Gradient-based Sparse Approximation for Computed Tomography

Elham Sakhaee, Manuel Arreola and Alireza Entezari

University of Florida

esakhaee@cise.ufl.edu

Tomographic Reconstruction

2

§  Recover the image given X-ray measurements

X-ray source

X-ray detector

Sinogram

Motivation

§  X-ray Exposure Reduction

§  ill-posed problem

Half-Detector

A x

b

Limited-Angle Few-View

Imag

es c

ourte

sy o

f Pan

et.a

l [1]

3

Sparse CT

§  Least-squares solution:

§  Regularize the solution:

§  R(u) can be sparsity promoting regularizer

A f

p

4

tomographic system matrix

intensity image

sinogram data

f̂ = argminu2RN

kAu� pk22

f̂ = argminu2RN

kAu� pk22 + �R(u)

Related Work (Sparsity)

§  X-let sparsity: -  Wavelet [Rantala 2006]

-  Curvelet [Hyder & Sukanesh, 2011]

§  Adaptive sparsity via dictionary learning

-  K-SVD [Liao & Sapiro 2008, Sakhaee & Entezari 2014]

§  Besov space priors: -  Bayesian inversion [Siltanen et al. 2012]

§  TV minimization: -  Very promising for biomedical images -  ASD-POCS [Pan & Sidky 2009]

5

Gradient Domain Sparsity

§  TV-based reconstruction:

6

f̂ = argminu2RN

kAu� pk22 + �(kDx

uk1 + kDy

uk1)

Seek a solution with sparse gradient magnitude

Gradient Components are Sparser

§  Gradient Magnitude (TV image):

§  Horizontal and vertical partial derivatives:

7 Horizontal Derivative Vertical Derivative

Method: Recovering Partial Derivatives

§  Horizontal derivative:

§  Vertical derivative:

§  May result in a non-integrable vector field

8

[f̂x

, f̂y

]T

f̂y = argminuy2RN

kAuy � pyk22 + �kuyk1

f̂x

= argminu

x

2RN

kAux

� px

k22 + �kux

k1

Method: Curl-free Constraint

§  For a vector field to be gradient field,

it must be curl-free (zero curl):

§  Adds a prior knowledge to the ill-posed problem

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curl(rf) = Dx

fy

�Dy

fx

= 0

Method: incorporating the curl constraint

§  Recover the gradient components simultaneously

§  Consider integrability constraint at recovery stage

10

f̂x

, f̂y

= argminu

x

,uy

2RN

kAux

� px

k22 + kAuy

� py

k22+

�(kux

k1 + kuy

k1) + µkDx

uy

�Dy

ux

k22

Method: LASSO Formulation

§  Define:

§  Reformulate as minimization:

11

`1

[f̂x

, f̂y

]T = argminv2R2N

kGv � p0k22 + �kvk1

G =

2

4A 00 A

µDy

�µDx

3

5 , p0 =

2

4px

py

0

3

5

Method: Final Image Reconstruction

§  Given the gradient vector field

§  Recover the final image by Poisson Equation [Perez

et al., 2003]:

12

r2f̂ = Dx

f̂x

+Dy

f̂y

[f̂x

, f̂y

]T

§  Q: Given , find:

and

§  A: Projection-slice theorem:

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s = S✓(F{f}) = F{P✓?(f)}

Method: derivation of X-ray measurements

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py = P✓?(fy)px

= P✓

?(fx

)

p = P✓?(f)

px

=P ✓

?(fx)

p =P ✓

?(f)

py=P ✓

?(fy

)

§  Fourier transform properties:

§  From projection-slice theorem:

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!F{fy} = j!yF{f}F{f

x

} = j!x

F{f}

Method: derivation of X-ray measurements

S✓

(F{fx

})(!) = S✓

(j!x

F{f})(!) = cos(✓)j!s(!)

S✓(F{fy})(!) = S✓(j!yF{f})(!) = sin(✓)j!s(!)

!x

= cos(✓)!

!y=

sin(✓)!

✓

Method: derivation of X-ray measurements

§  Intuitively:

px

= P✓

?(fx

) = cos(✓)DP✓

?(f)

py = P✓?(fy) = sin(✓)DP✓?(f)

Results: 15 projection views (4% of full range)

TV minimization SNR: 26.45 dB

Proposed SNR: 30.15 dB 16

FBP, SNR: 13.03 dB Ground Truth

Results: 15 projection views (4% of full range)

TV minimization SNR: 23.08 dB

Proposed SNR: 23.59 dB 17

FBP, SNR: 2.61 dB Ground Truth

Results: 15 projection views (4% of full range)

TV minimization Proposed

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Results: 10 projection views (2.7% of full range)

TV minimization SNR: 23.50 dB

Proposed SNR: 27.02 dB

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Separate Recovery SNR: 23.42 dB

8 12 15 20 24 30 36 4520

25

30

35

projection angles

SN

R (

db

)

SGF(proposed)

Separate Recovery

TV minimization

Results: Accuracy Comparison

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§  Accuracy vs. number of projection angles for Catphan dataset:

Results: Noisy Data (15 projection angles)

TV minimization SNR: 14.15 dB

Proposed SNR: 23.58 dB

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Separate Recovery SNR: 16.78 dB

Summary

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§  We propose: -  Leveraging higher sparsity of individual gradient

components

-  Enforcing curl-free constraint at recovery stage

-  Leveraging interdependency of partial derivatives

§  Provided a recipe for deriving of X-ray measurements corresponding to derivative images

Future Work

§  Application to 3D CT reconstruction

§  Robustness against other types of noise

§  Analytical derivation of X-ray measurements corresponding to derivative images using box-splines

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References §  Rantala, M., Vanska, S., Jarvenpaa, S., Kalke, M., Lassas, M., Moberg, J., & Siltanen, S.

(2006). Wavelet-based reconstruction for limited-angle X-ray tomography. Medical Imaging, IEEE Transactions on, 25(2), 210-217.

§  Hyder, S. Ali, and R. Sukanesh. "An efficient algorithm for denoising MR and CT images using digital curvelet transform." Software Tools and Algorithms for Biological Systems. Springer New York, 2011. 471-480.

§  Liao, H., Sapiro, G.: Sparse representations for limited data tomography. In Biomedical Imaging: From Nano to Macro, 2008. ISBI 2008. 5th IEEE International Symposium on. (2008) 1375–1378

§  Sakhaee, E., Entezari, A.: Learning splines for sparse tomographic reconstruction. Advances in visual computing, (Proc. of ISVC) Springer Lecture Notes, pp1-14, 2014.

§  Muller, J.L. and Siltanen, S., Linear and Nonlinear Inverse Problems with Practical Applications. Society for Industrial and Applied Mathematics, USA, 2012.

§  Pan, X., Sidky, E.Y., Vannier, M.: Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction? Inverse Problems 25 (2009)

§  Patel, V.M., Maleh, R., Gilbert, A. C., and Chellappa R., Gradient-based image recovery methods from incom- plete fourier measurements. Image Processing, IEEE Transactions on, vol. 21, no. 1, pp. 94–105, 2012.

§  Perez, P., Gangnet, M., and Blake, A., Poisson Image Editing, ACM transactions on Graphics, Vol 22, no. 3, pp313-318, 2003.

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Acknowledgements

§  This research was supported in part by the ONR grant N00014-14-1- 0762 and the NSF grant CCF/CIF-1018149.

§  We thank the imaging physicists at Shands hospital for providing the catphan phantom scan.

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26

Thank you …

Questions?

Related Work (MRI)

§  Derive partial Fourier measurements corresponding to [Patel et al. 2012]:

-  Horizontal partial derivative:

-  Vertical partial derivative:

§  Recover each component separately

§  Fit an integrable field to the recovered non-integrable field.

§  Reconstruct the image 27

Ffy = (1� e�2⇡i!y/N )Ff

Ffx

= (1� e�2⇡i!x

/N )Ff

Results: Noisy Data (27 projection angles)

TV minimization SNR: 13.66 dB

Proposed SNR: 15.85 dB

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Separate Recovery SNR: 14.17 dB

Objective

§  Leverage higher sparsity of partial derivatives, to reduce required measurements.

§  Given sinogram data, recover the gradient components simultaneously.

§  Enforce integrability constraint at recovery stage, as opposed to post-processing.

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