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

  • * Corresponding author: Jiansheng Yang

    * Corresponding author: Jiansheng Yang 
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  • 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.

    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).

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

    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).

    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 $[1, 2]$.

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

    Table 1.  Parameter settings of numerical and physical experiments.

    Forbild head Chicken, normal dose Chicken, low dose
    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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