Gradient-based Sparse Approximation for Computed Tomography Elham - - PowerPoint PPT Presentation

Gradient-based Sparse Approximation

for

Computed Tomography

Elham Sakhaee, Manuel Arreola and

and Alireza Entezari

University of Florida esakhaee@cise.ufl.edu

Tomographic Reconstruction

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§ 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

Images courtesy of Pan et.al [1]

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Sparse CT

§ Least-squares solution: § Regularize the solution: § R(u) can be sparsity promoting regularizer

A f p

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tomographic system matrix intensity image sinogram data

ˆ f = arg min

u∈RN

kAu pk2

2

ˆ f = arg min

u∈RN

kAu pk2

2 + λR(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]

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Gradient Domain Sparsity

§ TV-based reconstruction:

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ˆ f = arg min

u∈RN

kAu pk2

2 + λ(kDxuk1 + kDyuk1)

Seek a solution with sparse gradient magnitude

Gradient Components are Sparser

§ Gradient Magnitude (TV image): § Horizontal and vertical partial derivatives:

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Horizontal Derivative Vertical Derivative

Method: Recovering Partial Derivatives

§ Horizontal derivative: § Vertical derivative: § May result in a non-integrable vector field

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[ˆ fx,ˆ fy]T

ˆ fy = arg min

uy∈RN

kAuy pyk2

2 + λkuyk1

ˆ fx = arg min

ux∈RN

kAux pxk2

2 + λkuxk1

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) = Dxfy Dyfx = 0

Method: incorporating the curl constraint

§ Recover the gradient components simultaneously § Consider integrability constraint at recovery stage

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ˆ fx,ˆ fy = arg min

ux,uy∈RN

kAux pxk2

2 + kAuy pyk2 2+

λ(kuxk1 + kuyk1) + µkDxuy Dyuxk2

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Method: LASSO Formulation

§ Define: § Reformulate as minimization:

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`1

[ˆ fx,ˆ fy]T = arg min

v2R2N

kGv p0k2

2 + λkvk1

G =   A A µDy −µDx   , p0 =   px py  

Method: Final Image Reconstruction

§ Given the gradient vector field § Recover the final image by Poisson Equation [Perez

et al., 2003]:

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r2ˆ f = Dxˆ fx + Dyˆ fy

[ˆ fx,ˆ fy]T

§ Q: Given , find:

and

§ A: Projection-slice theorem:

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s = Sθ(F{f}) = F{Pθ⊥(f)}

Method: derivation of X-ray measurements

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py = Pθ⊥(fy) px = Pθ⊥(fx)

p = Pθ⊥(f)

p

x

= P

θ

⊥

( f

x

)

p = Pθ⊥(f)

py = Pθ⊥(fy)

§ Fourier transform properties: § From projection-slice theorem:

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ω F{fy} = jωyF{f}

F{fx} = jωxF{f}

Method: derivation of X-ray measurements

Sθ(F{fx})(ω) = Sθ(jωxF{f})(ω) = cos(θ)jωs(ω) Sθ(F{fy})(ω) = Sθ(jωyF{f})(ω) = sin(θ)jωs(ω)

ωx = cos(θ)ω

ωy = sin(θ)ω

θ

Method: derivation of X-ray measurements

§ Intuitively:

px = Pθ⊥(fx) = cos(θ)DPθ⊥(f) py = Pθ⊥(fy) = sin(θ)DPθ⊥(f)

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

TV minimization SNR: 26.45 dB Proposed SNR: 30.15 dB

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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

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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 45 20 25 30 35 projection angles SNR (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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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

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Ffy = (1 − e−2πiωy/N)Ff

Ffx = (1 − e−2πiωx/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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