restoration of missing data in limited angle tomography based … · 2017-01-30 · 7. bilateral...

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


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

4

Siemens Artis zee multi-purpose system


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


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


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


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,


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


● 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


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)


7. Regression problem

13

Curve fitting problem, 𝑛 = 100


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


9. Limit angle regression problem

● When n is even:

where 𝑁 is the number of projections

● When n is odd:

15

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 ) ..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( ) 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


9. Ill-posedness of limited angle tomography

● Regression problem[1,2]:

● Advantage: convenient to analyze its ill-posedness

16

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


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.


11. Artifacts caused by regression errors

● The regression artifacts appear as small streaks

18

Recondtrution of restored sinogram, 𝑓𝐻𝐿𝐶𝐶


Method: Fusion in Frequency Domain


1. Fourier transform of 𝑼𝒏(𝒔)

20

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


● 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)


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

23

,

, ,

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


4. Frequency domain representation

● More orders, more high frequency components, less ringing artifacts

24

Fourier representation and reconstructed images when the number of orders 𝑛𝑟 increases


5. Missing double wedge region

25

Limited angle reconstruction

Limited angle sinogram, 160º

measured

Frequency components of the reconstruction

FBP central slice theorem

2D FFT

2D IFFT


6. Fusion in frequency domain

● Use the restored frequency components to fill in the missing double

wedge region only

26

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


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

27

fused2 limited HLCC,BF( , ) ( , ) ( , ) ( , ) (1 ( , ))F w F w M w F w M w


III. Results and Discussion


a. Shepp-Logan phantom, noise-free


1. Estimation of moment curves

30

Estimated moment curve, e.g. n = 100, r = 0.86


1. Estimation of moment curves

● Linear correlation coefficients, 720 orders

31


2. Restored sinogram

32

Limited angle sinogram, 160º Restored sinogram, 360º


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


RMSE = 310 HU 189 HU 143 HU 178 HU 136 HU


b. Shepp-Logan phantom, Poisson noise


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


3. Preliminary experiment on clinical data


Reconstructed images

Top row window: [-1200, 2000] HU, bottom row window: [-200, 300] HU

38

RMSE = 310 HU 189 HU 143 HU 178 HU 136 HU


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


Thank you for your attention!

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restoration of missing data in limited angle tomography based … · 2017-01-30 · 7. bilateral...

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