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May 19, 2004

Spiral CT Image Reconstruction Using Alternating Minimization Methods

Shenyu YanThesis Advisor: Dr. O’Sullivan

Washington University, St. Louis, MissouriElectronic Systems & Signals Research Laboratory

May 19, 2004

CT introduction

Alternating Minimization method

Pose search of high-density object

Incomplete projection data

Conclusion and future work

Content

May 19, 2004

CT Development

May 19, 2004

Attenuation function

ρρµµ ×= ),(),( ZEZE Z is the atomic number of the

material or tissue

)(),(),( xcZEEx ⋅= µµ

May 19, 2004

X

Det

ecto

rs

1

768Source Position 14081

Y

Data collection

May 19, 2004

Attenuation Function and Beer’s Law

)(),(),( xcZEEx ⋅= µµ

),( θtp212121 )sincos(),,(),( dxdxtxxExxtp −+= ∫ ∫

+∞

∞−θθδµθ

),(0 )()( θtpeyIyI −⋅=

Radon Transform

Beer’s Law

1x

2x

t

),,(),(

321

21

xxxxxxx

== 2D space

3D space

θ

May 19, 2004

Standard reconstruction method

Central slice theorem

∫∞

∞−

+−⋅= dtetpP xxj )sincos(2 21),(),( θθπρθθρ

1D FT of projection data

2D FT of attenuation function

∫ ∫∞

∞−

∞

∞−

+−⋅=Μ 21)(2

21212211),(),( dxdxexxuu uxuxj πµ

( )21,),( uuP Μ=θρ

Filtered back projection

θρρθρµπ πρ ddePxx tj∫ ∫ ⎟

⎠⎞⎜

⎝⎛=

∞

∞−

−

0

221 ||),(),(

May 19, 2004

Multi-slice Spiral CT

y

z

)(qη

q = 3 2 1 0

β

γ

y

focus

detector

Sdp =Pitch:

x

Advantages VS conventional CT

•Rapid scanning

•Reduce Motion Artifacts

d: the table feed per rotation

S: the total width of the collimated beam

May 19, 2004

Image Space X:

Data Space Y:

2-D

21 XXX ×=

DSY ×=

Source S position is specified by fan angle β.

Detector D specified only by γ.

Image and Measurement Spaces

3-D Spiral

321 XXXX ××=

DSY ×=

Source S is specified by β and Pitch

Detector D specified by γ and detector row index q.

May 19, 2004

Point Spread Function

x

y

z

1D

2D3D

4D

focusF

1D2D

3D

4Dpoint spread function h(y|x): average path of the cone-beam through the voxel

x indexes the image voxel

y indexes the source-detector pair, ),,( qγβ

),,( 321 xxx

May 19, 2004

Conventional reconstruction method for spiral CT data

Additional step: Z-interpolation

Reconstruct the images in 2D plane

Cause the stair-step artifacts

Nonlinear effects

May 19, 2004

CT introduction

Alternating Minimization method

Pose search of high-density object

Incomplete projection data

Conclusion and future Work

Content

May 19, 2004

References

J. A. Fessler. Statistical image reconstruction methods for transmission tomography. In Handbook of Medical Imaging, Volume 2. Medical Image Processing and Analysis, ch. 15, SPIE 2000.

J. A. O’Sullivan and J. Benac. Alternating minimization algorithms for transmission tomography. Submitted to IEEE Trans. Med. Imaging.

May 19, 2004

E: Energies ranging from 19-120 keV

I0(y,E): Mean number of source photons

µ(x,E): Attenuation function; the image we are trying to estimate

SourceI0(y,E)

Objectµ(x,E)

Detector, g(y)

E

I0(y,E)

)(),()|(exp),()( 0 yExxyhEyIygE

xσµ +

⎭⎬⎫

⎩⎨⎧−= ∫ ∑

Model for Transmission CT

May 19, 2004

Optimization Problem: minc I(d || g)

Two families:

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−==Ε ∑

x

xcxyhEEyIcEyqq )()|()(exp),():,(: 0 µ

⎭⎬⎫

⎩⎨⎧

=≥= ∑ )(),(:0),()( ydEypEypdLE

AM Algorithm

∑ =E

cygcEyq ):():,(

Statistical Model

{ }[ ]∑∈

−−=Yy

ydcygcygydcdl )!(ln):():(ln)():(

( ) ∑∈

⎥⎦

⎤⎢⎣

⎡−−=

Yy

cygydcyg

ydydcygdI ):()():(

)(ln)():(||

(1)

(2)

(3)

(4)

Rewrite problem

( ) )||(minmin)(||min)(

qpIcgdIdLpEqc ∈∈

= (5)

May 19, 2004

AM Algorithm

⎥⎦

⎤⎢⎣

⎡−= ∑

∈ Xx

kk ExxyhEyIcEyq ),(ˆ)|(exp),():,(ˆ )(0

)( µ

⎥⎦

⎤⎢⎣

⎡−= ∑

∈ Xx

k xcExyhEyI )(ˆ)()|(exp),( )(0 µ

∑≠

+=

0'

)()()(

)():',(ˆ)():,(ˆ):,(ˆ

E

kkk

ycEyqydcEyqcEyp

σ

∑ ∑∈

=Yy E

kk cEypxyhExb ):,(ˆ)|()()(~ )()( µ

∑ ∑∈

=Yy E

kk cEyqxyhExb ):,(ˆ)|()()(ˆ )()( µ

(1)

(2)

(3)

(4)

(5)

Iterative update of the estimate

)(ˆ)(~

ln)(

1)(ˆ)(ˆ)(

)()()1(

xbxb

xZxcxc

k

kkk −=+

where 1)(

)()|(≤∑

∈ Xx xZExyh µ

May 19, 2004

Image space: 128x128x36 mm

Projection data space (per rotation): 580 (source) x 168 (detector)

Detector row number: 8

Collimation(slice) width: 0.75 mm

Pitch: 2

Travel of table per scan rotation: 12 mm

Rotation number: 2

PMMA Cylinder with Low-density Objects1- PMMA: 0.0229/mm

2- Nylon: 0.02095/mm

3- Teflon: 0.0423/mm

Viewing window size [0.016 0.02]

Experiments for Spiral CT

PMMA

Telflon

Nylon

May 19, 2004

15th 16th

17th 18th

19th 20th

21th 22th

Reconstructed Results from Low-density noiseless Synthetic data after AM 200 (8 OS) Iterations

May 19, 2004

Reconstructed Results from Low-density Real data after AM 50 Iterations

18th 19th 20th

15th 16th 17th

Too coarse sampling interval, and high pitch value

Voxel sphere not real sphere in synthetic data

May 19, 2004

Reconstructed Results from High-density noiseless Synthetic data after AM 800/5000 (8 OS) Iterations

17th 18th

20th19th

17th 18th

20th19th

May 19, 2004

CT introduction

Alternating Minimization method

Pose search of high-density object

Incomplete projection data

Conclusion and future work

Content

May 19, 2004

References

D. L. Snyder, J. A. O’Sullivan, R. J. Murphy et al.

Deblurring subject to nonnegativity constraints when

known functions are present, with application to object-

constrained computerized tomography. IEEE Trans.

Med. Imaging, 20(10): 1009-1017, Oct. 2001.

May 19, 2004

Prior Knowledge Assumption:

Attenuation coefficients/geometry are known

Exact pose (position and orientation) is known

One rigid object (individual parts are fixed)

Application Examples

Hip prostheses,

Brachytherapy applicators,

Dental fillings,

Prostate seeds, etc.

Incorporating Prior Knowledge

May 19, 2004

Oracle Test for 3D Synthetic Data with High-density Objects

AM 800 Iteration, (8 OS) AM Oracle 800, (8 OS)

Oracle method:

At each iteration, use AM algorithm to solve for the image, then substitute in the known voxel values.

May 19, 2004

Define “true” pixel value: c(x) = acknown(x:θ)+ (1-a)cunknown(x);= ca(x :θ) + cb(x)

Rederive algorithm with constraint: c*(x) ≥ ca(x)

)(ˆ)(~

ln)(

1)(ˆ)()(

)()(

xbxb

xZxcxc

k

kkAM −=Solution:

[ ]):(),(max):(ˆ )1( θθ xcxcxc aAMk =+For some θ :

Search the optimal θ which results in the best image

AM with Pose Search

( )[ ]):(ˆ||minarg )1(

)2(θθ

θxcgdI k

SE

+

∈=

May 19, 2004

Mathematical Description of the Pose ( 2-dimensional Case)

Rotate & Translate

),,( 21 ϕθ tt

⎥⎦

⎤⎢⎣

⎡−

=ϕϕϕϕ

cossinsincos

: Rwhere

Tttt ),( 21=ϖ{ }

txRxRTtT

SOR

ϖϖ

ϖ

+⋅=⋅=

×

)}({~)2(2

⎥⎦

⎤⎢⎣

⎡=∈

10tR

Vϖ

θ ( )matrix⋅× 33

{ } { }

bBaxBAbaxAB

xATaTBTbT

++=++⋅=

⋅⋅⋅

ϖϖ )(

)(}{}{⎥⎦

⎤⎢⎣

⎡ +=⎥

⎦

⎤⎢⎣

⎡×⎥⎦

⎤⎢⎣

⎡101010

bBaBAaAbB⇔

Two consecutive operations:

May 19, 2004

Calculate the gradients of the pose parameters in SE(2) space( Search direction )

Gradient Search Method

,cossinsincos EeR ϕ

ϕϕϕϕ

=⎥⎦

⎤⎢⎣

⎡−

= ⎥⎦

⎤⎢⎣

⎡ −=

0110

Ewhere

htFhtF

xFf ii

hi

ti

)()(lim0

−+=

∂∂

=→

( ) ( ) ( ) ( )εε

ϕεϕ

ε

ε

ε

EEE

ReFeFRFFf −

=−

=+

→→

)(

00limRelim

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎦

⎤⎢⎣

⎡=∇

2

1

000000

00

)(

t

t

R

t

R

ff

f

ff

F θ

May 19, 2004

Select the step size for each parameter, written in matrix S

Gradient Search Method (Continued)

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=t

R

t

t

R

SS

SS

SS

00

000000

2

1

Get the change for pose

⎥⎦

⎤⎢⎣

⎡ −=∆

−

10

Ttt

EfS fSe RR

θ

Update the new poseoldnew θθθ ⋅∆=

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡ −

101010oldold

Ttt

EfSnewnew tRfSetR RR

May 19, 2004

Pose Search Results

May 19, 2004

AMPS Results

True Phantom FBP AM 200 (22 OS), SYN

AMPS 200 (22 OS), SYN AMPS 200 (22 OS), REAL

May 19, 2004

CT introduction

Alternating Minimization method

Pose search of high-density object

Incomplete projection data

Conclusion and future work

Content

May 19, 2004

Mask for high-density object

AM 200 (22 OS), Incomplete data, REAL

AMPS 200 (22 OS), REAL

May 19, 2004

Scanned object outside the scanner FOV

FBP AM 50 (22 OS)

May 19, 2004

Scanned object outside of FOV

FBP AM 50 (22 OS)

May 19, 2004

CT introduction

Alternating Minimization method

Pose search of high-density object

Incomplete projection data

Conclusion and future work

Content

May 19, 2004

Conclusion and future work

Avoid the interpolation step when reconstructing the image from

spiral CT data

Great improvement in convergence rate when the AM algorithm

include prior information

pose search in three-dimensional image reconstruction

More real data experiments for spiral CT

May 19, 2004

Thank You !