Post on 04-Aug-2020
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 !