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
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
I. Motivation
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 Siemens Artis zee multi-purpose system 4
2. Parallel-beam limited angle tomography measured Complete sinogram Limited angle sinogram, 160º Shepp-Logan phantom Limited angle reconstruction, FBP 5
Method: Regression in Sinogram Domain
1. Chebyshev polynomials of the second kind Definition: sin(( n 1)arccos( s)) sin(( n 1) t) Un( s), if s cos( t) 1 sin( t) 2 s U n (s) is a polynomial of degree n with only even/odd monomials U ( s) 1; 0 U ( s) 2 s;... 1 U s s 2 2( ) 4 1; U s s s 3 3( ) 8 4 ; U n (s) (Wolfram MathWorld) 7
2. Moment curves n th order moment curve of the parallel-beam sinogram p(s, θ) 1 an( ) p( s, ) Un( s) ds 1 U n (s) -1 0 1 s complete sinogram an( ) 8
3. Fourier transform of the moment curve Fourier transform: 2 1 im an, m e an( ) d 2 0 HLCC: anm, 0 m n or n m is odd -1 0 1 s m n Complete sinogram an( ) Checkerboard pattern, a nm, 9
4. Invertible transform Orthogonal set with weight W s = (1 s 2 ) 1/2 : 1 1 W ( s) U ( s) U ( s) ds n m 0, / 2, n m n m Sinogram restoration: 2 p( s, ) an( )( W ( s) Un( s)) U n (s) -1 0 1 s 0 complete sinogram a ( ) n 10
5. Sinogram restoration, complete data Restoration of sinogram using 0 n r orders nr 2 p ( s, ) a ( )( W ( s) U ( s)) nr n n 0 Restored sinograms when the number of orders n r increases 11
6. n th moment curve, limited angle data Chebyshev transform, n th moment curve: 1 an( ) p( s, ) Un( s) ds 1 U n (s) Limited angle parallel-beam sinogram, 160º measured a ( ) n 12
7. Regression problem Curve fitting problem, n = 100 13
8. Analytical Form of a n (θ) Based on HLCC, the analytical form of a ( ) n is known If n is even: a ( ) b b cos(2 ) c sin(2 ) b cos(4 ) c sin(4 )... b cos( n ) c sin( n ) n 0 2 2 4 4 n n If n is odd: a ( ) b cos( ) c sin( ) b cos(3 ) c sin(3 )... b cos( n ) c sin( n ) n 1 1 3 3 n n In both cases, n + 1 unknown coefficients
9. Limit angle regression problem When n is even: 1 cos(2 ) sin(2 ) cos(4 ) sin(4 )... cos( n ) sin( n ) 2 0 0 0 0 0 0 n 0 1 cos(2 1) sin(2 1) cos(4 1) sin(4 1)... cos( n 1) sin( n 1) c2 an( 1) b 4 1 cos(2 2) sin(2 2) cos(4 2) sin(4 2)... cos( n 2) sin( n 2) an( 2) c 4 1 cos(2 ) sin(2 ) cos(4 ) sin(4 )... cos( n ) sin( n ) a ( ) N 1 N 1 N 1 N 1 N1 N1 n N1 b n where N is the number of projections b0 b a ( ) c n When n is odd: b1 cos( 0) sin( 0) cos(3 0) sin(3 0)... cos( n 0) sin( n 0) c 1 an( 0) cos( 1) sin( 1) cos(3 1) sin(3 1)... cos( n 1) sin( n 1) b3 an( 1) cos( 2) sin( 2) cos(3 2) sin(3 2)... cos( n 2) sin( n 2) c3 an( 2) cos( N 1) sin( N 1) cos(3 N 1) sin(3 N 1)... cos( n N 1) sin( n N 1) bn an( N 1) c n 15
9. Ill-posedness of limited angle tomography Regression problem [1,2] : X y Advantage: convenient to analyze its ill-posedness [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 16
10. Ill-posed regression problem Various existing algorithms to solve ill-posed regression problems Lasso regression: 1 2 X y arg min 2 2 1 solved by the iterative soft thresholding algorithm [3] Only low orders are estimated correctly, regression errors in higher orders cause artifacts [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. 17
11. Artifacts caused by regression errors The regression artifacts appear as small streaks Recondtrution of restored sinogram, f HLCC 18
Method: Fusion in Frequency Domain
1. Fourier transform of U n (s) F( W ( s) U ( s))( w) 1 sin(( n1)arccos( s)) 1 s 1 2 iws 1 s e d s ( s cos t) 2 sin(( n1) t) sin sin t iwcost t e sin tdt iwcost iwcost cos(( 2) ) d cos( ) d 0 0 ( J ' ( iw) J n2 n iwcost sin t sin(( n 1) t) e dt n t e t nt e t ' ( iw n )) where J n (z) is the modified Bessel function: 1 zcost ' n( ) cos( ) d 0 J z nt e t 20
2. Fourier transform of U n (s) examples A Bessel function rapidly tends to zero when the argument becomes less than the order n FFT of U 50 (s) 21
2. Fourier transform of U n s examples A Bessel function rapidly tends to zero when the argument becomes less than the order n FFT of U 200 (s) 22
3. High-pass filter U n s W(s) can be regarded as a high-pass filter 0, 0 w wcn,, F( W ( s) Un( s))( w) 0, w w w L, n H, n where w, n rad/s cn Sinogram restoration: 2 p ( s, ) a ( )( W ( s) U ( s)) nr n n 0 the missing orders higher than n r only contribute to the frequency range above w c,nr the frequency range [0, w c,nr ] should be complete a circular area with radius w c,nr is restored 23
4. Frequency domain representation More orders, more high frequency components, less ringing artifacts Fourier representation and reconstructed images when the number of orders n r increases 24
5. Missing double wedge region measured FBP Limited angle sinogram, 160º central slice theorem 2D FFT 2D IFFT Limited angle reconstruction Frequency components of the reconstruction 25
6. Fusion in frequency domain F ( w, ) F ( w, ) M ( w, ) F ( w, ) (1 M ( w, )) fused limited HLCC Use the restored frequency components to fill in the missing double wedge region only 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. 26
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 F ( w, ) F ( w, ) M ( w, ) F ( w, ) (1 M ( w, )) fused2 limited HLCC,BF 27
III. Results and Discussion
a. Shepp-Logan phantom, noise-free
1. Estimation of moment curves Estimated moment curve, e.g. n = 100, r = 0.86 30
1. Estimation of moment curves Linear correlation coefficients, 720 orders 31
2. Restored sinogram Limited angle sinogram, 160º Restored sinogram, 360º 32
3. Reconstructed images In the final fused image, streaks are reduced, only minor streaks remain, without reintroducing regression artifacts Limited angle reconstruction, f limited Recondtrution of restored sinogram, f HLCC Fusion with bilateral filter f fused2 33
RMSE = 310 HU 189 HU 143 HU 178 HU 136 HU 34
b. Shepp-Logan phantom, Poisson noise
Reconstructed images, Poisson noise Poisson noise is suppressed in f HLCC Streaks are reduced, although Poisson noise is back in f fused2 RMSE = 361 HU 184 HU 210 HU Limited angle reconstruction, f limited Recondtrution of restored sinogram, f HLCC Fusion with bilateral filter f fused2 36
3. Preliminary experiment on clinical data
Reconstructed images RMSE = 310 HU 189 HU 143 HU 178 HU 136 HU Top row window: [-1200, 2000] HU, bottom row window: [-200, 300] HU 38
IV. Conclusion
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 40
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