{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T05:15:43Z","timestamp":1772774143997,"version":"3.50.1"},"reference-count":59,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T00:00:00Z","timestamp":1687305600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T00:00:00Z","timestamp":1687305600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2023,8]]},"DOI":"10.1007\/s10915-023-02268-5","type":"journal-article","created":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T18:05:25Z","timestamp":1687370725000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Convergent Iterative Support Shrinking Algorithm for Non-Lipschitz Multi-phase Image Labeling Model"],"prefix":"10.1007","volume":"96","author":[{"given":"Yijie","family":"Yang","sequence":"first","affiliation":[]},{"given":"Yutong","family":"Li","sequence":"additional","affiliation":[]},{"given":"Chunlin","family":"Wu","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2433-3056","authenticated-orcid":false,"given":"Yuping","family":"Duan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,21]]},"reference":[{"issue":"2","key":"2268_CR1","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1109\/TPAMI.2011.130","volume":"34","author":"S Alpert","year":"2011","unstructured":"Alpert, S., Galun, M., Brandt, A., Basri, R.: Image segmentation by probabilistic bottom-up aggregation and cue integration. IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 315\u2013327 (2011)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"2268_CR2","doi-asserted-by":"crossref","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-\u0141ojasiewicz inequality. Math. Oper. Res 35(2), 438\u2013457 (2010)","DOI":"10.1287\/moor.1100.0449"},{"issue":"1\u20132","key":"2268_CR3","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s10107-011-0484-9","volume":"137","author":"H Attouch","year":"2013","unstructured":"Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program. 137(1\u20132), 91\u2013129 (2013)","journal-title":"Math. Program."},{"issue":"1","key":"2268_CR4","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1007\/s11263-010-0406-y","volume":"92","author":"E Bae","year":"2011","unstructured":"Bae, E., Yuan, J., Tai, X.: Global minimization for continuous multiphase partitioning problems using a dual approach. Int. J. Comput. Vis. 92(1), 112\u2013129 (2011)","journal-title":"Int. J. Comput. Vis."},{"issue":"3","key":"2268_CR5","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s11263-017-0995-9","volume":"123","author":"R Bergmann","year":"2017","unstructured":"Bergmann, R., Fitschen, J.H., Persch, J., Steidl, G.: Iterative multiplicative filters for data labeling. Int. J. Comput. Vis. 123(3), 435\u2013453 (2017)","journal-title":"Int. J. Comput. Vis."},{"issue":"3","key":"2268_CR6","doi-asserted-by":"crossref","first-page":"1718","DOI":"10.1137\/120864908","volume":"23","author":"W Bian","year":"2013","unstructured":"Bian, W., Chen, X.: Worst-case complexity of smoothing quadratic regularization methods for non-$$\\text{ L }$$ipschitzian optimization. SIAM J. Optim. 23(3), 1718\u20131741 (2013)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"2268_CR7","doi-asserted-by":"crossref","first-page":"2294","DOI":"10.1137\/140985639","volume":"8","author":"W Bian","year":"2015","unstructured":"Bian, W., Chen, X.: Linearly constrained non-Lipschitz optimization for image restoration. SIAM J. Imaging Sci. 8(4), 2294\u20132322 (2015)","journal-title":"SIAM J. Imaging Sci."},{"issue":"1","key":"2268_CR8","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s10107-014-0753-5","volume":"149","author":"W Bian","year":"2015","unstructured":"Bian, W., Chen, X., Ye, Y.: Complexity analysis of interior point algorithms for non-Lipschitz and nonconvex minimization. Math. Program. 149(1), 301\u2013327 (2015)","journal-title":"Math. Program."},{"key":"2268_CR9","doi-asserted-by":"crossref","first-page":"1313","DOI":"10.1007\/s10915-017-0402-2","volume":"72","author":"X Cai","year":"2017","unstructured":"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. 72, 1313\u20131332 (2017)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"2268_CR10","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1137\/120867068","volume":"6","author":"X Cai","year":"2013","unstructured":"Cai, X., Chan, R., Zeng, T.: A two-stage image segmentation method using a convex variant of the Mumford\u2013Shah model and thresholding. SIAM J. Imaging Sci. 6(1), 368\u2013390 (2013)","journal-title":"SIAM J. Imaging Sci."},{"issue":"2","key":"2268_CR11","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TIT.2005.862083","volume":"52","author":"EJ Cand\u00e8s","year":"2006","unstructured":"Cand\u00e8s, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theor. 52(2), 489\u2013509 (2006)","journal-title":"IEEE Trans. Inf. Theor."},{"issue":"5","key":"2268_CR12","doi-asserted-by":"crossref","first-page":"1632","DOI":"10.1137\/040615286","volume":"66","author":"TF Chan","year":"2006","unstructured":"Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632\u20131648 (2006)","journal-title":"SIAM J. Appl. Math."},{"issue":"2","key":"2268_CR13","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1109\/83.902291","volume":"10","author":"TF Chan","year":"2001","unstructured":"Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266\u2013277 (2001)","journal-title":"IEEE Trans. Image Process."},{"issue":"1","key":"2268_CR14","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1137\/15M1052834","volume":"55","author":"X Chen","year":"2017","unstructured":"Chen, X., Guo, L., Lu, Z., Ye, J.J.: An augmented lagrangian method for non-Lipschitz nonconvex programming. SIAM J. Numer. Anal. 55(1), 168\u2013193 (2017)","journal-title":"SIAM J. Numer. Anal."},{"issue":"12","key":"2268_CR15","doi-asserted-by":"crossref","first-page":"4709","DOI":"10.1109\/TIP.2012.2214051","volume":"21","author":"X Chen","year":"2012","unstructured":"Chen, X., Ng, M.K., Zhang, C.: Non-Lipschitz $$\\ell _p$$-regularization and box constrained model for image restoration. IEEE Trans. Image Process. 21(12), 4709\u20134721 (2012)","journal-title":"IEEE Trans. Image Process."},{"issue":"3","key":"2268_CR16","doi-asserted-by":"crossref","first-page":"1528","DOI":"10.1137\/120871390","volume":"23","author":"X Chen","year":"2013","unstructured":"Chen, X., Niu, L., Yuan, Y.: Optimality conditions and a smoothing trust region $$\\text{ N }$$ewton method for non-$$\\text{ L }$$ipschitz optimization. SIAM J. Optim. 23(3), 1528\u20131552 (2013)","journal-title":"SIAM J. Optim."},{"issue":"5","key":"2268_CR17","doi-asserted-by":"crossref","first-page":"2832","DOI":"10.1137\/090761471","volume":"32","author":"X Chen","year":"2010","unstructured":"Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of $$\\ell _2-\\ell _p$$ minimization. SIAM J. Sci. Comput. 32(5), 2832\u20132852 (2010)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2268_CR18","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1137\/080740167","volume":"3","author":"X Chen","year":"2010","unstructured":"Chen, X., Zhou, W.: Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 3(4), 765\u2013790 (2010)","journal-title":"SIAM J. Imaging Sci."},{"issue":"3","key":"2268_CR19","doi-asserted-by":"crossref","first-page":"395","DOI":"10.4208\/csiam-am.2020-0029","volume":"2","author":"YG ChunlinWu","year":"2021","unstructured":"ChunlinWu, Y.G.: A general non-lipschitz joint regularized model for multi-channel\/modality image reconstruction. CSIAM Trans. Appl. Math. 2(3), 395\u2013430 (2021)","journal-title":"CSIAM Trans. Appl. Math."},{"issue":"2","key":"2268_CR20","doi-asserted-by":"crossref","first-page":"337","DOI":"10.3934\/ipi.2015.9.337","volume":"9","author":"B Coll","year":"2015","unstructured":"Coll, B., Duran, J., Sbert, C.: Half-linear regularization for nonconvex image restoration models. Inverse Probl. Imaging 9(2), 337 (2015)","journal-title":"Inverse Probl. Imaging"},{"issue":"3","key":"2268_CR21","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1007\/s11263-013-0617-0","volume":"104","author":"NY El-Zehiry","year":"2013","unstructured":"El-Zehiry, N.Y., Grady, L.: Combinatorial optimization of the discretized multiphase $$\\text{ M }$$umford-$$\\text{ S }$$hah functional. Int. J. Comput. Vis. 104(3), 270\u2013285 (2013)","journal-title":"Int. J. Comput. Vis."},{"issue":"5","key":"2268_CR22","doi-asserted-by":"crossref","first-page":"2399","DOI":"10.1109\/TIP.2011.2182522","volume":"21","author":"Y Gu","year":"2012","unstructured":"Gu, Y., Wang, L., Tai, X.: A direct approach toward global minimization for multiphase labeling and segmentation problems. IEEE Trans. Image Process. 21(5), 2399\u20132411 (2012)","journal-title":"IEEE Trans. Image Process."},{"issue":"1","key":"2268_CR23","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/s10107-019-01435-7","volume":"185","author":"L Guo","year":"2021","unstructured":"Guo, L., Chen, X.: Mathematical programs with complementarity constraints and a non-$$\\text{ L }$$ipschitz objective: optimality and approximation. Math. Program. 185(1), 455\u2013485 (2021)","journal-title":"Math. Program."},{"issue":"3","key":"2268_CR24","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1007\/s10851-020-01001-3","volume":"63","author":"X Guo","year":"2021","unstructured":"Guo, X., Xue, Y., Wu, C.: Effective two-stage image segmentation: a new non-lipschitz decomposition approach with convergent algorithm. J. Math. Imaging Vis. 63(3), 356\u2013379 (2021)","journal-title":"J. Math. Imaging Vis."},{"issue":"7","key":"2268_CR25","doi-asserted-by":"crossref","first-page":"3556","DOI":"10.1109\/TIP.2018.2825112","volume":"27","author":"J Han","year":"2018","unstructured":"Han, J., Song, K.S., Kim, J., Kang, M.G.: Permuted coordinate-wise optimizations applied to $$l_{p}$$-regularized image deconvolution. IEEE Trans. Image Process. 27(7), 3556\u20133570 (2018)","journal-title":"IEEE Trans. Image Process."},{"issue":"3","key":"2268_CR26","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1137\/110854746","volume":"6","author":"M Hinterm\u00fcller","year":"2013","unstructured":"Hinterm\u00fcller, M., Wu, T.: Nonconvex $$\\text{ TV}^{q}$$ -models in image restoration: analysis and a trust-region regularization-based superlinearly convergent solver. SIAM J. Imaging Sci. 6(3), 1385\u20131415 (2013)","journal-title":"SIAM J. Imaging Sci."},{"issue":"3","key":"2268_CR27","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1007\/s10589-016-9854-9","volume":"65","author":"Y Huang","year":"2016","unstructured":"Huang, Y., Liu, H.: Smoothing projected $$\\text{ B }$$arzilai-$$\\text{ B }$$orwein method for constrained non-$$\\text{ L }$$ipschitz optimization. Comput. Optim. Appl. 65(3), 671\u2013698 (2016)","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"2268_CR28","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s11263-015-0809-x","volume":"115","author":"JH Kappes","year":"2015","unstructured":"Kappes, J.H., Andres, B., Hamprecht, F.A., Schn\u00f6rr, C., Nowozin, S., Batra, D., Kim, S., Kausler, B.X., Kr\u00f6ger, T., Lellmann, J., et al.: A comparative study of modern inference techniques for structured discrete energy minimization problems. Int. J. Comput. Vis. 115(2), 155\u2013184 (2015)","journal-title":"Int. J. Comput. Vis."},{"issue":"1","key":"2268_CR29","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1007\/s10915-015-0129-x","volume":"68","author":"A Lanza","year":"2016","unstructured":"Lanza, A., Morigi, S., Sgallari, F.: Constrained $$\\text{ TV}_p-\\ell _2$$ model for image restoration. J. Sci. Comput. 68(1), 64\u201391 (2016)","journal-title":"J. Sci. Comput."},{"issue":"333","key":"2268_CR30","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1090\/mcom\/3655","volume":"91","author":"C Li","year":"2022","unstructured":"Li, C., Chen, X.: Isotropic non-$$\\text{ L }$$ipschitz regularization for sparse representations of random fields on the sphere. Math. Comput. 91(333), 219\u2013243 (2022)","journal-title":"Math. Comput."},{"key":"2268_CR31","doi-asserted-by":"crossref","first-page":"678","DOI":"10.1016\/j.neunet.2021.08.001","volume":"143","author":"W Li","year":"2021","unstructured":"Li, W., Bian, W.: Smoothing neural network for L0 regularized optimization problem with general convex constraints. Neural Netw. 143, 678\u2013689 (2021)","journal-title":"Neural Netw."},{"issue":"9","key":"2268_CR32","doi-asserted-by":"crossref","first-page":"3361","DOI":"10.1109\/TNNLS.2019.2944388","volume":"31","author":"W Li","year":"2019","unstructured":"Li, W., Bian, W., Xue, X.: Projected neural network for a class of non-$$\\text{ L }$$ipschitz optimization problems with linear constraints. IEEE Trans. Neural Netw. Learn. Syst. 31(9), 3361\u20133373 (2019)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"2268_CR33","doi-asserted-by":"crossref","first-page":"7061","DOI":"10.1109\/TIP.2020.2997524","volume":"29","author":"Y Li","year":"2020","unstructured":"Li, Y., Wu, C., Duan, Y.: The $$\\text{ TV}_{p}$$ regularized $$\\text{ M }$$umford-$$\\text{ S }$$hah model for image labeling and segmentation. IEEE Trans. Image Process. 29, 7061\u20137075 (2020)","journal-title":"IEEE Trans. Image Process."},{"issue":"3","key":"2268_CR34","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1137\/20M1382817","volume":"14","author":"J Ma","year":"2021","unstructured":"Ma, J., Wang, D., Wang, X.P., Yang, X.: A characteristic function-based algorithm for geodesic active contours. SIAM J. Imaging Sci. 14(3), 1184\u20131205 (2021)","journal-title":"SIAM J. Imaging Sci."},{"issue":"5","key":"2268_CR35","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1002\/cpa.3160420503","volume":"42","author":"D Mumford","year":"1989","unstructured":"Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577\u2013685 (1989)","journal-title":"Commun. Pure Appl. Math."},{"issue":"3","key":"2268_CR36","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1137\/040619582","volume":"4","author":"M Nikolova","year":"2005","unstructured":"Nikolova, M.: Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares. Multiscale Model. Simul. 4(3), 960\u2013991 (2005)","journal-title":"Multiscale Model. Simul."},{"issue":"12","key":"2268_CR37","doi-asserted-by":"crossref","first-page":"3073","DOI":"10.1109\/TIP.2010.2052275","volume":"19","author":"M Nikolova","year":"2010","unstructured":"Nikolova, M., Ng, M.K., Tam, C.P.: Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans. Image Process. 19(12), 3073\u20133088 (2010)","journal-title":"IEEE Trans. Image Process."},{"issue":"1","key":"2268_CR38","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1137\/070692285","volume":"1","author":"M Nikolova","year":"2008","unstructured":"Nikolova, M., Ng, M.K., Zhang, S., Ching, W.K.: Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 1(1), 2\u201325 (2008)","journal-title":"SIAM J. Imaging Sci."},{"issue":"6","key":"2268_CR39","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1109\/TCYB.2016.2546965","volume":"47","author":"L Niu","year":"2016","unstructured":"Niu, L., Zhou, R., Tian, Y., Qi, Z., Zhang, P.: Nonsmooth penalized clustering via $$l_{p}$$ regularized sparse regression. IEEE Trans. Cybern. 47(6), 1423\u20131433 (2016)","journal-title":"IEEE Trans. Cybern."},{"issue":"1","key":"2268_CR40","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1007\/s11042-019-08179-8","volume":"79","author":"Y Ren","year":"2020","unstructured":"Ren, Y., Tang, L.: A nonconvex and nonsmooth anisotropic total variation model for image noise and blur removal. Multimed. Tools Appl. 79(1), 1445\u20131473 (2020)","journal-title":"Multimed. Tools Appl."},{"issue":"8","key":"2268_CR41","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1007\/s10851-019-00893-0","volume":"61","author":"M Roberts","year":"2019","unstructured":"Roberts, M., Spencer, J.: Chan-Vese reformulation for selective image segmentation. J. Math. Imaging Vis. 61(8), 1173\u20131196 (2019)","journal-title":"J. Math. Imaging Vis."},{"issue":"4","key":"2268_CR42","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1007\/s10898-017-0507-z","volume":"68","author":"T Sun","year":"2017","unstructured":"Sun, T., Jiang, H., Cheng, L.: Global convergence of proximal iteratively reweighted algorithm. J. Glob. Optim. 68(4), 815\u2013826 (2017)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"2268_CR43","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1023\/A:1020874308076","volume":"50","author":"LA Vese","year":"2002","unstructured":"Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50(3), 271\u2013293 (2002)","journal-title":"Int. J. Comput. Vis."},{"key":"2268_CR44","doi-asserted-by":"crossref","first-page":"2660","DOI":"10.1109\/TSP.2020.2985298","volume":"68","author":"C Wang","year":"2020","unstructured":"Wang, C., Yan, M., Rahimi, Y., Lou, Y.: Accelerated schemes for the $$ l_1\/l_2 $$ minimization. IEEE Trans. Signal Process. 68, 2660\u20132669 (2020)","journal-title":"IEEE Trans. Signal Process."},{"key":"2268_CR45","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1016\/j.jcp.2017.08.020","volume":"350","author":"D Wang","year":"2017","unstructured":"Wang, D., Li, H., Wei, X., Wang, X.P.: An efficient iterative thresholding method for image segmentation. J. Comput. Phys. 350, 657\u2013667 (2017)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"2268_CR46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-018-0752-4","volume":"90","author":"W Wang","year":"2022","unstructured":"Wang, W., Chen, Y.: An accelerated smoothing gradient method for nonconvex nonsmooth minimization in image processing. J. Sci. Comput. 90(1), 1\u201328 (2022)","journal-title":"J. Sci. Comput."},{"key":"2268_CR47","doi-asserted-by":"crossref","unstructured":"Wang, W., Tian, N., Wu, C.: Two-phase image segmentation by nonconvex nonsmooth models with convergent alternating minimization algorithms. J. Comput. Math. in press (2022)","DOI":"10.4208\/jcm.2108-m2021-0057"},{"issue":"1","key":"2268_CR48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-018-0752-4","volume":"83","author":"W Wang","year":"2020","unstructured":"Wang, W., Wu, C., Tai, X.C.: A globally convergent algorithm for a constrained non-$$\\text{ L }$$ipschitz image restoration model. J. Sci. Comput. 83(1), 1\u201329 (2020)","journal-title":"J. Sci. Comput."},{"key":"2268_CR49","volume":"410","author":"T Wu","year":"2021","unstructured":"Wu, T., Ng, M.K., Zhao, X.L.: Sparsity reconstruction using nonconvex tgpv-shearlet regularization and constrained projection. Appl. Math. Comput. 410, 126170 (2021)","journal-title":"Appl. Math. Comput."},{"issue":"5","key":"2268_CR50","doi-asserted-by":"crossref","first-page":"1587","DOI":"10.1109\/TIP.2015.2401430","volume":"24","author":"J Xiao","year":"2015","unstructured":"Xiao, J., Ng, M.K.P., Yang, Y.F.: On the convergence of nonconvex minimization methods for image recovery. IEEE Trans. Image Process. 24(5), 1587\u20131598 (2015)","journal-title":"IEEE Trans. Image Process."},{"issue":"7","key":"2268_CR51","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1109\/TNNLS.2012.2197412","volume":"23","author":"Z Xu","year":"2012","unstructured":"Xu, Z., Chang, X., Xu, F., Zhang, H.: $${L}_{1\/2}$$ regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst. 23(7), 1013\u20131027 (2012)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"3","key":"2268_CR52","doi-asserted-by":"crossref","first-page":"653","DOI":"10.3934\/ipi.2019030","volume":"13","author":"S Yan","year":"2019","unstructured":"Yan, S., Liu, J., Huang, H., Tai, X.C.: A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation. Inverse Probl Imaging 13(3), 653\u2013677 (2019)","journal-title":"Inverse Probl Imaging"},{"issue":"2","key":"2268_CR53","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1007\/s10915-018-0801-z","volume":"78","author":"J You","year":"2019","unstructured":"You, J., Jiao, Y., Lu, X., Zeng, T.: A nonconvex model with minimax concave penalty for image restoration. J. Sci. Comput. 78(2), 1063\u20131086 (2019)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"2268_CR54","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1007\/s10851-018-0830-0","volume":"61","author":"C Zeng","year":"2019","unstructured":"Zeng, C., Jia, R., Wu, C.: An iterative support shrinking algorithm for non-$$\\text{ L }$$ipschitz optimization in image restoration. J. Math. Imaging Vis. 61(1), 122\u2013139 (2019)","journal-title":"J. Math. Imaging Vis."},{"issue":"2","key":"2268_CR55","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/17M1123687","volume":"56","author":"C Zeng","year":"2018","unstructured":"Zeng, C., Wu, C.: On the edge recovery property of noncovex nonsmooth regularization in image restoration. SIAM J. Numer. Anal. 56(2), 1168\u20131182 (2018)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"2268_CR56","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1007\/s10444-018-9629-1","volume":"45","author":"C Zeng","year":"2019","unstructured":"Zeng, C., Wu, C.: On the discontinuity of images recovered by noncovex nonsmooth regularized isotropic models with box constraints. Adv. Comput. Math. 45(2), 589\u2013610 (2019)","journal-title":"Adv. Comput. Math."},{"issue":"1","key":"2268_CR57","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/18M119611X","volume":"30","author":"C Zhang","year":"2020","unstructured":"Zhang, C., Chen, X.: A smoothing active set method for linearly constrained non-$$\\text{ L }$$ipschitz nonconvex optimization. SIAM J. Optim. 30(1), 1\u201330 (2020)","journal-title":"SIAM J. Optim."},{"key":"2268_CR58","doi-asserted-by":"crossref","first-page":"3132","DOI":"10.1109\/TIP.2019.2957925","volume":"29","author":"H Zhang","year":"2019","unstructured":"Zhang, H., Qian, J., Zhang, B., Yang, J., Gong, C., Wei, Y.: Low-rank matrix recovery via modified schatten-$$ p $$ norm minimization with convergence guarantees. IEEE Trans. Image Process. 29, 3132\u20133142 (2019)","journal-title":"IEEE Trans. Image Process."},{"issue":"12","key":"2268_CR59","doi-asserted-by":"crossref","DOI":"10.1088\/1361-6420\/abc793","volume":"36","author":"Z Zheng","year":"2020","unstructured":"Zheng, Z., Ng, M., Wu, C.: A globally convergent algorithm for a class of gradient compounded non-$$\\text{ L }$$ipschitz models applied to non-additive noise removal. Inverse Probl. 36(12), 125017 (2020)","journal-title":"Inverse Probl."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02268-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-023-02268-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02268-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T08:12:40Z","timestamp":1689581560000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-023-02268-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,21]]},"references-count":59,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["2268"],"URL":"https:\/\/doi.org\/10.1007\/s10915-023-02268-5","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,21]]},"assertion":[{"value":"16 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 May 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 June 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"47"}}