Mumford-Shah-TV functional with application in X-ray interior tomography

Zhenhua Zhao; Yining Zhu; Jiansheng Yang; Ming Jiang

doi:10.3934/ipi.2018015

Article Contents

2018, Volume 12, Issue 2: 331-348. Doi: 10.3934/ipi.2018015

This issue Previous Article Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation Next Article The factorization method for cracks in elastic scattering

Mumford-Shah-TV functional with application in X-ray interior tomography

1.
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
2.
School of Mathematical Sciences, Capital Normal University and Beijing Higher Institution Engineering Research Center of Testing and Imaging, Beijing 100048, China
3.
Beijing Advanced Innovation Center for Imaging Technology, Capital Normal University, Beijing 100048, China
4.
Beijing International Center for Mathematical Research, Beijing 100871, China
5.
China Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Jiansheng Yang
* Corresponding author: Jiansheng Yang

Received: December 2016

Revised: October 2017

Published: April 2018

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Abstract

Both total variation (TV) and Mumford-Shah (MS) functional are broadly used for regularization of various ill-posed problems in the field of imaging and image processing. Incorporating MS functional with TV, we propose a new functional, named as Mumford-Shah-TV (MSTV), for the object image of piecewise constant. Both the image and its edge can be reconstructed by MSTV regularization method. We study the regularizing properties of MSTV functional and present an Ambrosio-Tortorelli type approximation for it in the sense of Γ-convergence. We apply MSTV regularization method to the interior problem of X-ray CT and develop an algorithm based on split Bregman and OS-SART iterations. Numerical and physical experiments demonstrate that high-quality image and its edge within the ROI can be reconstructed using the regularization method and algorithm we proposed.

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Mathematics Subject Classification: Primary: 92C55, 62P10; Secondary: 44A12.

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Figure 4. Reconstruction results of normal-dose projection data. (a)-(d): reconstructed images with display window of [0, 0.03]; (e), (f): edge images with display window [0.1, 0.9];(g)-(k): subfigures indicated by the rectangular in Fig. 3(a) and Fig. 4(a)-(d).

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Figure 3. Reconstructed images using non-truncated projection data. The display window is [0, 0.03]. The ROI is indicated by a circle.

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Figure 5. Reconstruction results of low-dose projection data. (a)-(d): reconstructed images with display window of $[0, 0.03]$; (e), (f): edge images with display window of $[0.3, 1.0]$; (g)-(k): sub-figures indicated by the rectangular in Fig. 3(b) and Fig. 5(a)-Fig. 5(d).

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Figure 1. Reconstructed results of Forbild head. (a): Forbild head phantom; (b)-(e): reconstructed images with display window of $[0, 2]$; (f), (g): edge images with display window of $[0, 0.8]$; (h): left to right, sub-figures indicated by the rectangular in (a)-(e) with display window of $[1, 2]$.

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Figure 2. Curves of $E_{\rm rec}(u^k)$ and $E_{\rm SSIM}(u^k)$ from the 4th iteration.

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Table 1. Parameter settings of numerical and physical experiments.

	Forbild head			Chicken, normal dose			Chicken, low dose
	MS	TV	MSTV	MS	TV	MSTV	MS	TV	MSTV
α	0.5	1e-2	1e-2	0.4	5e-4	5e-4	0.4	9e-4	9e-4
β	5e-3	*	1e-3	3e-6	*	2e-6	3e-4	*	4e-6
a	0	*	0	0	*	0	0	*	0
b	3	*	3	1	*	1	1	*	1
c	*	*	+∞	*	*	+∞	*	*	+∞

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