Spline-based Sparse Tomographic Reconstruction with Besov Priors - - PowerPoint PPT Presentation

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Spline-based Sparse Tomographic Reconstruction with Besov Priors Elham Sakhaee and and Alireza Entezari University of Florida esakhaee@cise.ufl.edu Tomographic Reconstruction Recover the image given X-ray measurements X-ray detector

SLIDE 1

Spline-based Sparse Tomographic Reconstruction with Besov Priors

Elham Sakhaee and

and Alireza Entezari

University of Florida esakhaee@cise.ufl.edu

SLIDE 2

Tomographic Reconstruction

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§ Recover the image given X-ray measurements

X-ray source X-ray detector Sinogram

SLIDE 3

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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SLIDE 4

Sparse CT

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

A x b

ˆ x = min

x

||Ax − b||2

2

ˆ x = min

x

||Ax − b||2

2 + λR(x)

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tomographic system matrix intensity image sinogram data

SLIDE 5

Related Work (Sparsity)

§ TV minimization:

Very promising for piece-wise constant images
ASD-POCS [Pan & Sidky 2009]

§ 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]

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SLIDE 6

Bayesian Inversion

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§ Find posterior distribution P(x|b), given:

Likelihood distribution
Prior distribution
Bayes formula:
Zero-mean Gaussian noise:
x is maximum a posteriori estimate:

A x b

P(x|b) ∝P(b|x) P(x)

P(b|x) = P✏(Ax − b) ∼ exp(− 1 2σ2 ||Ax − b||2

xMAP = argmax

x

P(x|b)

SLIDE 7

Discretization Invariance [Siltanen, 2012]

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Image courtesy of www.siltanen- research.net/MarseilleDiscInv.pdf

SLIDE 8

Discretization Invariance [Siltanen, 2012]

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Image courtesy of www.siltanen- research.net/MarseilleDiscInv.pdf

SLIDE 9

Discretization Invariance [Siltanen, 2012]

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Image courtesy of www.siltanen- research.net/MarseilleDiscInv.pdf

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Discretization Invariance [Siltanen, 2012]

§ Posterior estimate must converge as n & k tend to

infinity

§ Otherwise:

Adding number of measurements is not helpful
A higher resolution image does not converge to the

true image

§ TV is not discretization invariant

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SLIDE 11

Besov Space Priors

§ Bayesian inversion using Besov priors is

discretization invariant [Lassas et. al. 2008]

§ Besov space Bs

p,q : the space of functions with

certain level of smoothness

§ B1

1,1 : space of functions with (up to) first (weak)

derivatives in L1

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SLIDE 12

Common Pixel Representation

vs. Continuous object Finite grid reconstruction

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Image courtesy of C.G. Koay, https://science.nichd.nih.gov

SLIDE 13

Expansion Sets

§ Alternatives for pixel-basis

Blob functions [Lewitt 1990]
Kaiser-Bessel functions
Higher-order box-splines
Tensor-product linear B-spline
Tensor-product cubic B-spline
Zwart-Powell function

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f(x) =

N

X

n=1

cnϕ(x − xn)

SLIDE 14

Spline Formulation

§ Besov Prior: § Norm in B1

1,1 : equivalent to weighted Haar wavelet

coefficients

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||c||B1

1,1 = |w0| +

N−1

X

i=0 2i−1

X

j=0

2i/2|wi,j|

P(c) = Cexp(−λ||c||B1

1,1)

SLIDE 15

Spline Formulation

§ Posterior distribution of spline coefficients: § MAP estimate for higher order splines:

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P(c|b) = Cexp(−1 2||Hc − b||2

2 − λ||c||B1

1,1)

Spline System Matrix Spline Coefficients

cMAP = arg min

c

1 2||Hc − p||2

2 + λ||c||B1

1,1

SLIDE 16

Results: pixel-basis vs. Cubic box-spline

Pixel-basis (first-order box-spline) SNR: 14.64 dB Cubic (fourth-order box-spline) SNR: 16.67 dB

16 FBP SNR: 11.92 dB

§ 45 projection views (12.5% of full range):

SLIDE 17

Results: The impact of Besov Prior

Without Besov Prior SNR: 17.85 dB With Besov Prior SNR: 19.07 dB

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§ 60 projection views (16.6% of full range):

SLIDE 18

Results: TV minimization vs. Besov priors

TV minimization SNR: 15.50 dB Cubic box-spline w/ Besov prior SNR: 19.94 dB

18 Ground Truth

§ 120 projection views (33.3% of full range):

SLIDE 19

Results: Resilience to reduction of views

19 45 views SNR: 14.67 dB 60 views SNR: 14.67 dB 90 views SNR: 14.67 dB

SLIDE 20

Results: Accuracy Comparison

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§ Accuracy of tensor-product Box, Linear and Cubic &

non-separable Zwart-Powel

30 45 60 90 12 14 16 18 20 22

number of projection angles SNR (dB)

pixel−basis Linear Zwart−Powell Cubic

SLIDE 21

Summary

§ Recovering sparse spline coefficients with Besov

space priors:

§ Advantages:

Higher approximation order
Edge-preserving prior
Discretization invariance

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SLIDE 22

Future Work

§ Mixed spline representations § Analysis of approximation error as a function of

grid resolution

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References

§

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)

§

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

§

Muller, J.L. and Siltanen, S., Linear and Nonlinear Inverse Problems with Practical

Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA,

2012.

§

Kolehmainen, V., Lassas, M., Niinimaki, K., Siltanen, S.: Sparsity-promoting bayesian

inversion. Inverse Problems 28 (2012)

§

Saksman M.L., Siltanen S., Discretization-invariant bayesian inversion and Besov space priors, arXiv preprint (2009)

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