ELEG 479 Lecture #8 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware

ELEG 479Lecture #8

Mark Mirotznik, Ph.D.Associate Professor

The University of Delaware

Summary of Last LectureX-ray Radiography

Overview of different systems for projection radiography Instrumentation

Overall system layout X-ray sources grids and filters detectors

Imaging Equations Basic equations Geometrical distortions More complicated imaging equations

Hounsfield’s Experimental CT

Lets look at how CT works!

x

y

= xray attenuation of 2.5

= xray attenuation of 5

= xray attenuation of 0

Example

l

),0( lg

Our First Projection

Our First Projection

l

),0( lg

l

l

45o

l

),45( lg o

Rotate and Take Another Projection

l

90ol

),90( lg o

Rotate and Take Another Projection

l

This is called a sinogram

Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


Sinogram

l


The sinogram is what is measured by a CT machine. The real trick is how do we reconstruct the unknown image from the sinogram data?

l


Radon Transform)ln(),(

o

d

I

Ilg Given and ),(),( yxyxf

dssysxflg

slsy

slsx

))(),((),(

)cos()sin()(

)sin()cos()(

In CT we measure )ln(),(o

d

I

Ilg

and need to find ),(),( yxyxf

)cos()sin()(

)sin()cos()(

))(),((),(

slsy

slsx

dssysxflgusing

Radon Transform

In CT we measure ),( lg

and need to find ),( yxf

We use

dxdylyxyxflg )sincos(),(),(

Reconstruction


The Problem

In imaging we measure g(l,q) and need to determine f(x,y)

q

l0

p

g( ,q l)x

y

f(x,y)

??

Back Projection MethodA little trick that almost works!

ObjectBack Projection

Back Projection MethodA little trick that almost works!

Object

Back Projection

We do this for every angle and then add together all the back projected images

Back Projection MethodStep #1: Generate a complete an image for each projection (e.g. for each angle q)

)),sin()cos((),( yxgyxb

Step #2: Add all the back projected images together

These are called back projected images

0

),(),( dyxbyxfb

Back Projection Method

Kind of worked but we need to do better than this. Need to come up with a better reconstruction algorithm.

Reconstructed ImageOriginal object Reconstructed object

Projection-Slice TheoremThis is a very important theorem in CT imaging

First take the 1D Fourier transform a projection g(l, )q

dlelglgG ljD

21 ),(,(),(

),( lg



dlelglgG ljD

21 ),(,(),(

Next we substitute the Radon transform for g(l, )q

dldxdyelyxyxfG lj 2)sincos(),(),(




dlelglgG ljD

21 ),(,(),(

Next we substitute the Radon transform for g(l, )q

dldxdyelyxyxfG lj 2)sincos(),(),(


Next we do a little rearranging

dxdydlelyxyxfG lj 2)sincos(),(),(

Projection-Slice TheoremThis is a very important theorem in CT imagingNext we do a little rearranging

dxdydlelyxyxfG lj 2)sincos(),(),(

Applying the properties of the delta function

dxdyeyxfG

dxdyeyxfG

yxj

yxj

sincos2

sincos2

),(),(

),(),(

What does this look like?


dxdyeyxfG

dxdyeyxfG

yxj

yxj

sincos2

sincos2

),(),(

),(),(

What does this look like?

This looks a lot like

dxdyeyxfvuF yvxuj 2),(),(

with )sin(),cos( vu


))sin(),cos((),( FG

dxdyeyxfG yxj sincos2),(),(

So what does this mean?


))sin(),cos((),( FG


Question: So what does this mean?

Answer: If I take the 1D FT of a projection at an angle q the result is the same asa slice of the 2D FT of the original object f(x,y)


))sin(),cos((),( FG


So what does this mean?

If I take the 1D FT of a projection at an angle q the result is the same asa slice of the 2D FT of the original object f(x,y)

Projection-Slice TheoremIf I take the 1D FT of a projection at an angle q the result is the same asa slice of the 2D FT of the original object f(x,y)

f(x,y) F(u,v)

2D FT

dlelglgG ljooDo

21 ),(),(),(

),( olg qo

qo

)sin(),cos( vu

u

vvu

vu

122 tan,

)sin(),cos(

The Fourier Reconstruction Method

Take projections at all angles q.Take 1D FT of each projection to build F(u,v) one slice at a time.Take the 2D inverse FT to reconstruct the original object based on F(u,v)

f(x,y) F(u,v)

2D IFT

dlelglgG ljooDo

21 ),(),(),(

),( olg qo

q

),(),( 12 Gyxf D

)sin(),cos( vu

Image Reconstruction Using Filtered Backprojection

t

),( lg

Filter

Backprojection

Filtered Back Projection

The Fourier method is not widely used in CT because of the computational issues with creating the 2D FT from projections. However, the method does lead to a popular technique called filtered back projection.

2

0

sincos2)sin,cos(),( ddeFyxf yxj

In polar coordinates the inverse Fourier transform can be written as

dudvevuFyxf yvxuj 2),(),(

with )sin(),cos( vu


The Fourier method is not widely used in CT because of the computational issues with creating the 2D FT from projections. However, the method does lead to a popular technique called filtered back projection.

2

0

sincos2)sin,cos(),( ddeFyxf yxj

In polar coordinates the inverse Fourier transform can be written as

with )sin(),cos( vu

From the projection theorem ))sin(),cos((),( FG

2

0

sincos2),(),( ddeGyxf yxjWe can write this as

Filtered Back ProjectionThe Fourier method is not widely used in CT because of the computational issues with creating the 2D FT from projections. However, the method does lead to a popular technique called filtered back projection.

2

0

sincos2),(),( ddeGyxf yxjWe can write this as

Since ),(),( lglg you can show

0

sincos2),(),( ddeGyxf yxj

which can be rewritten as

0 )sin()cos(

2),(),( ddeGyxfyxl

lj

Filtered Back Projectionverses Back Projection

0 )sin()cos(

2),(),( ddeGyxfyxl

lj


0

),(),( dyxbyxfb

A. Back Projection

B. Filtered Back Projection

dlelglgG ljD

21 ),(,(),(

Filtered Back Projection MethodThis always works!

Object

Digital Filter1) take 1D FFT of projection2) multiply by ramp filter3) take 1D inverse FFT4) make a back projection


Filtered Back Projection MethodAlways works!

Object



Filtered Back Projection MethodAlways works!

Object



We do this for every angle and then add together all the filtered back projected images

Filtered Back Projectionverses Back Projection

0 )sin()cos(

2),(),( ddeGyxfyxl

lj


0

),(),( dyxbyxfb

A. Back Projection

B. Filtered Back Projection

dlelglgG ljD

21 ),(,(),(

Matlab DemoYour Assignment(b) Write a matlab function that reconstructs an image

using the filtered back projection method

Convolution Back Projection

It may be easier computationally to compute the inner 1D IFT using a convolution

0 )sin()cos(

2),(),( ddeGyxfyxl

lj

From the filtered back projection algorithm we get

)()()()( 212111 xfxfFFD recall

0)sin()cos(

11 )(),(),( dlgyxf

yxlD


0)sin()cos(

11 )(),(),( dlgyxf

yxlD

Let

0)sin()cos(

11

)(),(),(

)()(

dlclgyxf

lc

yxl

D


The problem is delc ljD

211 )()(

does not exist

0)sin()cos(

11

)(),(),(

)()(

dlclgyxf

lc

yxl

D


The problem is delc ljD

211 )()(

does not exist

0)sin()cos(

11

)(),(),(

)()(

dlclgyxf

lc

yxl

D

The solution deWWlc ljD

211 )())(()(~

where )(W is called a weighting function

0

)(~),(),( dlclgyxf

deWWlc ljD

211 )())(()(~


Common window functions

Hamming window Lanczos window (sinc function) Simple rectangular window Ram-Lak window Kaiser window Shepp-Logan window

0

)(~),(),( dlclgyxf

deWWlc ljD

211 )())(()(~

• Incorporated linear array of 30 detectors

• More data acquired to improve image quality (600 rays x 540 views)

• Shortest scan time was 18 seconds/slice

• Narrow fan beam allows more scattered radiation to be detected

• Number of detectors increased substantially (to more than 800 detectors)

• Angle of fan beam increased to cover entire patient– Eliminated

need for translational motion

• Mechanically joined x-ray tube and detector array rotate together

• Newer systems have scan times of ½ second

2G3G

Ring artifacts

• The rotate/rotate geometry of 3rd generation scanners leads to a situation in which each detector is responsible for the data corresponding to a ring in the image

• Drift in the signal levels of the detectors over time affects the t values that are backprojected to produce the CT image, causing ring artifacts

Ring artifacts

• Designed to overcome the problem of ring artifacts

• Stationary ring of about 4,800 detectors

• Designed to overcome the problem of ring artifacts

• Stationary ring of about 4,800 detectors

• Developed specifically for cardiac tomographic imaging

• No conventional x-ray tube; large arc of tungsten encircles patient and lies directly opposite to the detector ring

• Electron beam steered around the patient to strike the annular tungsten target

• Capable of 50-msec scan times; can produce fast-frame-rate CT movies of the beating heart

• Helical CT scanners acquire data while the table is moving• By avoiding the time required to translate the patient table, the total scan time

required to image the patient can be much shorter• Allows the use of less contrast agent and increases patient throughput• In some instances the entire scan be done within a single breath-hold of the

patient

Computer Assignment1. Write a MATLAB program that reconstructs an image from its projections using

the back projection method. Your program should allow the user to input a phantom object and a set (e.g. vector) of projection angle. Your program should then: (a) compute the sinogram of the object (you can use Matlab’s radon.m command to do this), (b) compute the reconstructed image from the sinogram and vector of projection angles, (c) try your program out for several different objects and several different ranges of projection angles

2. Do the same as #1 using the filter back projection method.

3. (grad students only) Do the same with the convolution back projection method

Documents

ELEG 479 Lecture #8 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware