Computer Tomography: Computational theory and methodsborko/pub/ct.pdfDenition Tomography: technique...

Computer Tomography:

Computational theory and methods

University of Bergen

15 October 2004, Bergen.

Borislav V. [email protected]

http://www.ii.uib.no/ �borkoDepartment of computer science

University of Bergen, Norway

Computer Tomography: Computational theory and methods – p.1/28

http://www.ii.uib.no/~borko

Outline

�

Introduction�

Fields of applications�

Principles of X-ray CT�

Reconstruction techniques in 2D- Transform based methods (FBP)- Algebraic reconstruction techniques�

Con-beam reconstructions- Circular- Helical

�

Conclusions


Definition

Tomography: technique for imaging cross-sectionsor slices from a set of external measurements of aspatially varying function.


Fields of applications

�

X-ray CT- medical imaging- industrial nondestructive evaluation- airport screening- microtomography

Magnetic Resonance Imaging (MRI)- electromagnetic waves

Transmission electron microscopy- an electron beem

CT principles are also applied in:- oceanography- astronomy- geophysics- optics



�

X-ray CT- medical imaging- industrial nondestructive evaluation- airport screening- microtomography�

Magnetic Resonance Imaging (MRI)- electromagnetic waves





�


Magnetic Resonance Imaging (MRI)- electromagnetic waves�





�


Magnetic Resonance Imaging (MRI)- electromagnetic waves�

Transmission electron microscopy- an electron beem�



Some hisrory

�

Radon 1917- reconstruction of a function from its projections

Hounsfield 1972- patented the first CT scanner

Cormack 1963 - 1964- contributionto mathematics of X-ray CT

Hounsfield and Cormack shared the 1979 Nobel Prize for medicine

since then the field is steadily evolving and givis rise to never ceasing flow of newalgorithms


Some hisrory

�

Radon 1917- reconstruction of a function from its projections�

Hounsfield 1972- patented the first CT scanner





Some hisrory

�


Hounsfield 1972- patented the first CT scanner�





Some hisrory

�



Cormack 1963 - 1964- contributionto mathematics of X-ray CT�




Some hisrory

�



Cormack 1963 - 1964- contributionto mathematics of X-ray CT�

Hounsfield and Cormack shared the 1979 Nobel Prize for medicine�



Principles of X-ray CT


Principles of X-ray CT

�

X-ray traverses anobject along stight line�

attenuated signals fromvarious directions arerecorded�

reconstruction algorithmsare used to convert theprojections


Projections

Let� �� represents the density of a two-dimensional object� � applied intensity form the X-ray source� � � the attenuated intensity of a ray as it propagates throught the object

along the line

� � � ��

The line integral can be written asRadon Transform

where

is the Dirac delta function and

(the equation of the X-ray)


Projections

Let� �� represents the density of a two-dimensional object� � applied intensity form the X-ray source� � � the attenuated intensity of a ray as it propagates throught the object

along the line

� � � ��

then

� � ��

where

�

- projection angle�

- detector position


where




Projections

��


where




Projections

��


��

� � � �� ! ��" ��

where!


" �� #$% � & �% ' � � � �



Fourier Slice Theorem

Consider the two-dimensional FT of the object function

( ��) � * � � � �

� � � �� + ,- . / �0 132 4 5 � � � �

and the one-dimensional FT of the projection

� � � � �

6 � ��7 � � � � �� + ,- . 8 �9� �



Consider the two-dimensional FT of the object function

( ��) � * � � � �

� � � �� + ,- . / �0 132 4 5 � � � �

and the one-dimensional FT of the projection

� � � � �

6 � ��7 � � � � �� + ,- . 8 �9� �

Simplest case, when * � :

(

� � :)

( ��) � : � � � �

� � � �� + ,- . �0 � � � � � � �

; � � � �� < + ,- . �0 � �

� � �

;� � �� < + ,- . �0 � � � � � �� = > �� + ,- . �0 � � � 6 � = > �) �



In general for any anglr

�6 � ��7 � � ( ��7 � � � � ( � 7 #$% � � 7 % ' � � �@?


Fourier reconstruction algorithm

�

Take projection of an object function at angles

� A � � - � ? ? ? � � B�

Fourier transform each projection to obtain the values of( �) � * ��

Recover the object function

� �� by using the inverse FT

� ��

� � ( �) � * ��+ ,- . / �0 12 4 5 � ) � *

Discretising on the square

where is even integer.

The values of F(u,v) on a square grid are obtained from the available values along the

radial lines by interpolation.



�


� A � � - � ? ? ? � � B�




� ��

� � ( �) � * ��+ ,- . / �0 12 4 5 � ) � *

Note: infinite number of projections on an interval with lenght C allows exactreconstuction.

Discretising on the square

where is even integer.


radial lines by interpolation.



�


� A � � - � ? ? ? � � B�




� ��

� � ( �) � * ��+ ,- . / �0 12 4 5 � ) � *

Discretising on the square � D EF G � G D E FIH � D E F G � G D E F

� �� J KD-L M-

N = L M-L M-

� = L M-( O PD � Q D R + ,- . / / N MS 5 0 1 / � MS 5 4 5 �

where

T

is even integer.


radial lines by interpolation.Computer Tomography: Computational theory and methods – p.9/28

Reconstruction for Parallel Projections

� ��

� � ( �) � * � + ,- . / �0 132 4 5 � ) � *

� - .>

�> ( � 7 � � ��+ ,- . 8

UV WX Y�� #$% � & �% ' � � � 7 � 7 � �



� ��

� � ( �) � * � + ,- . / �0 132 4 5 � ) � *

� - .>

�> ( � 7 � � ��+ ,- . 8

UV WX Y�� #$% � & �% ' � � � 7 � 7 � �

� .>

� � 6 � � 7 �Z 7 Z + ,- . 8 �� 7X Y V W[]\ / � 5

� �



� ��

� � ( �) � * � + ,- . / �0 132 4 5 � ) � *

� - .>

�> ( � 7 � � ��+ ,- . 8

UV WX Y�� #$% � & �% ' � � � 7 � 7 � �

� .>

� � 6 � � 7 �Z 7 Z + ,- . 8 �� 7X Y V W[]\ / � 5

� � � .>

� �

� � �� ^ ��+ ,- . 8_ � ^ Z 7 Z + ,- . 8 �� 7 � �



� ��

� � ( �) � * � + ,- . / �0 132 4 5 � ) � *

� - .>

�> ( � 7 � � ��+ ,- . 8

UV WX Y�� #$% � & �% ' � � � 7 � 7 � �

� .>

� � 6 � � 7 �Z 7 Z + ,- . 8 �� 7X Y V W[]\ / � 5

� � � .>

� �

� � �� ^ ��+ ,- . 8_ � ^ Z 7 Z + ,- . 8 �� 7 � �

� .>

`abacabaed�

� �f� ��^ � � � Z 7 Z + ,- . 8 / � _ 5 � 7X Y V Wg / � _ 5

� ^h

ibicibiej � � � .>

; � � �� ^ �k � � � ^ �� ^ < � �



� ��

� � ( �) � * � + ,- . / �0 132 4 5 � ) � *

� - .>

�> ( � 7 � � ��+ ,- . 8

UV WX Y�� #$% � & �% ' � � � 7 � 7 � �

� .>

� � 6 � � 7 �Z 7 Z + ,- . 8 �� 7X Y V W[]\ / � 5

� � � .>

� �

� � �� ^ ��+ ,- . 8_ � ^ Z 7 Z + ,- . 8 �� 7 � �

� .>

`abacabaed�

� �f� ��^ � � � Z 7 Z + ,- . 8 / � _ 5 � 7X Y V Wg / � _ 5

� ^h


; � � �� ^ �k � � � ^ �� ^ < � �

� .> � �f� l k � �� #$% � & �% ' � � �� .> m � �� #$% � & �% ' � � ��



� ��

� � ( �) � * � + ,- . / �0 132 4 5 � ) � *

� - .>

�> ( � 7 � � ��+ ,- . 8

UV WX Y�� #$% � & �% ' � � � 7 � 7 � �

� .>

� � 6 � � 7 �Z 7 Z + ,- . 8 �� 7X Y V W[]\ / � 5

� � � .>

� �

� � �� ^ ��+ ,- . 8_ � ^ Z 7 Z + ,- . 8 �� 7 � �

� .>

`abacabaed�

� �f� ��^ � � � Z 7 Z + ,- . 8 / � _ 5 � 7X Y V Wg / � _ 5

� ^h


; � � �� ^ �k � � � ^ �� ^ < � �

� .> � �f� l k �X Y V W

Filtering�� #$% � & �% ' � � ��

X Y V W

Backprojection

� .> m � �� #$% � & �% ' � � ��


The FBP algorithm

Step 1: Filtering m � � � � � � � 6 � ��7 �Z 7 Z + ,- . 8 �� 7

Superior results are obtained by deemphasizing the high frequencies with theHamming window function .

where is the inverse DFT of .

Step 2: BackprojectionReconstruc by the formula

where is the number of the projection angles .Note The values of in Step 2 can be approxiamted from the values obtained in Step 1by interpolation (linear).

Computational cost: Backprojection summation —>

Fast backprojections (Nilsson’96) —>


The FBP algorithm


Use badlimited FT to compute

6 � � 7 �

i.e.

6 � ��7 � J 6 � n P F oT p � KF oL M- A

B = L M-�� nq F o p + ,- . / N B M L 5 � P � � TF � ? ? ? � TF �

where

o

is a frequency band limit andT

is some (large) number corresponding to thenumber of the sampling points. (FFT)

Superior results are obtained bydeemphasizing the high frequencies with theHamming window function .







The FBP algorithm


Use badlimited FT to compute

6 � � 7 �

i.e.

6 � ��7 � J 6 � n P F oT p � KF oL M- A

B = L M-�� nq F o p + ,- . / N B M L 5 � P � � TF � ? ? ? � TF �

where

o

is a frequency band limit andT

is some (large) number corresponding to thenumber of the sampling points. (FFT)Therefore

m � nq F o p J n F oT p L M-N = L M-

6 � n P F oT p rsrsrbr P F oT rsrsrbr + ,- . / N B M L 5 � q � � TF � ? ? ? � TF ?

Inverse DFT of the product

6 � � P � F o E T � �

and

Z P � F o E T �Z

.

Superior results areobtained by deemphasizing the high frequencies with theHamming window function .







The FBP algorithm

Step 1: FilteringSuperior results are obtained by deemphasizing the high frequencies with theHamming window function

t

.

m � nq F o p J n F oT p L M-N = L M-

6 � n P F oT p rsrsrbr P F oT rsrsrbr t n P F oT p + ,- . / N B M L 5

Superior results are obtained by deemphasizing the high frequencies with theHamming window function .







The FBP algorithm


t

.

m � nq F o p J n F oT p �� nq F o p l u nq F o p �where

u �q Ev o �

is the inverse DFT of

Z P � F o E T �Z t � P � F o E T � �

.






The FBP algorithm


t

.


u �q Ev o �


Z P � F o E T �Z t � P � F o E T � �

.

Step 2: BackprojectionReconstruc

� �� by the formula

� �� J C wx

= A m �zy �� #$% � & �% ' � � �

where

w

is the number of the projection angles

� .Note The values of

m � in Step 2 can be approxiamted from the values obtained in Step 1by interpolation (linear).




The FBP algorithm


t

.


u �q Ev o �


Z P � F o E T �Z t � P � F o E T � �

.

Step 2: BackprojectionReconstruc

� �� by the formula

� �� J C wx

= A m �zy �� #$% � & �% ' � � �

where

w

is the number of the projection angles

� .Note The values of

m � in Step 2 can be approxiamted from the values obtained in Step 1by interpolation (linear).


{ � T | �


{ � T- �$ } T �


Reconstruction for Fan ProjectionsEquiangular Rays

Let

~f� ��

be a fan projection

�

is the angle between the sourceand a reference axis.

� gives the location of a ray

� �|SO| source to origin dist.

�

- source to point

�� dist.In polar coordinates

� �" � u ��

� � � & �" % ' � � � - & �" #$% � � - ,where � � � � u

.

where



� �� .>

� � �f� � � �k �� #$% � & �% ' � � � � � � ��

� �" � u � � KF - .>

� � �� k �" #$% � � � u � � � ��

where



� �� .>

� � �f� � � �k �� #$% � & �% ' � � � � � � ��

� �" � u � � KF - .>

� � �� k �" #$% � � � u � � � ��

� ? ? ?� - .>

K�- m � � � � ��

where

m � � � � � ~ �� l� ��

~ �� ~� �� #$% �

� � � � � KF n �% ' � �p - k � � �


Weighted Backprojection AlgorithmEquiangular Rays

Let

� is the sampling interval of the projection� are the angles at which projections are taken

Step 1 For each

~f� y � Q � � � Q � �

, generate the corresponding modified projection~ �� y � Q � � � ~� � Q � �� #$% Q �

Step 2 Convolve with to generate thecorresponding filtered projection (FFT)

where is a smoothing filter.

Step 3 Perform a weighted backprojection

where is the angle of a ray that passes through the point and .



Let


Step 1 For each

~f� y � Q � � � Q � �


Step 2 Convolve

~ �� y � Q � � with� � Q � � � A- � �� - k � Q � � to generate thecorresponding filtered projection (FFT)m � y � Q � � � ~ �� y � Q � � l� � Q � �

Step 2 Convolve

with to generate the corresponding filtered projection (FFT)

where is a smoothing filter.





Let


Step 1 For each

~f� y � Q � � � Q � �


Step 2 Convolve

~ �� y � Q � � with� � Q � � � A- � �� - k � Q � � to generate thecorresponding filtered projection (FFT)m � y � Q � � � ~ �� y � Q � � l� � Q � � lq � Q � � �

where

q � Q � � is a smoothing filter.





Let


Step 1 For each

~f� y � Q � � � Q � �


Step 2 Convolve

~ �� y � Q � � with� � Q � � � A- � �� - k � Q � � to generate thecorresponding filtered projection (FFT)m � y � Q � � � ~ �� y � Q � � l� � Q � � lq � Q � � �

where

q � Q � � is a smoothing filter.

Step 3 Perform a weighted backprojection� �� J � � = A w K�- �� A � m � y ��

where � �

is the angle of a ray that passes through the point

�� and

� � � F C E w

.


Reconstruction for Fan ProjectionsEqually Spaced Detectors

Let

~f� � � � be a fan projection

�

is the angle between the sourceand a reference axis.

� is the distance along the detector

� �|SO| source to origin dist.

�

- the dist. source to projectionof a pixel on the central ray

In polar coordinates

� �" � u ��

� � � & " % ' � � � � u ��

where


Reconstruction for Fan ProjectionsEqually Spaced Detectors

� �� .>

� � �� k �� #$% � & �% ' � � � � � � ��

� �" � u � � KF - .>

� � �f� � � � k �" #$% � � � u � � � ��

� ? ? ?� - .>

K�- m � � � � � � � �

where

m � � � � � ~ �� l � � � �

~ �� ~� � � �� - & �-

� � � � � KF k � � �


Algebraic reconstruction techniques

Advantages

- easy to handle different rays geometry- more adaptable to missing data - better image quality- metal artifacts are reduced- less radiation dose

Disadvantage

Higher computational cost!


Image and Projection Representation

The idea: Assume that the cross section consists of array of unknowns

- total number of cellsis the value of

in the -th cell- the -th line integral ray-sum

whereis the total number of rays

represents the contribution

of the -th cell to the -th ray integral




�

- total number of cells�� is the value of

� ¡£¢¥¤ ¦ §

in the

¨ -th cell©«ª - the

¬-th line integral ray-sum

whereis the total number of rays

represents the contribution

of the -th cell to the -th ray integral




�

- total number of cells�� is the value of

� ¡£¢¥¤ ¦ §

in the

¨ -th cell©«ª - the

¬-th line integral ray-sum

�¯® ° ± ª � �� © ª¤ ¬ � ²¤ ³¤ ´ ´ ´¤ µ

whereµ

is the total number of rays

± ª � represents the contribution

of the¨ -th cell to the

¬-th ray integralComputer Tomography: Computational theory and methods – p.16/28

Projection methods

Therefore we have to solve the following linear system of equations

7 A A � A & 7 A- �- &� � � & 7 A L � L � ¶ A

7 - A � A & 7 - - �- &� � � & 7 - L � L � ¶-...7 · A � A & 7 ·- �- &� � � & 7 · L � L � ¶ ·

for large values of

¸

and

T

.

Use iterative methods - projestions (Kaczmarz’37, Tanabe’71)

Let is the initial guess

where


Projection methods

Therefore we have to solve the following linear system of equations

7 A A � A & 7 A- �- &� � � & 7 A L � L � ¶ A

7 - A � A & 7 - - �- &� � � & 7 - L � L � ¶-...7 · A � A & 7 ·- �- &� � � & 7 · L � L � ¶ ·

for large values of

¸

and

T

.

Use iterative methods - projestions (Kaczmarz’37, Tanabe’71)

Let

� / > 5 � O � / > 5A � � / > 5- � ? ? ? � / > 5L Ris the initial guess

� / 5 � � / A 5 � ¹�º »y¯¼ ½ ¾À¿ 8y Áy Â8y ¿ 8y 7 Ã � K � F � ? ? ? �

where 7 � � 7 A � 7 ÅÄ - � ? ? ? 7 L �


Projection methods

Theorem (Tanabe’71)If there exists a unique solution

� /ÇÆ 5

to the system of equations, then

� 'IÈBÉ � � / B · 5 � � /ÇÆ 5

The rate of convergence depends of the angles between the hyperplanes

- Full orthogonalization is computationally not feasible

- Pairwise orthogonalization scheme (Ramakrishnan’79)

- Carefully choose the order of the hyperplanes (Hounsfield’72)

When the process oscillate in the neighborhood of the intersections of thehyperplanes

When the process converges to a solution , such thatis minimized.


Projection methods


� /ÇÆ 5


� 'IÈBÉ � � / B · 5 � � /ÇÆ 5The rate of convergence depends of the angles between the hyperplanes



- Carefully choose the order of the hyperplanes (Hounsfield’72)

When the process oscillate in the neighborhood of the intersections of thehyperplanes

When the process converges to a solution , such thatis minimized.


Projection methods


� /ÇÆ 5


� 'IÈBÉ � � / B · 5 � � /ÇÆ 5The rate of convergence depends of the angles between the hyperplanes



- Carefully choose the order of the hyperplanes (Hounsfield’72)Ê When

¸Ë T

the process oscillate in the neighborhood of the intersections of thehyperplanesÊ When

¸ G T

the process converges to a solution

� � /ÇÆ 5

, such thatZ � / > 5 � � � /ÇÆ 5 Z

is minimized.


Projection methods

Consider again � / 5 � � / A 5 � ¹�º »y¯¼ ½ ¾À¿ 8y Áy Â8y ¿ 8y 7 Ã � K � F � ? ? ? �

It can also be written as

� / 5, � � / A 5, & ¶ � Ì Í LB = A 7 - B 7 , Î � K � F � ? ? ? � T � Ã � K � F � ? ? ? �

where Ì � � / A 5 � 7 � Í LB = A � / A 5B 7 B ?

Therefore � / 5, � � / A 5, & � � / 5, �

where � � / 5, � Áy ÏyÐ ÑÓÒ9Ô ½ 8 Õy Ò7 , ?

The main computational diffuculty is in the calculation, storage and retrieval of

the weight coefficients .


Projection methods

Consider again � / 5 � � / A 5 � ¹�º »y¯¼ ½ ¾À¿ 8y Áy Â8y ¿ 8y 7 Ã � K � F � ? ? ? �

It can also be written as

� / 5, � � / A 5, & ¶ � Ì Í LB = A 7 - B 7 , Î � K � F � ? ? ? � T � Ã � K � F � ? ? ? �

where Ì � � / A 5 � 7 � Í LB = A � / A 5B 7 B ?

Therefore � / 5, � � / A 5, & � � / 5, �

where � � / 5, � Áy ÏyÐ ÑÓÒ9Ô ½ 8 Õy Ò7 , ?

The main computational diffuculty is in the calculation, storage and retrieval of

the weight coefficients 7 , .Computer Tomography: Computational theory and methods – p.19/28

Algebraic Reconstruction Tech.(ART)

Ê Replace 7 , by 0 or 1, depending wether the center of the

Î -th cell is within the

Ã-th ray.Thus we can use � � / 5, � ¶ � Ì Tfor all cells whos centers are within the

Ã -th ray.T � Í LB = A 7 - B is the number of cells whose centers are within the

Ã -th ray.

Superior approxiamtion is

where is the lenght of the -th ray trought the reconstruction region.

- ART suffer from salt and pepper noise, caused by the approximations used for .

- It is possible to reduce the noce byrelaxation (i.e. update a pixel by )

SIRT - change the cell values after going through all of the equations







Ã -th ray.Ê Superior approxiamtion is � � / 5, � ¶ � � Ì T �

where

� is the lenght of the

Ã-th ray trought the reconstruction region.

- ART suffer from salt and pepper noise, caused by the approximations used for .










where



- ART suffer from salt and pepper noise, caused by the approximations used for 7 , .










where



- ART suffer from salt and pepper noise, caused by the approximations used for 7 , .- It is possible to reduce the noce by

relaxation (i.e. update a pixel by

Ö � � / 5, � Ö G K

)



SART (Simultaneous ART)

First reported by Anderson and Kak ’84Ê Reduce the error in the approximation of the ray integrals

bilinear elements insted of the traditional pixel basis

Correction terms are simultaneously applied to all the raysas for the SIRT methods

Hamming windowemphasizes the corrections applied near the middle of a ray relative to thoseapplied near the ends.

The overall SART iterative scheme is

where is the -th row of and are relaxation coefficients.



First reported by Anderson and Kak ’84Ê Reduce the error in the approximation of the ray integralsInsted of solving

¶ � L, = A 7 , � , � Ã � K � F � ? ? ? � ¸?

We use again the Radon Transform¶ � ~ � ��

� � � �� ! ��" ��

where " �� :

is the equation of theÃ-th ray and

~ is the projection operator.









¶ � L, = A 7 , � , � Ã � K � F � ? ? ? � ¸?

We use again the Radon Transform¶ � ~ � ��

� � � �� ! ��" ��

where " �� :

is the equation of theÃ-th ray and

~ is the projection operator.

Let

×Ø , �� Ù L, = A be basis functions and let� �� J Ú� �� Í L, = A � , Ø , ��

then¶ � ~ � �� J ~ Ú� �� L

, = A� , Û ,

wher Û , � ~ Ø , �� .









¶ � L, = A 7 , � , � Ã � K � F � ? ? ? � ¸?

We solve

¶ � L, = A� , Û , � Ã � K � F � ? ? ? � ¸?

- The two systems are the same for the pixel basis

Ø , �� ÜÞÝß

Kinside the

Î -th pixel:

everywhere else

- Superior reconstructions are obtained by using bilinear elementspyramid shaped basis








First reported by Anderson and Kak ’84Ê Reduce the error in the approximation of the ray integralsbilinear elements insted of the traditional pixel basis







First reported by Anderson and Kak ’84Ê Reduce the error in the approximation of the ray integralsbilinear elements insted of the traditional pixel basisÊ Correction terms are simultaneously applied to all the raysas for the SIRT methods






First reported by Anderson and Kak ’84Ê Reduce the error in the approximation of the ray integralsbilinear elements insted of the traditional pixel basisÊ Correction terms are simultaneously applied to all the raysas for the SIRT methodsÊ Hamming windowemphasizes the corrections applied near the middle of a ray relative to thoseapplied near the ends.





First reported by Anderson and Kak ’84Ê Reduce the error in the approximation of the ray integralsbilinear elements insted of the traditional pixel basisÊ Correction terms are simultaneously applied to all the raysas for the SIRT methodsÊ Hamming windowemphasizes the corrections applied near the middle of a ray relative to thoseapplied near the ends.


� / B 1 A 5, � � / B 5, & Ö BÍ · = A Û ,·

= A Û , ¶ � Û � � / B 5Í L, = A Û , � Î � K � F � ? ? ? � T �

where Û is the

Ã-th row of

Dand

Ö B Ë :

are relaxation coefficients.


More iterative algorithmsCensor and Elfving’02The idea: Use generalized projections and diagonal weighting matrices to reflect thesparsity of the

D

matrix.

Definition: A family

×à Ù · = Aof diagonal

Tâá T

matrices with diagonal elements� , Ë :

is called Sparsity Pattern Oriented (SPO) with respect to

¸ á Tmatix

Dif:

1)

Í · = A à �

2)� , � :

iff Û , � :for every

Ã � K � F � ? ? ? � ¸

.

Special case

if

if

where is the number of the nonzero elements in column of .Component Averaging (CAV)

Jiang and Wang’03 ART, SART, CAV, DWE are special cases of the Landweber method



D

matrix.



Tâá T



¸ á Tmatix

Dif:

1)

Í · = A à �

2)� , � :


Ã � K � F � ? ? ? � ¸

.

Diagonal Weighting (DWE)

� / B 1 A 5, � � / B 5, & Ö B ·y Ô ½äãyå æÔ ç

Û , ¶ � Û � � / B 5

Í Léè Ô ½ ãy è æÔ ç ê Õy èëy è � Î � K � F � ? ? ? � T?

Special case

if

if

where is the number of the nonzero elements in column of .Component Averaging (CAV)




D

matrix.



Tâá T



¸ á Tmatix

Dif:

1)

Í · = A à �

2)� , � :


Ã � K � F � ? ? ? � ¸

.

Special case

� , � ÜÞÝß

AÆå if Û , ì � :

:if Û , � :

where � , is the number of the nonzero elements in column

Î

of

D

.Component Averaging (CAV)

� / B 1 A 5, � � / B 5, & Ö B · = A Û , ¶ � Û � � / B 5

Í Lîí = A � í Û- í � Î � K � F � ? ? ? � T?




D

matrix.



Tâá T



¸ á Tmatix

Dif:

1)

Í · = A à �

2)� , � :


Ã � K � F � ? ? ? � ¸

.

Special case

� , � ÜÞÝß

AÆå if Û , ì � :

:if Û , � :

where � , is the number of the nonzero elements in column

Î

of

D

.Component Averaging (CAV)

� / B 1 A 5, � � / B 5, & Ö B · = A Û , ¶ � Û � � / B 5

Í Lîí = A � í Û- í � Î � K � F � ? ? ? � T?

Ê Jiang and Wang’03 ART, SART, CAV, DWE are special cases of the Landweber method� / B 1 A 5 � � / B 5 & Ö B ï A D ð o � ¶ � D� / B 5 �


3D Reconstructions

� Slice by SliceDisadvantages

- low speed- discrete nature- higher radiation dose

Cone-Beam Reconstructions- Circular- Helical


3D Reconstructions

� Slice by SliceDisadvantages

- low speed- discrete nature- higher radiation dose

� Cone-Beam Reconstructions- Circular- Helical


Circular Cone-Beam Reconstructions

The Idea: Apply fan-beam FBP toeach plane in the cone

Feldkamp, Davis, Kress ’84 (FDK)

A ray from the cone is described by

wherelocate a ray in a fanspecify the location ot the fan

- projection angle- fan angle

—> (rotation by degrees around the -axes)


Circular Cone-Beam Reconstructions

The Idea: Apply fan-beam FBP toeach plane in the cone

Feldkamp, Davis, Kress ’84 (FDK)

A ray from the cone is described by� � � #$% � & �% ' � �" � � � � � % ' � � & � #$% � �% ' � �&òñ #$% �

where� � � � �

locate a ray in a fan�" � � �

specify the location ot the fan�

- projection angle� - fan angle

�� ñ �

—>

� � � � � ñ �(rotation by

�degrees around the ñ -axes)


The FDK Algorithm

Based on the FBP Algorithm for equispatial rays� � � � � � ñ � � KF - .>

�-� � � � � -�

� ~� � ¶ � ó � �ô �- & ó- & ¶- k n � �� ¶ p � ¶� � �

where

� � Z 6õ Z

and

ó � öö Æ ñ .

The Algorithm

Step 1 Compute the modified projection

Step 2 Convolve with

Step 3 Backproject over the 3D reconstruction grid


The FDK Algorithm

Based on the FBP Algorithm for equispatial rays� � � � � � ñ � � KF - .>

�-� � � � � -�

� ~� � ¶ � ó � �ô �- & ó- & ¶- k n � �� ¶ p � ¶� � �

where

� � Z 6õ Z

and

ó � öö Æ ñ .

The Algorithm

Step 1 Compute the modified projection~ �� ¶ � ó � � �ô �- & ó- & ¶- ~� � ¶ � ó �

Step 2 Convolve

~ �� ¶ � ó �

with

k � ¶ � EFm � � ¶ � ó � � ~ �� ¶ � ó � l KF k � ¶ �

Step 3 Backproject over the 3D reconstruction grid� � � � � � ñ � � - .>

�-� � � � � - m � � � �� ñ� � � ��


The Tuy–Smith sufficiency condition

Exact reconstruction is possible if all planes intersecting theobject also intersect the source trajectory at least once.

Circular trajectory does not satisfay this condition since aplane parallel to the trajectory may also intersect the object.

Exact reconstruction form helical cone-beam data ispossible. Tam’95 Kudo, Noo, Defrise ’98


Conclusions

The challenges:

� Three dimentional images

� Better quality

� Faster algorithms

÷ Restricted in time


References

�

K. Tanabe, Projection Method for Solving a Singular Systems of Linear Equationsand its Applications, Numer. Math., 17, 203-214 (1971).�

H. Tuy, An Inversion Formula for Cone-Beam Reconstruction, SIAM J. Appl. Math.42 546–552 (1983).�

L. Feldkamp, L. Davis and J. Kress, Practical Cone-Beam Algorithm, Journal of theOptical Society of America 1, 612-619 (1984).�

K. Tam, Three-Dimensional Computerized Tomography Scanning Method andSystem for Large Objects with Smaller Area Detectors, US Patent 5,390,112, Feb14.(1995).�

S. Nilsson, Fast Backprojection, Technical report LiTH-ISY-R-1865, LinköpingUniversity (1996).�

A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press.(1998).�

H.Kudo, F. Noo and M. Defris, Cone-Beam Filtered Backprojection Algorithm forTruncated Helical Data, Physics in Medicine and Biology 43, 2885–2909 (1998).


References

�

G.Wang and M. Vannier, Computerized Tomography, (1998) available at:http://dolphin.radiology.uiowa.edu/ge/Teaching/ct/ct.html�

Y. Saad, H. van der Vorst, Iterative Solution of Linear Systems in the 20th Century,J. Comput. Appl. Math. 123, 1–33 (2000).�

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction,SIAM, (2001).�

H. Turbell, Cone-Beam Reconstruction Using Filtered Backprojectio, DissertationNo.672, Linköping Sudies in Science and Technology (2001).�

Y. Censor and T. Elfving, Block-Iterative Algorithms with Diagonally Scaled ObliqueProjections for the Linear Feasibility Problems, Siam J. Matrix Anal. Appl., 24,40-58 (2002).�

M. Jiang and G. Wang, Convergence Studies on Iterative Algorithms for ImageReconstruction, IEEE Transactions on Medical Imaging , 22, 569-579 (2003).


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