12 Image Reconstruction

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ENGG 167

MEDICAL IMAGING

Lecture 12: Friday, Oct. 19

Image Reconstruction from Projections

Reference: Chapters 12 & 13, The Essential Physics of Medical Imaging , Bushberg

Computed Tomography, Kalender, Verlag, 2000.

Chapter 12, Intermediate Physics for Medicine and Biology, 3 rd Ed., Hobbie.

Chapter 3, Principles of Computerized Tomographic Imaging, Kak and Slaney,

IEEE Press

Medical Physics and Biomedical Engineering, Brown, et al, IoP Publishing.

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Image Reconstruction

Ref: Hobbie

1) 2-D Fourier transform review

2) Filtering in space and frequency domain

2) Backprojection

3) Filtered Backprojection


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1) Basics Fourier Transform of an image f(x,y)

Ref: Hobbie

Fourier amplitude

Real space

Transformed image is described in k-space,where kx= 2/x, ky=2/y are spatial frequencies

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1) What does the DFT image represent ?

Ref: Hobbie

Highest Frequencies

kx,max = (Nx/Sx) = 1/[2(pixel width)]

ky,max = (Ny/Sy)

lowest frequencies

kx,min = 1/Sxky,min = 1/Sy

S size of image (mm)

N number of pixels

Maximal negative frequency

kx,max = - (Nx/Sx) = -1/[2(pixel width)]

ky,max = - (Ny/Sy)


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2) Fourier Slice Theorem example

F=imread('abdomen visible human gray.jpg');

imagesc((F));colorbar; colormap('bone');

X=abs(fft2(F));imagesc(log(X(:,:,1)));colorbar;colormap('bone');

X=abs(fft2(F));imagesc(fftshift(log(X(:,:,1))));colorbar;colormap('bone'); P=angle(fft2(F));imagesc(fftshift(P(:,:,1)));colorbar

X=abs(fft2(F));imagesc((X(:,:,1)));colorbar;colormap('bone');

Note DFT has

same number of

pixels, but two images

magnitude and phase

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3) Fourier Transform of an image f(x,y) in MATLAB

F=imread('kpaulsen.jpg');imagesc((F));colorbar

X=abs(fft2(F));imagesc( fftshift(log(X(:,:,1))));colorbar

P=angle(fft2(F));imagesc( fftshift(P(:,:,1)));colorbar

X=abs(fft2(F));imagesc(log(X(:,:,1)));colorbar

P=angle(fft2(F));imagesc(P(:,:,1));colorbar

K-space image with

K=0 at center

Note DFT

image has both

amplitude

and phase

information


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1) Review: Fourier Transform of an image f(x,y) in MATLAB

A=fft2(F);F1=uint8(abs(fft2(A./20500)));

F2=imrotate(F1,180);imagesc(F2)

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2) Filtering space domain convolution

Convolution integral

g(x) = filtered data

g(x) = initial data

h(x-x) = filter function

'( ) ( ') ( ') 'g x g x h x x dx

=

'( ) ( ) ( )g x g x h x=

g=[1 3 4 2 3 5 6 7 8 9 5 6 7 8 9 7 5 4 3 2 3 2 4 2 1 1 1 ];

plot(g);

h=[1 2 1];

gp=conv(g,h);Figure; plot(gp)

[1 2 1] =

h=[.25 .5 .25];

gp=conv(g,h);

plot(gp)

[0.25 0.5 0.25] =

smoothing

filter


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2) Filtering space domain convolution


g(x) = filtered datag(x) = initial data

h(x-x) = filter function

'( ) ( ') ( ') 'g x g x h x x dx

=

'( ) ( ) ( )g x g x h x=

g=[1 3 4 2 3 5 6 7 8 9

5 6 7 8 9 7 5 4 3 2

3 2 4 2 1 1 1 ];

plot(g);

h=[1 2 1];

gp=conv(g,h);

Figure; plot(gp)

=

[-1 2 -1]

=

=[-1 2 -1]

[-1 4 -1]

g=[1 2 2 2 2 7 7 7

7 7 7 7 7 4 4 4

4 4 2 2 2 2 2 2 2 2 1 ];

plot(g)

gp=conv(g,h);

plot(gp)

h=[-1 4 -1];gp=conv(g,h);

plot(gp)

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2) Filtering of Images space domain


g(x,y) = image

f(x,y) = test object

h(x-x,y-y) = filter function

'( , ) ( ', ') ( ', ') ' 'g x y g x y h x x y y dx dy

=

'( , ) ( , ) ( , )g x y g x y h x y=

1 2 1

2 4 2

1 2 1=

h = fspecial(type,parameters) - predefined filters of any shape

X=imread('happyface.jpg');

imagesc(X);

B=[1 2 1; 2 4 2; 1 2 1];

Y=conv2(X,B);

Imagesc(Y);colormap(bone);

-1 -2 -1

-2 8 -2

-1 -2 -1

=

1 1 1

0 0 0

-1 -1 -1=

imagesc(X);

B=[-1 -2 -1; -2 8 -2; -1 -2 -1];

Y=conv2(X,B);


imagesc(X);

B=[1 2 1; 0 0 0; -1 -1 -1];

Y=conv2(X,B);


Prewitt Filter


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2) Filtering of Images frequency domain

'( , ) ( , ) ( , )g x y g x y h x y=

'( , ) ( , ) ( , )x y x y x y

g k k g k k h k k =

The Fourier transform of a

convolution is the multiplicationof the transformed functions

1 2 1

2 4 2

1 2 1=

X=imread('happyface.jpg');

imagesc(X);

B=[1 2 1; 2 4 2; 1 2 1];

Y=conv2(X,B);


'( ) ( ') ( ') 'g x g x h x x dx

= '( ) ( ) ( )g x g x h x=

'( ) ( ) ( )x x xg k g k h k =

This is also true in 2-D (images)

X =

FFT2 FFT2 FFT2

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3) Image Reconstruction : projection data

Ref: Kalendar


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3) Backprojection - Linear single backprojection

SourceDetector

Source

Detectorinverse = 1 back projection

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3) Backprojection two linear projections

Source

Detector 2

Source

Detector 1+

Detector 1

Detector 2


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3) Backprojection Multiple linear projections

3 projections 4 projections many projections

Originalobject

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3) Image Reconstruction concept of backprojection

Ref: Bushberg


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3) Image Reconstruction Algebraic ReconstructionTechnique (ART)

Ref: Bushberg

A

B

C

D

1 1 0 0

1 0 1 0

1 0 0 1

0 1 1 0

0 1 0 1

0 0 1 1

=

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6

5

9

8

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K x = b

KTK x = KTb

x = [KTK]-1 KTb

Over-determined non-square matrix K

Multiply by KT first

Invert square matrix

Larger problems must be solved iterativelyusing standard methods for solving large

matrix operation problems.

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3) Relating to x-y coordinate back-projection

Ref: Hobbie


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3) Filtered Backprojection - parallel and cone beams

Ref: Kak and Slaney

Optional Read through equations from Chapter 3in book by Kak and Slaney.

Available on the web at:http://rvl4.ecn.purdue.edu/~malcolm/pct/pct-toc.html

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3) The Radon Transform Kak and Slaney

Ref: Kak and Slaney


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3) Fourier Slice Theorem Kak and Slaney

Ref: Kak and Slaney

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3) Fourier Slice Theorem Kak and Slaney

Ref: Kak and Slaney


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2) Fourier Slice Theorem what does the DFT image represent ?

Ref: Hobbie

Projectionprofile

Integrate intensities

along x-direction

1-D

DFT

DFT of

projection

profile

Creates line

Central slice of

Entire DFT image

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Ref: Hobbie

Projection

profiles

Integrate intensities

along x-direction

1-D

DFTs

Create lines in

central slice of

entire DFT image

The more angles

used, the better

the Fourier space

image is filled


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Ref: Hobbie

1-D

DFTs

at each

projection

DFT image represents integration of

original projections DFT transformed and

summed together.

This is the fast way to create the DFT

image from projection data. The moreprojections taken, the more complete the

sampling.

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3) Radon Transform sinogram data backprojection

Ref: Brown et al.


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3) Backprojection of Radon Transform

Ref: Brown et al.

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3) Backprojection filtering required to bring out high

Ref: Brown et al.


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3) Image Reconstruction - concept ofbackprojection and filtered backprojection

Ref: Bushberg

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3) Reconstruction

Ref: Kalendar


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3) Parallel Beams and Fan Beams

Ref: Kak and Slaney

Chapter 3 in book by Kak and Slaney.

http://rvl4.ecn.purdue.edu/~malcolm/pct/pct-toc.html

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Filtered Backprojection - Review


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MATLAB implementation of a test phantom

Ref: Matlab Help Guide

P = phantom('Modified Shepp-Logan',200); imshow(P)

Shepp-Logan phantom is a standard head CT simulation.

Described in detail in Kak & Slaney, pg 102.


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MATLAB implementation of the Radon transform (i.e. projection data)


Imaging a square

iptsetpref('ImshowAxesVisible','on')

I = zeros(100,100);

I(25:75,25:75) = 1;

theta = 0:180;

[RI,xp] = radon(I,theta);

imshow(theta,xp,RI,[],'notruesize'), colormap(bone), colorbar

R = radon(I,theta)If you omit theta, it defaults to 0:179.

Imaging the Modified Shepp-Logan phantom

P = phantom('Modified Shepp-Logan',200); imshow(P)

[RP,xp] = radon(P,theta);

imshow(theta,xp,RP,[],'notruesize'), colormap(bone), colorbar

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MATLAB implementation of the Inverse Radon transform

(i.e. image reconstruction from projection data)


I = iradon(P,theta)

I = iradon(P,theta,interp,filter,d,n)

interp specifies the type of interpolation to use in the backprojection. The available options are listed in order of

increasing accuracy and computational complexity:

'nearest' - nearest neighbor interpolation

'linear' - linear interpolation (default)

'spline' - spline interpolation

filter specifies the filter to use for frequency domain filtering. filter is a string that specifies any of the following

standard filters:

'Ram-Lak' - The cropped Ram-Lakor ramp filter (default). The frequency response of this filter is | f |. Because this

filter is sensitive to noise in the projections, one of the filters listed below may be preferable. These filters multiply

the Ram-Lak filter by a window that de-emphasizes high frequencies.

'Shepp-Logan' - The Shepp-Logan filter multiplies the Ram-Lak filter by a sinc function.

'Cosine' - The cosine filter multiplies the Ram-Lak filter by a cosine function.

'Hamming' - The Hamming filter multiplies the Ram-Lak filter by a Hamming window.

'Hann' - The Hann filter multiplies the Ram-Lak filter by a Hann window.

d is a scalar in the range (0,1] that modifies the filter by rescaling its frequency axis. The default is 1. If d is less than

1, the filter is compressed to fit into the frequency range [0,d], in normalized frequencies; all frequencies above d are

set to 0.

n is a scalar that specifies the number of rows and columns in the reconstructed image. If n is not specified, the size is

determined from the length of the projections.

n = 2*floor(size(P,1)/(2*sqrt(2)))


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MATLAB implementation of the Inverse Radon transform(i.e. image reconstruction from projection data)


I = iradon(P,theta)I = iradon(P,theta,interp,filter,d,n)

KI = iradon(PI,theta)

imshow(theta,xp,KI,[],'notruesize'), colormap(bone), colorbar

KP = iradon(P,theta)

imshow(theta,xp,KP,[],'notruesize'), colormap(bone), colorbar

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Effect of projection number


P = phantom('Modified Shepp-Logan',200);[RP,xp] = radon(P,theta);KP = iradon(RP,theta);

imshow(theta,xp,KP,[],'notruesize'), colormap(bone), colorbar

theta = 0:40:170 theta = 0:20:170 theta = 0:10:170 theta = 0:1:170


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Effect of noise in the projection data


sd=0 sd=0.05 sd=0.10 sd=0.20 sd=0.50

imshow(theta,xp,RN1,[],'notruesize'), colormap(jet), colorbar;

P = phantom('Modified Shepp-Logan',200);

[RP,xp] = radon(P,theta);

RN1=RP.*(1+sd*randn(size(RP)));

KP1 = iradon(RN1,theta);

imshow(KP1), colormap(jet), colorbar

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