An Efficient Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation

Abstract

In this paper, we design an efficient, multi-stage image segmentation framework that incorporates a weighted difference of anisotropic and isotropic total variation (AITV). The segmentation framework generally consists of two stages: smoothing and thresholding, thus referred to as smoothing-and-thresholding (SaT). In the first stage, a smoothed image is obtained by an AITV-regularized Mumford-Shah (MS) model, which can be solved efficiently by the alternating direction method of multipliers (ADMMs) with a closed-form solution of a proximal operator of the

{\ell }_1-\alpha {\ell }_2

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regularizer. The convergence of the ADMM algorithm is analyzed. In the second stage, we threshold the smoothed image by

K

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-means clustering to obtain the final segmentation result. Numerical experiments demonstrate that the proposed segmentation framework is versatile for both grayscale and color images, efficient in producing high-quality segmentation results within a few seconds, and robust to input images that are corrupted with noise, blur, or both. We compare the AITV method with its original convex TV and nonconvex TV

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counterparts, showcasing the qualitative and quantitative advantages of our proposed method.

Keywords

Image segmentation / Non-convex optimization / Mumford-Shah (MS) model / Alternating direction method of multipliers (ADMMs) / Proximal operator

References

[1.] Alpert, S., Galun, M., Basri, R., Brandt, A.: Image segmentation by probabilistic bottom-up aggregation and cue integration. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 315–327 (2007)

[2.] Ambrosio L, Tortorelli VM. Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math., 1990, 43(8): 999-1036

[3.] Arthur, D., Vassilvitskii, S.: k-means++: the advantages of careful seeding. In: 2007 ACM-SIAM Symposium on Discrete Algorithms (SODA’07), pp. 1027–1035 (2007)

[4.] Bar, L., Chan, T.F., Chung, G., Jung, M., Kiryati, N., Mohieddine, R., Sochen, N., Vese, L.A.: Mumford and Shah model and its applications to image segmentation and image restoration. In: Scherzer, O. (ed.) Handbook of Mathematical Methods in Imaging, pp. 1095–1157. Springer, New York, NY (2011). https://doi.org/10.1007/978-0-387-92920-0_25

[5.] Beck A. . First-Order Methods in Optimization, 2017 Philadelphia SIAM

[6.] Benninghoff H, Garcke H. Efficient image segmentation and restoration using parametric curve evolution with junctions and topology changes. SIAM J. Imaging Sci., 2014, 7(3): 1451-1483

[7.] Birkholz H. A unifying approach to isotropic and anisotropic total variation denoising models. J. Comput. Appl. Math., 2011, 235(8): 2502-2514

[8.] Boyd S, Parikh N, Chu E, Peleato B, Eckstein J, et al.. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends® Mach. Learn., 2011, 3(1): 1-122

[9.] Bui K, Park F, Lou Y, Xin J. A weighted difference of anisotropic and isotropic total variation for relaxed Mumford-Shah color and multiphase image segmentation. SIAM J. Imaging Sci., 2021, 14(3): 1078-1113

[10.] Burdescu, D.D., Brezovan, M., Ganea, E., Stanescu, L.: A new method for segmentation of images represented in a HSV color space. In: Advanced Concepts for Intelligent Vision Systems: 11th International Conference, ACIVS 2009, Bordeaux, France, September 28–October 2, 2009. Proceedings 11, pp. 606–617. Springer (2009)

[11.] Cai X, Chan R, Nikolova M, Zeng T. A three-stage approach for segmenting degraded color images: smoothing, lifting and thresholding (SLaT). J. Sci. Comput., 2017, 72(3): 1313-1332

[12.] Cai X, Chan R, Zeng T. A two-stage image segmentation method using a convex variant of the Mumford-Shah model and thresholding. SIAM J. Imaging Sci., 2013, 6(1): 368-390

[13.] Cao W, Sun J, Zongben X. Fast image deconvolution using closed-form thresholding formulas of L q ( q = 1 2 , 2 3 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q (q= \frac{1}{2}, \frac{2}{3})$$\end{document} regularization. J. Vis. Commun. Image Represent., 2013, 24(1): 31-41

[14.] Chambolle A. Finite-differences discretizations of the Mumford-Shah functional. ESAIM: Math. Model. Numer. Anal., 1999, 33(2): 261-288

[15.] Chambolle A, Dal Maso G. Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: Math. Model. Numer. Anal., 1999, 33(4): 651-672

[16.] Chambolle A, Pock T. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis., 2011, 40(1): 120-145

[17.] Chan R, Yang H, Zeng T. A two-stage image segmentation method for blurry images with Poisson or multiplicative gamma noise. SIAM J. Imaging Sci., 2014, 7(1): 98-127

[18.] Chan RH, Ng MK. Conjugate gradient methods for Toeplitz systems. SIAM Rev., 1996, 38(3): 427-482

[19.] Chan TF, Esedoglu S, Nikolova M. Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math., 2006, 66(5): 1632-1648

[20.] Chan TF, Vese LA. Active contours without edges. IEEE Trans. Image Process., 2001, 10(2): 266-277

[21.] Chan TF, Yezrielev Sandberg B, Vese LA. Active contours without edges for vector-valued images. J. Vis. Commun. Image Represent., 2000, 11(2): 130-141

[22.] Chang H, Lou Y, Duan Y, Marchesini S. Total variation-based phase retrieval for Poisson noise removal. SIAM J. Imaging Sci., 2018, 11(1): 24-55

[23.] Chang H, Lou Y, Ng MK, Zeng T. Phase retrieval from incomplete magnitude information via total variation regularization. SIAM J. Sci. Comput., 2016, 38(6): A3672-A3695

[24.] Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 3869–3872. IEEE (2008)

[25.] Chen, T.-W., Chen, Y.-L., Chien, S.-Y.: Fast image segmentation based on K-means clustering with histograms in HSV color space. In: 2008 IEEE 10th Workshop on Multimedia Signal Processing, pp. 322–325. IEEE (2008)

[26.] Chen X, Ng MK, Zhang C. Non-Lipshitz ℓ p \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _p$$\end{document}-regularization and box constrained model for image reconstruction. IEEE Trans. Image Process., 2012, 21(12): 4709-4721

[27.] Clarke F. . Functional Analysis, Calculus of Variations and Optimal Control, 2013 Heidelberg Springer

[28.] Condat L. Discrete total variation: new definition and minimization. SIAM J. Imaging Sci., 2017, 10(3): 1258-1290

[29.] Deng W, Yin W. On the global and linear convergence of the generalized alternating direction method of multipliers. J. Sci. Comput., 2016, 66(3): 889-916

[30.] Dice LR. Measures of the amount of ecologic association between species. Ecology, 1945, 26(3): 297-302

[31.] Ding L, Han W. α ℓ 1 - β ℓ 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ell _1 -\beta \ell _2$$\end{document} regularization for sparse recovery. Inverse Probl., 2019, 35(12)

[32.] Esedoglu S, Tsai Y-HR. Threshold dynamics for the piecewise constant Mumford-Shah functional. J. Comput. Phys., 2006, 211(1): 367-384

[33.] Esser E, Zhang X, Chan TF. A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci., 2010, 3(4): 1015-1046

[34.] Ge H, Chen W, Ng MK. New restricted isometry property analysis for ℓ 1 - ℓ 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1-\ell _2$$\end{document} minimization methods. SIAM J. Imaging Sci., 2021, 14(2): 530-557

[35.] Getreuer P. Chan-Vese segmentation. Image Process. Line, 2012, 2: 214-224

[36.] Gobbino M. Finite difference approximation of the Mumford-Shah functional. Commun. Pure Appl. Math., 1998, 51(2): 197-228

[37.] Goldstein T, Bresson X, Osher S. Geometric applications of the split Bregman method: segmentation and surface reconstruction. J. Sci. Comput., 2010, 45(1–3): 272-293

[38.] Goldstein T, Osher S. The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci., 2009, 2(2): 323-343

[39.] Gu S, Xie Q, Meng D, Zuo W, Feng X, Zhang L. Weighted nuclear norm minimization and its applications to low level vision. Int. J. Comput. Vis., 2017, 121(2): 183-208

[40.] Han D, Sun D, Zhang L. Linear rate convergence of the alternating direction method of multipliers for convex composite programming. Math. Oper. Res., 2018, 43(2): 622-637

[41.] Hartigan JA, Wong MA. A k-means clustering algorithm. J. R. Stat. Soc.: Ser. C (Appl. Stat.), 1979, 28(1): 100-108

[42.] Hintermüller M, Tao W. Nonconvex TV q \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ TV}^q$$\end{document}-models in image restoration: analysis and a trust-region regularization-based superlinearly convergent solver. SIAM J. Imaging Sci., 2013, 6(3): 1385-1415

[43.] Hou, Z.: A review on MR image intensity inhomogeneity correction. Int. J. Biomed. Imaging 2006, 049515 (2006)

[44.] Huang, Z.-K., Liu, D.-H.: Segmentation of color image using EM algorithm in HSV color space. In: 2007 International Conference on Information Acquisition, pp. 316–319. IEEE (2007)

[45.] Jung M. Piecewise-smooth image segmentation models with L 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} data-fidelity terms. J. Sci. Comput., 2017, 70(3): 1229-1261

[46.] Jung M, Kang M, Kang M. Variational image segmentation models involving non-smooth data-fidelity terms. J. Sci. Comput., 2014, 59(2): 277-308

[47.] Lanza A, Morigi S, Sgallari F. Constrained TV p - ℓ 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ TV}_p-\ell _2$$\end{document} model for image restoration. J. Sci. Comput., 2016, 68(1): 64-91

[48.] Le Thi HA, Pham Dinh T. DC programming and DCA: thirty years of developments. Math. Program., 2018, 169(1): 5-68

[49.] Li C, Kao C-Y, Gore JC, Ding Z. Minimization of region-scalable fitting energy for image segmentation. IEEE Trans. Image Process., 2008, 17(10): 1940-1949

[50.] Li F, Ng MK, Zeng TY, Shen C. A multiphase image segmentation method based on fuzzy region competition. SIAM J. Imaging Sci., 2010, 3(3): 277-299

[51.] Li F, Osher S, Qin J, Yan M. A multiphase image segmentation based on fuzzy membership functions and L1-norm fidelity. J. Sci. Comput., 2016, 69(1): 82-106

[52.] Li F, Zhu Y. Smoothing and clustering guided image decolorization. Image Anal. Stereol., 2021, 40(1): 17-27

[53.] Li P, Chen W, Ge H, Ng MK. ℓ 1 - α ℓ 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1- \alpha \ell _2$$\end{document} minimization methods for signal and image reconstruction with impulsive noise removal. Inverse Probl., 2020, 36(5): 055009

[54.] Li X, Yang X, Zeng T. A three-stage variational image segmentation framework incorporating intensity inhomogeneity information. SIAM J. Imaging Sci., 2020, 13(3): 1692-1715

[55.] Li Y, Wu CL, Duan Y. The TV p \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ TV}_p$$\end{document} regularized Mumford-Shah model for image labeling and segmentation. IEEE Trans. Image Process., 2020, 29: 7061-7075

[56.] Lou, Y., Osher, S., Xin, J.: Computational aspects of constrained L 1 - L 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1-L_2$$\end{document} minimization for compressive sensing. In: Le, T., Hoai, A., Pham, D., Tao, N., Ngoc, T. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 169–180. Springer (2015)

[57.] Lou Y, Yan M. Fast L1–L2 minimization via a proximal operator. J. Sci. Comput., 2018, 74(2): 767-785

[58.] Lou Y, Yin P, He Q, Xin J. Computing sparse representation in a highly coherent dictionary based on difference of L 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} and L 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document}. J. Sci. Comput., 2015, 64(1): 178-196

[59.] Lou Y, Zeng T, Osher S, Xin J. A weighted difference of anisotropic and isotropic total variation model for image processing. SIAM J. Imaging Sci., 2015, 8(3): 1798-1823

[60.] Luong, Q.-T.: Color in computer vision. In: Chen, C.H., Pau, L.F., Wang, P.S.P. (eds.) Handbook of Pattern Recognition and Computer Vision, pp. 311–368. World Scientific, Singapore (1993)

[61.] Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Eighth IEEE International Conference on Computer Vision, 2001. ICCV 2001. Proceedings, vol. 2, pp. 416–423. IEEE (2001)

[62.] Merriman B, Bence JK, Osher SJ. Motion of multiple junctions: a level set approach. J. Comput. Phys., 1994, 112(2): 334-363

[63.] Mumford D, Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math., 1989, 42(5): 577-685

[64.] Ng MK, Chan RH, Tang W-C. A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J. Sci. Comput., 1999, 21(3): 851-866

[65.] Ono S. L 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}_0$$\end{document} gradient projection. IEEE Trans. Image Process., 2017, 26(4): 1554-1564

[66.] Paschos G. Perceptually uniform color spaces for color texture analysis: an empirical evaluation. IEEE Trans. Image Process., 2001, 10(6): 932-937

[67.] Pham Dinh T, Le Thi HA. Convex analysis approach to DC programming: theory, algorithms and applications. Acta Math. Vietnam, 1997, 22(1): 289-355

[68.] Pham Dinh T, Le Thi HA. A DC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim., 1998, 8(2): 476-505

[69.] Pock, T., Chambolle, A., Cremers, D., Bischof, H.: A convex relaxation approach for computing minimal partitions. In: IEEE Conference on Computer Vision and Pattern Recognition, 2009. CVPR 2009, pp. 810–817. IEEE (2009)

[70.] Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the Mumford-Shah functional. In: 2009 IEEE 12th International Conference on Computer Vision, pp. 1133–1140. IEEE (2009)

[71.] Potts, R.B.: Some generalized order-disorder transformations. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 48, pp. 106–109. Cambridge University Press, Cambridge (1952)

[72.] Rahimi Y, Wang C, Dong H, Lou Y. A scale-invariant approach for sparse signal recovery. SIAM J. Sci. Comput., 2019, 41(6): A3649-A3672

[73.] Rockafellar RT, Wets RJ-B. . Variational Analysis, 2009 Heidelberg Springer

[74.] Storath M, Weinmann A. Fast partitioning of vector-valued images. SIAM J. Imaging Sci., 2014, 7(3): 1826-1852

[75.] Sural, S., Qian, G., Pramanik, S.: Segmentation and histogram generation using the HSV color space for image retrieval. In: Proceedings. International Conference on Image Processing, vol. 2, pp. II–II. IEEE (2002)

[76.] Toure, S., Diop, O., Kpalma, K., Maiga, A.S.: Best-performing color space for land-sea segmentation. In: 2018 41st International Conference on Telecommunications and Signal Processing (TSP), pp. 1–5. IEEE (2018)

[77.] Vese LA, Chan TF. A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vision, 2002, 50(3): 271-293

[78.] Wang C, Tao M, Chuah C-N, Nagy J, Lou Y. Minimizing l 1 over l 2 norms on the gradient. Inverse Probl., 2022, 38(6)

[79.] Wang C, Tao M, Nagy JG, Lou Y. Limited-angle CT reconstruction via the L 1 / L 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1/L_2$$\end{document} minimization. SIAM J. Imaging Sci., 2021, 14(2): 749-777

[80.] Wang C, Yan M, Rahimi Y, Lou Y. Accelerated schemes for the L 1 / L 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1/L_2$$\end{document} minimization. IEEE Trans. Signal Process., 2020, 68: 2660-2669

[81.] Wang D, Wang X-P. The iterative convolution-thresholding method (ICTM) for image segmentation. Pattern Recogn., 2022, 130

[82.] Wang X-F, Huang D-S, Huan X. An efficient local Chan-Vese model for image segmentation. Pattern Recogn., 2010, 43(3): 603-618

[83.] Wang Y, Yang J, Yin W, Zhang Y. A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci., 2008, 1(3): 248-272

[84.] Wu T, Mao Z, Li Z, Zeng Y, Zeng T. Efficient color image segmentation via quaternion-based L 1 / L 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1/L_2$$\end{document} regularization. J. Sci. Comput., 2022, 93(1): 9

[85.] Wu T , Shao J, Gu X, Ng MK, Zeng T. Two-stage image segmentation based on nonconvex ℓ 2 - ℓ p \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2- \ell _p$$\end{document} approximation and thresholding. Appl. Math. Comput., 2021, 403: 126168

[86.] Wu T, Zhao Y, Mao Z, Shi L, Li Z, Zeng Y. Image segmentation via Fischer-Burmeister total variation and thresholding. Adv. Appl. Math. Mech., 2022, 14(4): 960-988

[87.] Xu Z, Chang X, Xu F, Zhang H. L 1 / 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1/2}$$\end{document} regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst., 2012, 23(7): 1013-1027

[88.] Yin P, Lou Y, He Q, Xin J. Minimization of ℓ 1 - 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1-2}$$\end{document} for compressed sensing. SIAM J. Sci. Comput., 2015, 37(1): A536-A563

[89.] You J, Jiao Y, Lyu XL, Zeng T. A nonconvex model with minimax concave penalty for image restoration. J. Sci. Comput., 2019, 78(2): 1063-1086

[90.] Zeng C, Wu CL. On the edge recovery property of noncovex nonsmooth regularization in image restoration. SIAM J. Numer. Anal., 2018, 56(2): 1168-1182

[91.] Zhang S, Xin J. Minimization of transformed L 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} penalty: closed form representation and iterative thresholding algorithms. Commun. Math. Sci., 2017, 15(2): 511-537

[92.] Zhang S, Xin J. Minimization of transformed L 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} penalty: theory, difference of convex function algorithm, and robust application in compressed sensing. Math. Program., 2018, 169(1): 307-336

Funding

Division of Mathematical Sciences(DMS-2151235); National Science Foundation(CAREER 1846690); Division of Mathematical Sciences(DMS-2219904)