pixels crossed by the mth ray.

True ray sum

Computed ray sum

m

mmn

mn q

pff 1

Approximations to the Kaczmarz method

Generic ART procedure:1. Prepare an initial estimate

2. Compute ray sum

3. Obtain difference between true ray sum and the computed ray sum and apply the correction.

4. Perform Steps 2 and 3 for all rays available.

5. Repeat Steps 2-4 as many times as required.

0f

mq

Original

2.

1.

178 angles

dt =

1 voxel width

3.

Original again

5.

6.

4.

Imaging with Diffraction Sources

Non ionizing radiationUltrasonicElectromagnetic (optical or thermal)

Refraction and diffraction

Fourier diffraction theorem

Imaging with Diffraction Sources

When an object, f(x,y), is illuminated with a plane wave the Fourier transform of the forward scattered fields measured on line TT’ gives the values of the 2-D transform, F(w1,w2), of the object along a circular arc in the frequency domain, as shown in the right half of the figure.

= measured attenuation coefficient. = attenuation coefficient of water When K = 1000 units are called Hounsfield Units

Air: -1000 HU Water: 0 HU Bone 1000 HU

Study 86 healthy infants aged 0-5 years White matter: 15 HU to 22 HU Gray matter: 23 HU to 30 HU Difference between grey and white matter exactly 8 HU (In all measurements) Boris P, Bundgaard F, Olsen A. Childs Nerv Syst. 1987;3(3):175-7

Display of CT Images

1

µ

µKHU

µµ

Microtomography

µ-scale CT Volume: few

Nanotomography already introduced.

Biomedical use: Both dead and alive (in-vivo) rat and mouse scanning. Human skin samples, small tumors, mice bone for

osteoporosis research.

3cm

Estimation of Tissue Components with CT

Manual segmentation of tumor by radiologist

Parametric model for the tissue composition Gaussian mixture model

Method to estimate the parameters of the model EM algorithm

Gaussian Mixture Model (i)

Fit M gaussian kernels to intensity histogram

Gaussian Mixture Model (ii)

Intensity value for voxel is a Gaussian random variable. Parameters for ith tissue: Probability that voxel belonging to that tissue gets value x

M number of different tissues in tumor = the fraction of belonging to ith tissue (probability).

Tumor as whole: PDF is a mixture of M Gaussians

2

2

2exp

2

1|

i

i

i

ii

µxxp

iii µ ,

i

M

i i11

MMi ,...,,,..., 1

M

i iii xpxp1

||

Gaussian Mixture Model (iii)

Tumor as whole: PDF is a mixture of M Gaussians

Probability of parameter set

If nothing is known about

Find that maximizes likelihood

MMi ,...,,,..., 1

M

i iii xpxp1

||

xpxpp

xp

||

|| xpcxp

N

jjxpxpxL

1

||ˆ|

Gaussian Mixture Model (iv)

Probability that jth voxel with value belongs to the ith tissue type

EM algorithm (iterative, chapter 8) ->

,

|

,

,||,|

j

iji

j

jj xp

xp

xp

ixpipxip

jx

newi

newi

newi ,,

Ending Remarks

Some image manipulation tasks can be performed in 1D in radon domain (edge detection etc.).

Reconstruction heavily dependent on reconstruction algorithm (method).

MRI images are usually reconstructed with Fourier method (according to book).

CT allows fast 3D imaging So does MRI. MRI has better sensitivity especially with

soft tissues.