Restoration of Missing Data in Limited Angle Tomography Based on Helgason- Ludwig Consistency Conditions Yixing Huang, Oliver Taubmann, Xiaolin Huang, Guenter Lauritsch, Andreas Maier 26.01. 2017 Pattern Recognition Lab (CS 5) Erlangen, Germany
Restoration of Missing Data in Limited
Angle Tomography Based on Helgason-
Ludwig Consistency Conditions
Yixing Huang, Oliver Taubmann, Xiaolin Huang,
Guenter Lauritsch, Andreas Maier
26.01. 2017
Pattern Recognition Lab (CS 5)
Erlangen, Germany
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
Outline
I. Motivation
II. Method
Regression in Sinogram Domain (Chebyshev Fourier transform)
Fusion in Frequency Domain (bilateral filtering in spatial domain)
III. Results and Discussion
Shepp-Logan phantom, noise-free and Poisson noise
Clinical data
IV. Conclusion
2
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
I. Motivation
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
1. Limited angle tomography
● Classical incomplete data reconstruction problem
● Gantry rotation is restricted by other system parts or external objects
● Parallel-beam: less than 180º;
Fan-beam and cone-beam: less than a short scan
4
Siemens Artis zee multi-purpose system
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
2. Parallel-beam limited angle tomography
5
Limited angle sinogram, 160º
Shepp-Logan phantom Limited angle reconstruction, FBP
Complete sinogram
measured
Method: Regression in Sinogram Domain
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
1. Chebyshev polynomials of the second kind
● Definition:
● 𝑈𝑛(𝑠) is a polynomial of degree 𝑛
with only even/odd monomials
7
2
sin(( 1)arccos( )) sin(( 1) )( ) , if cos( )
sin( )1n
n s n tU s s t
ts
Un(s) (Wolfram MathWorld)
0
1
2
2
3
3
( ) 1;
( ) 2 ;
( ) 4 1;
( ) 8 4 ;
...
U s
U s s
U s s
U s s s
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
2. Moment curves
● 𝑛th order moment curve of the parallel-beam sinogram 𝑝(𝑠, 𝜃)
8
1
1
( ) ( , ) ( )n na p s U s ds
complete sinogram ( )na
𝑈𝑛(𝑠)
-1 1 0 𝑠
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
3. Fourier transform of the moment curve
● Fourier transform:
● HLCC:
9
2
,
0
1( )
2
im
n m na e a d
Complete sinogram ( )na
m
n
-1 1 0 𝑠
,n ma
, 0 | | or | | is oddn ma m n n m
Checkerboard pattern,
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
4. Invertible transform
● Orthogonal set with weight 𝑊 𝑠 = (1 − 𝑠2)1/2:
● Sinogram restoration:
10
complete sinogram ( )na
𝑈𝑛(𝑠)
-1 1 0 𝑠
1
1
0,( ) ( ) ( )
/ 2,n m
n mW s U s U s ds
n m
0
2( , ) ( )( ( ) ( ))n np s a W s U s
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
● Restoration of sinogram using 0 – 𝑛𝑟 orders
11
5. Sinogram restoration, complete data
Restored sinograms when the number of orders 𝑛𝑟 increases
0
2( , ) ( )( ( ) ( ))
r
r
n
n n np s a W s U s
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
6. nth moment curve, limited angle data
● Chebyshev transform, 𝑛th moment curve:
12
1
1
( ) ( , ) ( )n na p s U s ds
Limited angle parallel-beam sinogram,
160º measured ( )na
𝑈𝑛(𝑠)
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
7. Regression problem
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Curve fitting problem, 𝑛 = 100
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
8. Analytical Form of 𝒂𝒏(𝜽)
● Based on HLCC, the analytical form of is known
● If 𝑛 is even:
● If 𝑛 is odd:
● In both cases, 𝑛 + 1 unknown coefficients
0 2 2 4 4( ) cos(2 ) sin(2 ) cos(4 ) sin(4 ) ... cos( ) sin( )n n na b b c b c b n c n
( )na
1 1 3 3( ) cos( ) sin( ) cos(3 ) sin(3 ) ... cos( ) sin( )n n na b c b c b n c n
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
9. Limit angle regression problem
● When n is even:
where 𝑁 is the number of projections
● When n is odd:
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0 0 0 0 0 0
1 1 1 1 1 1
2 2 2 2 2 2
1 1 1 1
1 cos(2 ) sin(2 ) cos(4 ) sin(4 ) ... cos( ) sin( )
1 cos(2 ) sin(2 ) cos(4 ) sin(4 ) ... cos( ) sin( )
1 cos(2 ) sin(2 ) cos(4 ) sin(4 ) ... cos( ) sin( )
1 cos(2 ) sin(2 ) cos(4 ) sin(4 ) ..N N N N
n n
n n
n n
0
2
0
2
1
4
2
4
1 1 1
( )
( )
( )
. cos( ) sin( ) ( )
n
n
n
N N n N
n
n
b
ba
ca
ba
c
n n ab
c
0 0 0 0 0 0
1 1 1 1 1 1
2 2 2 2 2 2
1 1 1 1 1
cos( ) sin( ) cos(3 ) sin(3 ) ... cos( ) sin( )
cos( ) sin( ) cos(3 ) sin(3 ) ... cos( ) sin( )
cos( ) sin( ) cos(3 ) sin(3 ) ... cos( ) sin( )
cos( ) sin( ) cos(3 ) sin(3 ) ... cos( ) sN N N N N
n n
n n
n n
n
1
01
3 1
3 2
1 1
( )
( )
( )
in( ) ( )
n
n
n
N n n N
n
b
ac
b a
c a
n b a
c
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
9. Ill-posedness of limited angle tomography
● Regression problem[1,2]:
● Advantage: convenient to analyze its ill-posedness
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X y
[1] Louis A K & Törnig W 1980 Picture reconstruction from projections in restricted range. Mathematical Methods in the Applied Sciences
[2] Louis A K & Natterer F1983 Mathematical problems of computerized tomography. Proceedings of the IEEE
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
10. Ill-posed regression problem
● Various existing algorithms to solve ill-posed regression problems
● Lasso regression:
solved by the iterative soft thresholding algorithm[3]
● Only low orders are estimated correctly, regression errors in higher
orders cause artifacts
17
212 12
arg min || || || || X y
[3] Ingrid Daubechies, Michel Defrise, and Christine De Mol 2004 An iterative thresholding algorithm for linear inverse problems with a
sparsity constraint. Communications on pure and applied mathematics.
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
11. Artifacts caused by regression errors
● The regression artifacts appear as small streaks
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Recondtrution of restored sinogram, 𝑓𝐻𝐿𝐶𝐶
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
Method: Fusion in Frequency Domain
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
1. Fourier transform of 𝑼𝒏(𝒔)
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where 𝐽′𝑛(𝑧) is the modified Bessel function:
1
2
21
cos
cos
cos cos
0 0
2
( ( ) ( ))( )
sin(( 1)arccos( ))1 d ( cos )
1
sin(( 1) )sin sin d
sin
sin sin(( 1) ) d
cos(( 2) ) d cos( ) d
( ' ( )
n
iws
iw t
iw t
iw t iw t
n
W s U s w
n ss e s s t
s
n tt e t t
t
t n t e t
n t e t nt e t
J iw J
F
' ( ))n iw
cos
0
1' ( ) cos( ) dz t
nJ z nt e t
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC 21
FFT of 𝑼𝟓𝟎(𝒔)
● A Bessel function rapidly tends to zero when the argument becomes
less than the order 𝑛
2. Fourier transform of 𝑼𝒏(𝒔) examples
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC 22
● A Bessel function rapidly tends to zero when the argument becomes
less than the order 𝑛
2. Fourier transform of 𝑼𝒏 𝒔 examples
FFT of 𝑼𝟐𝟎𝟎(𝒔)
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
3. High-pass filter
● 𝑈𝑛 𝑠 ∙ 𝑊(𝑠) can be regarded as a high-pass filter
● Sinogram restoration:
the missing orders higher than 𝑛𝑟 only contribute to the frequency range above 𝑤𝑐,𝑛𝑟
the frequency range [0, 𝑤𝑐,𝑛𝑟] should be complete
a circular area with radius 𝑤𝑐,𝑛𝑟 is restored
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,
, ,
0, 0 ,( ( ) ( ))( )
0,
c n
n
L n H n
w wF W s U s w
w w w
where , rad/sc nw n
0
2( , ) ( )( ( ) ( ))
rn n np s a W s U s
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
4. Frequency domain representation
● More orders, more high frequency components, less ringing artifacts
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Fourier representation and reconstructed images when the number of orders 𝑛𝑟 increases
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
5. Missing double wedge region
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Limited angle reconstruction
Limited angle sinogram, 160º
measured
Frequency components of the reconstruction
FBP central slice theorem
2D FFT
2D IFFT
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
6. Fusion in frequency domain
● Use the restored frequency components to fill in the missing double
wedge region only
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The black, blue, and green areas are the missing, measured, and HLCC estimated frequency
components, respectively, where the faded green area might be not correctly estimated.
fused limited HLCC( , ) ( , ) ( , ) ( , ) (1 ( , ))F w F w M w F w M w
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
7. Bilateral filtering in spatial domain before fusion
● High frequency components inside the double wedge region are still
missing or erroneously estimated
● A strong bilateral filter can reduce regression artifacts and partially
recover high frequency components associated with reliable high
contrast edges
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fused2 limited HLCC,BF( , ) ( , ) ( , ) ( , ) (1 ( , ))F w F w M w F w M w
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
III. Results and Discussion
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
a. Shepp-Logan phantom, noise-free
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
1. Estimation of moment curves
30
Estimated moment curve, e.g. n = 100, r = 0.86
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
1. Estimation of moment curves
● Linear correlation coefficients, 720 orders
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26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
2. Restored sinogram
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Limited angle sinogram, 160º Restored sinogram, 360º
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
3. Reconstructed images
In the final fused image, streaks are reduced, only minor streaks remain,
without reintroducing regression artifacts
33
Limited angle reconstruction,
𝑓𝑙𝑖𝑚𝑖𝑡𝑒𝑑
Recondtrution of restored sinogram,
𝑓𝐻𝐿𝐶𝐶
Fusion with bilateral filter
𝑓𝑓𝑢𝑠𝑒𝑑2
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC 34
RMSE = 310 HU 189 HU 143 HU 178 HU 136 HU
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
b. Shepp-Logan phantom, Poisson noise
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
Reconstructed images, Poisson noise
● Poisson noise is suppressed in 𝑓𝐻𝐿𝐶𝐶
● Streaks are reduced, although Poisson noise is back in 𝑓𝑓𝑢𝑠𝑒𝑑2
36
Limited angle reconstruction,
𝑓𝑙𝑖𝑚𝑖𝑡𝑒𝑑
Recondtrution of restored sinogram,
𝑓𝐻𝐿𝐶𝐶
Fusion with bilateral filter
𝑓𝑓𝑢𝑠𝑒𝑑2
RMSE = 361 HU 184 HU 210 HU
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
3. Preliminary experiment on clinical data
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
Reconstructed images
Top row window: [-1200, 2000] HU, bottom row window: [-200, 300] HU
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RMSE = 310 HU 189 HU 143 HU 178 HU 136 HU
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
IV. Conclusion
26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
Conclusion
● We propose a regression and fusion method to restore missing data in limited
angle tomography
● The regression formulation allows convenient ill-posedness analysis.
● Only low frequency components are correctly restored in the regression step
● Bilateral filtering is utilized to reduce artifacts caused by regression errors and
partially recover high frequency components
● We use the restored frequency components to fill in the missing double
wedge region only
● Streak artifacts are reduced and intensity offset is corrected without
reintroducing artifacts in the final fused image
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26.01.2017 | Yixing Huang | Pattern Recognition Lab (CS 5) | Missing Data Restoration Based on HLCC
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