CT Ch6 Part1

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Computed Tomography

(Part 1)

Yao Wang

Polytechnic School of Engineering

New York University, Brooklyn, NY 11201

Based on Prince and Links, Medical Imaging Signals and Systems and

Lecture Notes by Prince. Figures are from the book.

EL-GY 6813 / BE-GY 6203 / G16.4426

Medical Imaging

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Lecture Outline

Instrumentation

CT Generations

X-ray source and collimation

CT detectors

Image Formation

Line integrals

Parallel Ray Reconstruction

Radon transform

Back projection

Filtered backprojection

Convolution backprojection

Implementation issues

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Limitation of Projection Radiography

Projection radiography

Projection of a 2D slice along one direction only

Can only see the shadowof the 3D body

CT: generating many 1D projections in different angles

When the angle spacing is sufficiently small, can reconstruct the

2D slice very well

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1stGeneration CT: Parallel Projections

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2ndGeneration

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4G

FastCannot use

collimator atdetector, hence

affected byscattering

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5G: Electron Beam CT (EBCT)

Stationary source and detector.Used for fast (cine) whole heart imaging

Source of x-ray moves around by steering an electronbeam around X-ray tube anode.

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6G: Helical CT

Entire abdomen or chest can be completed in 30 sec.

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7G: Multislice

From http://www.kau.edu.sa/Files/0008512/Files/19500_2nd_presentation_final.pdf

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Reduced scan time and increased Z-resolution (thin slices)Most modern MSCT systems generates 64 slices per rotation, can image

whole body (1.5 m) in 30 sec.


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Generation Source Source Collimation Detector

1st Single X-ray Tube Pencil Beam Single

2nd Single X-ray Tube Fan Beam (not enough to cover FOV) Multiple

3rd Single X-ray Tube Fan Beam (enough to cover FOV) Many

4th Single X-ray Tube Fan Beam covers FOV

Stationary

Ring of

Detectors

5th

Many tungsten

anodes in single

large tube Fan Beam

Stationary

Ring of

Detectors

6th 3G/4G 3G/4G 3G/4G

7th Single X-ray Tube Cone Beam

Multiple

array of

detectors


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X-ray Source

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Dual Energy CT

Recall that linear attenuation coefficient depends on theX-ray energy

Dual energy CT enables simultanious acquisition to two

images of the organ, reflecting their attenuation profilescorresponding to different energies

The scanner has two x-ray tubes using two different

voltage levels to generate the x-rays.

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X-ray Detectors

Convert detected photons to lights

Convert light to electric current

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CT Measurement Model

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I_0 must be calibrated for each detector, measured in the absence of

attenuating objects

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CT Number

Need 12 bits to represent, h=-1000 for air..

Godfrey Hounsfield, together with Alan Cormack, invented X-ray CT!

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Parameterization of a Line

s

x

y

= l

Option 1 (parameterized bys):

Option 2:

Each projection line isdefined by (l,!)

A point on this line (x,y) can

be specified with two options

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Line Integral: parametric form

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Line Integral: set form

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Physical meaning of f

& g

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What is g(l,)?

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Example

Example 1: Consider an image slice which contains a

single square in the center. What is its projections along0, 45, 90, 135 degrees?

Example 2: Instead of a square, we have a rectangle.Repeat.

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Sinogram

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Bottom row: !=0 (vertical proj), middle !="/2 (horizontal proj), top !="-d!

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Backprojection

The simplest method for reconstructing an image from a

projection along an angle is by backprojection

Assigning every point in the image along the line defined by (l,!) theprojected value g(l, !), repeat for all l for the given !

s

x

y

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Example

Continue with the example of the image with a square in

the center. Determine the backprojected image fromeach projection and the reconstruction by summing

different number of backprojections

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Two Ways of Performing Backprojection

Option 1: assigning value of g(l, !) to all points on the line (l, !)

g(l, !) is only measured at certain l: ln=n #l If l is coarsely sampled (#l is large), many points in an image will not be

assigned a value

Many points on the line may not be a sample point in a digital image

Option 2: For each !, go through all sampling points (x,y) in animage, find its corresponding l=x cos !+y sin !, take the g valuefor (l, !)

g(l, !) is only measured at certain l: ln=n #l

must interpolate g(l, !) for any l from given g(ln, !)

Option 2 is better, as it makes sure all sample points in an image are

assigned a value

For more accurate results, the backprojected value at each pointshould be divided by the length of the underlying image in theprojection direction (if known)

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Backprojection Summation

Replaced by asum in practice

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Implementation Issues

From L. Parra at CUNY, http://bme.ccny.cuny.edu/faculty/parra/teaching/med-imaging/lecture4.pdf32

g(:,phi) stores the projection data at angle phi corresponding to excentricities stored in s_n

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Implementation: Projection

To create projection data using computers will have similar problems.Possible l and !are both quantized. If you first specify (l,!), then find (x,y)that are on this line. It is not easy. Instead, for given !, you can go throughall (x,y) and determine corresponding l, quantize l to one of those you wantto collect data.

Sample matlab code (for illustration purpose only)

f(x,y) stores the original image data

G(l,phi) stores projection data, ql is the desired quantization stepsize for l.

N=ceil(sqrt(I*I+J*J))+1; %(assume image size IxJ), N0 is maximum lateral distance

N0= floor((N-1)/2);ql=1;

G=zeros(N,180);

for phi=0:179for (x=-J/2:J/2-1; y=-I/2:I/2-1)

l=x*cos(phi*pi/180)+y*sin(phi*pi/180);

l=round(l/ql)+N0+1;If (l>=1) && (l

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Problems with Backprojection

!Blurring

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Projection Slice Theorem

The Fourier Transform of a projection at angle !is a line in theFourier transform of the image at the same angle.

If (l,!) are sampled sufficientlydense, then from g (l,!) weessentially know F(u,v) (on the polar coordinate), and by inverse

transform we can obtain f(x,y)!

dlljlgG }2exp{),(),( !"##" $=%&

&$

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Ill t ti f th P j ti Sli

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Illustration of the Projection Slice

Theorem

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Proof

Go through on the board

Using the set form of the line integral

See Prince&Links, P. 203 (2ndedition)

dlljlgG }2exp{),(),( !"##" $=%&

&$

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The Fourier Method

The projection slice theorem leads to the following

conceptuallysimple reconstruction method

Take 1D FT of each projection to obtain G($,!) for all !

Convert G($,!) to Cartesian grid F(u,v)

Take inverse 2D FT to obtain f(x,y)

Not widely used because

Difficult to interpolate polar data onto a Cartesian grid

Inverse 2D FT is time consuming

But is important for conceptual understanding

Take inverse 2D FT on G($,!) on the polar coordinate leads tothe widely used Filtered Backprojectionalgorithm

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Filtered Backprojection

Inverse 2D FT in Cartesian coordinate:

Inverse 2D FT in Polar coordinate:

Proof of filtered backprojection algorithm

Inverse FT

! ! +

= dudvevuFyxf yvxuj )(2),(),( "

! !>" #>"+

=

$

%%$& %&&%&%&

20 0

)sincos(2)sin,cos(),( ddeFyxf yxj

=l=G(",!)

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Filtered Backprojection Algorithm

Algorithm:

For each !

Take 1D FT of g(l,!) for each !-> G($,!)

Frequency domain filtering: G($,!) -> Q($,!)=|$|G($,!)

Take inverse 1D FT: Q($,!) -> q(l,!)

Backprojecting q(l,!) to image domain -> b!(x,y)

Sum of backprojected images for all !

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Function of the Ramp Filter

Filter response:

c($) =|$|

High pass filter

G($,!) is more denselysampled when $is small, and

vice verse The ramp filter compensate

for the sparser sampling athigher $

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Convolution Backprojection

The Filtered backprojection method requires taking 2 Fourier

transforms (forward and inverse) for each projection

Instead of performing filtering in the FT domain, perform convolutionin the spatial domain

Assuming c(l) is the spatial domain filter

|$| c(l)

|$|G($,!) c(l) * g(l,!)

For each !:

Convolve projection g(l,!) with c(l): q(l,!)= g(l,!) * c(l)

Backprojecting q(l,!) to image domain -> b!(x,y)

Add b!(x,y) to the backprojection sum

Much faster if c(l) is short

Used in most commercial CT scanners

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Step 1: Convolution

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Step 2: Backprojection

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Step 3: Summation

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Ramp Filter Design

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The Ram-Lak Filter (from [Kak&Slaney])

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Common Filters

Ram-Lak: using the rectangular window

Tend to amplify noise

Shepp-Logan: using a sinc window

Cosine: using a cosine window

Hamming: using a generalized Hamming window

See Fig. B.5 in A. Webb, Introduction to biomedicalimaging

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Practical Implementation

Projectionsg(l, !)are only measured at finite intervals

l=n#; #chosen based on maximum frequency in G(",!), W

1/#>=2W or #

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Matlab Implementation

MATLAB (image toolbox) has several built-in functions:

phantom: create phantom images of size NxNI = PHANTOM(DEF,N) DEF=Shepp-Logan,Modified Shepp-Logan

Can also construct your own phantom, or use an arbitrary image

radon: generate projection data from a phantom

Can specify sampling of !R = RADON(I,THETA)

The number of samples per projection angle = sqrt(2) N

iradon: reconstruct an image from measured projections

Uses the filtered backprojection method

Can choose different filters and different interpolation methods forperforming backprojection

[I,H]=IRADON(R,THETA,INTERPOLATION,FILTER,FREQUENCY_SCALING,OUTPUT_SIZE)

Use help radonetc. to learn the specifics

Other useful command:

imshow, imagesc, colormap

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Use MATLAB to illustrate projection and reconstruction

using different parameter settings

More vs. less projection angles

Different filters

I=phantom(def,N),

DEF='Shepp-Logan, 'Modified Shepp-Logan (default)

R = RADON(I,THETA), e.g. THETA=0:179 or 0:2:179

I = IRADON(R,THETA,INTERPOLATION,FILTER)

Interpoltation filters: used for backprojection

'nearest' - nearest neighbor interpolation,

'linear' - linear interpolation (default),

'spline' - spline interpolation

Reconstruction filter: Ram-Lak (default), 'Shepp-Logan, 'Cosine


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Top: 180 projectionsMiddle: 90 projections

Bottom: 45 projections

MATLAB code:

>> I=imread('lena.jpg');>> r180=radon(I,0:179); % 0:2:179;

>> p180=iradon(r180,0:179,spline,cosine);>> subplot(1,3,1),imagesc(I);

>> subplot(1,3,2),imagesc(r180);

>> subplot(1,3,3),imagesc(p180);

>> truesize>> colormap(gray)

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Summary

Different generations of CT machines:

Difference and pros and cons of each

X-ray source and detector design

Require (close-to) monogenic x-ray source

Relation between detector reading and absorption properties of theimaged slice

Line integral of absorption coefficients (Radon transform)

Reconstruction methods

Backprojection summation

Fourier method (projection slice theorem)

Filtered backprojection

Convolution backprojection

Impact of number of projection angles on reconstruction imagequality

Matlab implementations

Equivalent, but differ in

computation

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Reference

Prince and Links, Medical Imaging Signals and Systems,Chap 6.

Webb, Introduction to biomedical imaging, Appendix B.

Kak and Slanley, Principles of Computerized

Tomographic Imaging, IEEE Press, 1988. Chap. 3

Electronic copy available at

http://www.slaney.org/pct/pct-toc.html

Good description of different generations of CTmachines

http://www.kau.edu.sa/Files/0008512/Files/19500_2nd_presentation_final.pdf

http://bme.ccny.cuny.edu/faculty/parra/teaching/med-imaging/

lecture4.pdf

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Homework

Reading:

Prince and Links, Medical Imaging Signals and Systems,Chap6, Sec.6.1-6.3.3

Note down all the corrections for Ch. 6 on your copy ofthe textbook based on the provided errata.

Problems for Chap 6 of the text book:

1. P6.5

2. P6.14 (Note that you can leave the solution of part (a) in theform of the convolution of two functions without deriving theactual convolution. For part (b) you could describe in terms ofintuition.)

3. P6.17

4. The added problem in the following page

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Added problem

Consider a 4x4 discrete image that contains a diagonal lineI=[1,0,0,0;0,1,0,0;0,0,1,0;0,0,0,1];

a) determine its projections in the directions: 0, 45,90,135 degrees.

b) determine the backprojected image from each projection;

c) determine the reconstructed images by using projections in the 0 and 90degrees only.

d) determine the reconstructed images by using all projections. Comment onthe difference from c).

Note that for back-projection, you should normalize the backprojection ineach direction based on the total length of projection along that direction inthe square area.


Computer Assignment

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Computer AssignmentDue: Two weeks from lecture date

1. Learn how do phantom.radon,iradonwork; summarize theirfunctionalities. Type demoson the command line, then select toolbox -

> image processing -> transform -> reconstructing an image fromprojection data. Alternatively, you can use helpfor each particularfunction.

2. Write a MATLAB program that 1) generate a phantom image (you can usea standard phantom provided by MATLAB or construct your own), 2)produce projections in a specified number of angle, 3) reconstruct the

phantom using backprojection summation; Your program should allow theuser to specify the number of projection angle. Run your program withdifferent number of projections for the same view angle, and the differentview angles, and compare the quality. You should NOT use the radon( )and iradon()function in MATLAB.

3. Repeat 1 but uses filter backprojection method for step 3). In addition to

the number of projection angles, you should be able to specify the filteramong several filters provided by Matlab and the interpolation filters usedfor backprojection. Compare the reconstructed image quality obtained withdifferent filters and interpolation methods for the same view angle andnumber of projections. You can use the iradon()function in MATLAB

4. (Optional) Repeat 3 but uses convolution backprojection method. Youhave to do your own program

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CT Ch6 Part1