# American Institute of Mathematical Sciences

April  2018, 12(2): 331-348. doi: 10.3934/ipi.2018015

## 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

Received  December 2016 Revised  October 2017 Published  February 2018

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.

Citation: Zhenhua Zhao, Yining Zhu, Jiansheng Yang, Ming Jiang. Mumford-Shah-TV functional with application in X-ray interior tomography. Inverse Problems & Imaging, 2018, 12 (2) : 331-348. doi: 10.3934/ipi.2018015
##### References:
 [1] L. Ambrosio, Variational problems in SBV and image segmentation, Acta Appl. Math., 17 (1989), 1-40.  doi: 10.1007/BF00052492.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000.  Google Scholar [3] L. Ambrosio and V. M. Tortorelli, Approximation of functional depending on jumps by elliptic functional via Γ-convergence, Commun. Pur. Appl. Math., 43 (1990), 999-1036.  doi: 10.1002/cpa.3160430805.  Google Scholar [4] L. Bar, N. Sochen and N. Kiryati, Semi-blind image restoration via Mumford-Shah regularization, IEEE Trans. Image Process., 15 (2006), 483-493.  doi: 10.1109/TIP.2005.863120.  Google Scholar [5] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, 1982.  Google Scholar [6] D. P. Bertsekas, A. Nedi and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003.  Google Scholar [7] A. Blake and A. Zisserman, Visual Reconstruction, MIT press Cambridge, 1987.  Google Scholar [8] Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Trans. Pattern Anal. Mach. Intell., 23 (2001), 1222-1239.  doi: 10.1109/ICCV.1999.791245.  Google Scholar [9] A. Braides, Gamma-convergence for Beginners, Oxford University Press, 2002.  Google Scholar [10] A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. Appl. Math., 55 (1995), 827-863.  doi: 10.1137/S0036139993257132.  Google Scholar [11] T. F. Chan and L. Vese, Active contours without edges, IEEE Trans. Image Process., 10 (2001), 266-277.  doi: 10.1109/83.902291.  Google Scholar [12] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4.  Google Scholar [13] E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal., 108 (1989), 195-218.  doi: 10.1007/BF01052971.  Google Scholar [14] S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl. Math., 13 (2002), 353-370.   Google Scholar [15] A. Faridani, E. L. Ritman and K. T. Smith, Local tomography, SIAM J. Appl. Math., 52 (1992), 459-484.  doi: 10.1137/0152026.  Google Scholar [16] M. Fornasier and R. Ward, Iterative thresholding meets free-discontinuity problems, Found. Comput. Math., 10 (2010), 527-567.  doi: 10.1007/s10208-010-9071-3.  Google Scholar [17] T. Goldstein and S. Osher, The split bregman method for l1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar [18] C. Hamaker, K. Smith, D. Solmon and S. Wagner, The divergent beam X-ray transform, Rocky Mt. J. Math., 10 (1980), 253-283.  doi: 10.1216/RMJ-1980-10-1-253.  Google Scholar [19] K. Hohm, M. Storath and A. Weinmann, An algorithmic framework for Mumford-Shah regularization of inverse problems in imaging, Inverse Problems, 31 (2015), 115011-30pp.   Google Scholar [20] M. Jiang, P. Maass and T. Page, Regularizing properties of the Mumford-Shah functional for imaging applications, Inverse Problems, 30 (2014), 035007-17pp.   Google Scholar [21] Y. Kee and J. Kim, A convex relaxation of the Ambrosio-Tortorelli elliptic functionals for the Mumford-Shah functional, in CVPR, (2014), 4074-4081.  doi: 10.1109/CVPR.2014.519.  Google Scholar [22] E. Klann, A Mumford-Shah-like method for limited data tomography with an application to electron tomography, SIAM J. Imaging Sci., 4 (2011), 1029-1048.  doi: 10.1137/100817371.  Google Scholar [23] E. Klann and R. Ramlau, Regularization properties of Mumford-Shah-type functionals with perimeter and norm constraints for linear ill-posed problems, SIAM J. Imaging Sci., 6 (2013), 413-436.  doi: 10.1137/110858422.  Google Scholar [24] H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Phys. Med. Biol., 53 (2008), 2207-2231.   Google Scholar [25] A. K. Louis and A. Rieder, Incomplete data problems in X-ray computerized tomography, Numer. Math., 56 (1989), 371-383.  doi: 10.1007/BF01396611.  Google Scholar [26] P. Maass, The interior Radon transform, SIAM J. Appl. Math., 52 (1992), 710-724.  doi: 10.1137/0152040.  Google Scholar [27] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pur. Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar [28] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001.  Google Scholar [29] Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, vol. 87, Springer, 2004.  Google Scholar [30] T. Page, Simultaneous reconstruction and segmentation with the Mumford-Shah functional for X-ray tomography, master's thesis, Diplomarbeit University of Bremen, 2011. Google Scholar [31] T. Pock, A. Chambolle, D. Cremers and H. Bischof, A convex relaxation approach for computing minimal partitions, in CVPR, (2009), 810-817.  doi: 10.1109/CVPR.2009.5206604.  Google Scholar [32] E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $\mathbb{R}^2$ and $\mathbb{R}^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.  doi: 10.1137/0524069.  Google Scholar [33] R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys., 221 (2007), 539-557.  doi: 10.1016/j.jcp.2006.06.041.  Google Scholar [34] L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM: Control, Optimisation and Calculus of Variations, 6 (2001), 517-538.  doi: 10.1051/cocv:2001121.  Google Scholar [35] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D., 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [36] J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in CVPR, (1996), 136-142.  doi: 10.1109/CVPR.1996.517065.  Google Scholar [37] E. Y. Sidky, J. H. Jørgensen and X. Pan, Convex optimization problem prototyping with the Chambolle-Pock algorithm for image reconstruction in computed tomography Physics in Medicine & Biology, 57 (2012), arXiv: 1111.5632. doi: 10.1088/0031-9155/57/10/3065.  Google Scholar [38] C. R. Vogel, A multigrid method for total variation-based image denoising, in Computation and control Ⅳ, Springer, 20 (1995), 323-331.   Google Scholar [39] G. Wang and M. Jiang, Ordered-subset simultaneous algebraic reconstruction techniques (OS-SART), Journal of X-ray Science and Technology, 12 (2003), 957-961.  doi: 10.1109/TIP.2003.815295.  Google Scholar [40] J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 035013, 29pp.  Google Scholar [41] Y. Ye, H. Yu, Y. Wei and G. Wang, A general local reconstruction approach based on a truncated Hilbert transform Int. J. Biomed. Imaging, 2007 (2007), Article ID 63634, 8 pages. doi: 10.1155/2007/63634.  Google Scholar [42] H. Yu and G. Wang, Compressed sensing based interior tomography, Phys. Med. Biol., 54 (2009), 2791-2805.   Google Scholar [43] H. Yu, J. Yang, M. Jiang and G. Wang, Supplemental analysis on compressed sensing based interior tomography Phys. Med. Biol. , 54 (2009), N425. doi: 10.1088/0031-9155/54/18/N04.  Google Scholar [44] H. Yu, Y. Ye, S. Zhao and G. Wang, Local ROI reconstruction via generalized FBP and BPF algorithms along more flexible curves Int. J. Biomed. Imaging, 2006 (2006), Article ID 14989, 7 pages. doi: 10.1155/IJBI/2006/14989.  Google Scholar [45] Z. Zhao, J. Yang and M. Jiang, A fast algorithm for high order total variation minimization based interior tomography, J. X-ray Sci. Technol., 23 (2015), 349-364.  doi: 10.3233/XST-150494.  Google Scholar [46] Y. Zhu, M. Zhao and Y. Zhao, Noise reduction with low dose CT data based on a modified ROF model, Optics express, 20 (2012), 17987-18004.  doi: 10.1364/OE.20.017987.  Google Scholar

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##### References:
 [1] L. Ambrosio, Variational problems in SBV and image segmentation, Acta Appl. Math., 17 (1989), 1-40.  doi: 10.1007/BF00052492.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000.  Google Scholar [3] L. Ambrosio and V. M. Tortorelli, Approximation of functional depending on jumps by elliptic functional via Γ-convergence, Commun. Pur. Appl. Math., 43 (1990), 999-1036.  doi: 10.1002/cpa.3160430805.  Google Scholar [4] L. Bar, N. Sochen and N. Kiryati, Semi-blind image restoration via Mumford-Shah regularization, IEEE Trans. Image Process., 15 (2006), 483-493.  doi: 10.1109/TIP.2005.863120.  Google Scholar [5] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, 1982.  Google Scholar [6] D. P. Bertsekas, A. Nedi and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003.  Google Scholar [7] A. Blake and A. Zisserman, Visual Reconstruction, MIT press Cambridge, 1987.  Google Scholar [8] Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Trans. Pattern Anal. Mach. Intell., 23 (2001), 1222-1239.  doi: 10.1109/ICCV.1999.791245.  Google Scholar [9] A. Braides, Gamma-convergence for Beginners, Oxford University Press, 2002.  Google Scholar [10] A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. Appl. Math., 55 (1995), 827-863.  doi: 10.1137/S0036139993257132.  Google Scholar [11] T. F. Chan and L. Vese, Active contours without edges, IEEE Trans. Image Process., 10 (2001), 266-277.  doi: 10.1109/83.902291.  Google Scholar [12] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4.  Google Scholar [13] E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal., 108 (1989), 195-218.  doi: 10.1007/BF01052971.  Google Scholar [14] S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl. Math., 13 (2002), 353-370.   Google Scholar [15] A. Faridani, E. L. Ritman and K. T. Smith, Local tomography, SIAM J. Appl. Math., 52 (1992), 459-484.  doi: 10.1137/0152026.  Google Scholar [16] M. Fornasier and R. Ward, Iterative thresholding meets free-discontinuity problems, Found. Comput. Math., 10 (2010), 527-567.  doi: 10.1007/s10208-010-9071-3.  Google Scholar [17] T. Goldstein and S. Osher, The split bregman method for l1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar [18] C. Hamaker, K. Smith, D. Solmon and S. Wagner, The divergent beam X-ray transform, Rocky Mt. J. Math., 10 (1980), 253-283.  doi: 10.1216/RMJ-1980-10-1-253.  Google Scholar [19] K. Hohm, M. Storath and A. Weinmann, An algorithmic framework for Mumford-Shah regularization of inverse problems in imaging, Inverse Problems, 31 (2015), 115011-30pp.   Google Scholar [20] M. Jiang, P. Maass and T. Page, Regularizing properties of the Mumford-Shah functional for imaging applications, Inverse Problems, 30 (2014), 035007-17pp.   Google Scholar [21] Y. Kee and J. Kim, A convex relaxation of the Ambrosio-Tortorelli elliptic functionals for the Mumford-Shah functional, in CVPR, (2014), 4074-4081.  doi: 10.1109/CVPR.2014.519.  Google Scholar [22] E. Klann, A Mumford-Shah-like method for limited data tomography with an application to electron tomography, SIAM J. Imaging Sci., 4 (2011), 1029-1048.  doi: 10.1137/100817371.  Google Scholar [23] E. Klann and R. Ramlau, Regularization properties of Mumford-Shah-type functionals with perimeter and norm constraints for linear ill-posed problems, SIAM J. Imaging Sci., 6 (2013), 413-436.  doi: 10.1137/110858422.  Google Scholar [24] H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Phys. Med. Biol., 53 (2008), 2207-2231.   Google Scholar [25] A. K. Louis and A. Rieder, Incomplete data problems in X-ray computerized tomography, Numer. Math., 56 (1989), 371-383.  doi: 10.1007/BF01396611.  Google Scholar [26] P. Maass, The interior Radon transform, SIAM J. Appl. Math., 52 (1992), 710-724.  doi: 10.1137/0152040.  Google Scholar [27] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pur. Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar [28] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001.  Google Scholar [29] Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, vol. 87, Springer, 2004.  Google Scholar [30] T. Page, Simultaneous reconstruction and segmentation with the Mumford-Shah functional for X-ray tomography, master's thesis, Diplomarbeit University of Bremen, 2011. Google Scholar [31] T. Pock, A. Chambolle, D. Cremers and H. Bischof, A convex relaxation approach for computing minimal partitions, in CVPR, (2009), 810-817.  doi: 10.1109/CVPR.2009.5206604.  Google Scholar [32] E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $\mathbb{R}^2$ and $\mathbb{R}^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.  doi: 10.1137/0524069.  Google Scholar [33] R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys., 221 (2007), 539-557.  doi: 10.1016/j.jcp.2006.06.041.  Google Scholar [34] L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM: Control, Optimisation and Calculus of Variations, 6 (2001), 517-538.  doi: 10.1051/cocv:2001121.  Google Scholar [35] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D., 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [36] J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in CVPR, (1996), 136-142.  doi: 10.1109/CVPR.1996.517065.  Google Scholar [37] E. Y. Sidky, J. H. Jørgensen and X. Pan, Convex optimization problem prototyping with the Chambolle-Pock algorithm for image reconstruction in computed tomography Physics in Medicine & Biology, 57 (2012), arXiv: 1111.5632. doi: 10.1088/0031-9155/57/10/3065.  Google Scholar [38] C. R. Vogel, A multigrid method for total variation-based image denoising, in Computation and control Ⅳ, Springer, 20 (1995), 323-331.   Google Scholar [39] G. Wang and M. Jiang, Ordered-subset simultaneous algebraic reconstruction techniques (OS-SART), Journal of X-ray Science and Technology, 12 (2003), 957-961.  doi: 10.1109/TIP.2003.815295.  Google Scholar [40] J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 035013, 29pp.  Google Scholar [41] Y. Ye, H. Yu, Y. Wei and G. Wang, A general local reconstruction approach based on a truncated Hilbert transform Int. J. Biomed. Imaging, 2007 (2007), Article ID 63634, 8 pages. doi: 10.1155/2007/63634.  Google Scholar [42] H. Yu and G. Wang, Compressed sensing based interior tomography, Phys. Med. Biol., 54 (2009), 2791-2805.   Google Scholar [43] H. Yu, J. Yang, M. Jiang and G. Wang, Supplemental analysis on compressed sensing based interior tomography Phys. Med. Biol. , 54 (2009), N425. doi: 10.1088/0031-9155/54/18/N04.  Google Scholar [44] H. Yu, Y. Ye, S. Zhao and G. Wang, Local ROI reconstruction via generalized FBP and BPF algorithms along more flexible curves Int. J. Biomed. Imaging, 2006 (2006), Article ID 14989, 7 pages. doi: 10.1155/IJBI/2006/14989.  Google Scholar [45] Z. Zhao, J. Yang and M. Jiang, A fast algorithm for high order total variation minimization based interior tomography, J. X-ray Sci. Technol., 23 (2015), 349-364.  doi: 10.3233/XST-150494.  Google Scholar [46] Y. Zhu, M. Zhao and Y. Zhao, Noise reduction with low dose CT data based on a modified ROF model, Optics express, 20 (2012), 17987-18004.  doi: 10.1364/OE.20.017987.  Google Scholar
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).
Reconstructed images using non-truncated projection data. The display window is [0, 0.03]. The ROI is indicated by a circle.
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).
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 $[1, 2]$.
Curves of $E_{\rm rec}(u^k)$ and $E_{\rm SSIM}(u^k)$ from the 4th iteration.
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 * * +∞ * * +∞ * * +∞
 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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