# 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 and 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. [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000. [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. [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. [5] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, 1982. [6] D. P. Bertsekas, A. Nedi and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003. [7] A. Blake and A. Zisserman, Visual Reconstruction, MIT press Cambridge, 1987. [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. [9] A. Braides, Gamma-convergence for Beginners, Oxford University Press, 2002. [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. [11] T. F. Chan and L. Vese, Active contours without edges, IEEE Trans. Image Process., 10 (2001), 266-277.  doi: 10.1109/83.902291. [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. [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. [14] S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl. Math., 13 (2002), 353-370. [15] A. Faridani, E. L. Ritman and K. T. Smith, Local tomography, SIAM J. Appl. Math., 52 (1992), 459-484.  doi: 10.1137/0152026. [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. [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. [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. [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. [20] M. Jiang, P. Maass and T. Page, Regularizing properties of the Mumford-Shah functional for imaging applications, Inverse Problems, 30 (2014), 035007-17pp. [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. [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. [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. [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. [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. [26] P. Maass, The interior Radon transform, SIAM J. Appl. Math., 52 (1992), 710-724.  doi: 10.1137/0152040. [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. [28] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. [29] Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, vol. 87, Springer, 2004. [30] T. Page, Simultaneous reconstruction and segmentation with the Mumford-Shah functional for X-ray tomography, master's thesis, Diplomarbeit University of Bremen, 2011. [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. [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. [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. [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. [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. [36] J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in CVPR, (1996), 136-142.  doi: 10.1109/CVPR.1996.517065. [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. [38] C. R. Vogel, A multigrid method for total variation-based image denoising, in Computation and control Ⅳ, Springer, 20 (1995), 323-331. [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. [40] J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 035013, 29pp. [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. [42] H. Yu and G. Wang, Compressed sensing based interior tomography, Phys. Med. Biol., 54 (2009), 2791-2805. [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. [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. [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. [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.

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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. [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000. [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. [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. [5] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, 1982. [6] D. P. Bertsekas, A. Nedi and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003. [7] A. Blake and A. Zisserman, Visual Reconstruction, MIT press Cambridge, 1987. [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. [9] A. Braides, Gamma-convergence for Beginners, Oxford University Press, 2002. [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. [11] T. F. Chan and L. Vese, Active contours without edges, IEEE Trans. Image Process., 10 (2001), 266-277.  doi: 10.1109/83.902291. [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. [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. [14] S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl. Math., 13 (2002), 353-370. [15] A. Faridani, E. L. Ritman and K. T. Smith, Local tomography, SIAM J. Appl. Math., 52 (1992), 459-484.  doi: 10.1137/0152026. [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. [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. [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. [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. [20] M. Jiang, P. Maass and T. Page, Regularizing properties of the Mumford-Shah functional for imaging applications, Inverse Problems, 30 (2014), 035007-17pp. [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. [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. [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. [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. [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. [26] P. Maass, The interior Radon transform, SIAM J. Appl. Math., 52 (1992), 710-724.  doi: 10.1137/0152040. [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. [28] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. [29] Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, vol. 87, Springer, 2004. [30] T. Page, Simultaneous reconstruction and segmentation with the Mumford-Shah functional for X-ray tomography, master's thesis, Diplomarbeit University of Bremen, 2011. [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. [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. [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. [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. [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. [36] J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in CVPR, (1996), 136-142.  doi: 10.1109/CVPR.1996.517065. [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. [38] C. R. Vogel, A multigrid method for total variation-based image denoising, in Computation and control Ⅳ, Springer, 20 (1995), 323-331. [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. [40] J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 035013, 29pp. [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. [42] H. Yu and G. Wang, Compressed sensing based interior tomography, Phys. Med. Biol., 54 (2009), 2791-2805. [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. [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. [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. [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.
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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