A general framework of piecewise-polynomial Mumford–Shah model for image segmentation

References

• G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Vol. 147, Springer, New York, 2006.
   Google Scholar
• N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach, Commun. Comput. Phys. 7 (2010), pp. 759.
   Web of Science ®Google Scholar
• E. Bae and X.C. Tai, Graph cut optimization for the piecewise constant level set method applied to multiphase image segmentation, in Scale Space and Variational Methods in Computer Vision, Springer, Berlin, Heidelberg, 2009, pp. 1–13.
   Google Scholar
• Y. Boykov and G. Funka-Lea, Graph cuts and efficient N-D image segmentation, Int. J. Comput. Vis. 70 (2006), pp. 109–131. doi: 10.1007/s11263-006-7934-5
   Web of Science ®Google Scholar
• E.S. Brown, T.F. Chan and X. Bresson, Completely convex formulation of the Chan–Vese image segmentation model, Int. J. Comput. Vis. 98 (2012), pp. 103–121. doi: 10.1007/s11263-011-0499-y
   Web of Science ®Google Scholar
• V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Int. J. Comput. Vis. 22 (1997), pp. 61–79. doi: 10.1023/A:1007979827043
   Web of Science ®Google Scholar
• T.F. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math. 66 (2006), pp. 1632–1648. doi: 10.1137/040615286
   Web of Science ®Google Scholar
• T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, 2005.
   Google Scholar
• T.F. Chan and L.A. Vese, Active contours without edges, IEEE Trans. Image Process. 10 (2001), pp. 266–277. doi: 10.1109/83.902291
   PubMed Web of Science ®Google Scholar
• C. Chen and G. Xu, Construction of geometric partial differential equations for level sets, J. Comput. Math. 28 (2010), pp. 105–121. doi: 10.4208/jcm.2009.10-m1014
   Web of Science ®Google Scholar
• D. Comaniciu and P. Meer, Mean shift: A robust approach toward feature space analysis, IEEE Trans. Pattern Anal. Mach. Intell. 24 (2002), pp. 603–619. doi: 10.1109/34.1000236
   Web of Science ®Google Scholar
• D. Cremers, M. Rousson and R. Deriche, A review of statistical approaches to level set segmentation: Integrating color, texture, motion and shape, Int. J. Comput. Vis. 72 (2007), pp. 195–215. doi: 10.1007/s11263-006-8711-1
   Web of Science ®Google Scholar
• P.F. Felzenszwalb and D.P. Huttenlocher, Efficient graph-based image segmentation, Int. J. Comput. Vis. 59 (2004), pp. 167–181. doi: 10.1023/B:VISI.0000022288.19776.77
   Web of Science ®Google Scholar
• D. Freedman and T. Zhang, Interactive graph cut based segmentation with shape priors, in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005 (CVPR 2005), San Diego, CA, USA, Vol. 1, 2005, pp. 755–762.
   Google Scholar
• T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction, J. Scient. Comput. 45 (2010), pp. 272–293. doi: 10.1007/s10915-009-9331-z
   Web of Science ®Google Scholar
• T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imag. Sci. 2 (2009), pp. 323–343. doi: 10.1137/080725891
   Web of Science ®Google Scholar
• M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, Int. J. Comput. Vis. 1 (1988), pp. 321–331. doi: 10.1007/BF00133570
   Web of Science ®Google Scholar
• C. Li, C.Y. Kao, J.C. Gore and Z. Ding, Minimization of region-scalable fitting energy for image segmentation, IEEE Trans. Image Process. 17 (2008), pp. 1940–1949. doi: 10.1109/TIP.2008.2002304
   PubMed Web of Science ®Google Scholar
• F. Li, M.K. Ng and C. Li, Variational fuzzy Mumford–Shah model for image segmentation, SIAM J. Appl. Math. 70 (2010), pp. 2750–2770. doi: 10.1137/090753887
   Web of Science ®Google Scholar
• J. Lie, M. Lysaker and X.C. Tai, A binary level set model and some applications to Mumford–Shah image segmentation, IEEE Trans. Image Process. 15 (2006), pp. 1171–1181. doi: 10.1109/TIP.2005.863956
   PubMed Web of Science ®Google Scholar
• D. Mumford and J. Shah, Boundary detection by minimizing functionals, in IEEE Conference on Computer Vision and Pattern Recognition, San Francisco, CA, USA, 1985, pp. 22–26.
   Google Scholar
• D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math. 42 (1989), pp. 577–685. doi: 10.1002/cpa.3160420503
   Web of Science ®Google Scholar
• S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Vol. 153, Springer, New York, 2003.
   Google Scholar
• S. Osher and J.A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations, J. Comput. Phys. 79 (1988), pp. 12–49. doi: 10.1016/0021-9991(88)90002-2
   Web of Science ®Google Scholar
• D.K. Panjwani and G. Healey, Markov random field models for unsupervised segmentation of textured color images, IEEE Trans Pattern Anal. Mach. Intell. 17 (1995), pp. 939–954. doi: 10.1109/34.464559
   Web of Science ®Google Scholar
• N. Paragios and R. Deriche, Geodesic active regions: A new framework to deal with frame partition problems in computer vision, J. Visual Commun. Image Represent. 13 (2002), pp. 249–268. doi: 10.1006/jvci.2001.0475
   Web of Science ®Google Scholar
• L. Rada and K. Chen, A variational model and its numerical solution for local, selective and automatic segmentation, Numer. Algorithms 66 (2014), pp. 399–430. doi: 10.1007/s11075-013-9741-8
   Web of Science ®Google Scholar
• J. Shi and J. Malik, Normalized cuts and image segmentation, IEEE Trans. Pattern Anal. Mach. Intell. 22 (2000), pp. 888–905. doi: 10.1109/34.868688
   Web of Science ®Google Scholar
• J. Spencer and K. Chen, A convex and selective variational model for image segmentation, Commun. Math. Sci. 13 (2015), pp. 1453–1472. doi: 10.4310/CMS.2015.v13.n6.a5
   Web of Science ®Google Scholar
• X.C. Tai, O. Christiansen, P. Lin and I. Skjælaaen, Image segmentation using some piecewise constant level set methods with MBO type of projection, Int. J. Comput. Vis. 73 (2007), pp. 61–76. doi: 10.1007/s11263-006-9140-x
   Web of Science ®Google Scholar
• A. Tsai, A.R. Yezzi Jr and A.S. Willsky, Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification, IEEE Trans. Image Process. 10 (2001), pp. 1169–1186. doi: 10.1109/83.935033
   PubMed Web of Science ®Google Scholar
• L. Vese, Multiphase object detection and image segmentation, in Geometric Level Set Methods in Imaging, Vision, and Graphics, S. Osher and N. Paragios, eds., Springer, New York, 2003, 175–194.
   Google Scholar
• L.A. Vese and T.F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Comput. Vis. 50 (2002), pp. 271–293. doi: 10.1023/A:1020874308076
   Web of Science ®Google Scholar
• Y. Yang, C. Li, C. Kao and S. Osher, Split Bregman method for minimization of region-scalable fitting energy for image segmentation, in Advances in Visual Computing, G. Bebis, R. Boyle, B. Parvin, D. Koracin, R. Chung, R. Hammound, M. Hussain, T. Kar-Han, R. Crawfis, D. Thalmann, D. Kao and L. Avila, eds., Springer, Berlin, Heidelberg, 2010, 117–128.
   Google Scholar
• J. Ye and G. Xu, Geometric flow approach for region-based image segmentation, IEEE Trans. Image Process. 21 (2012), pp. 4735–4745. doi: 10.1109/TIP.2011.2162421
   PubMed Web of Science ®Google Scholar
• R. Zhang, X. Bresson, T. Chan and X.C. Tai, Four color theorem and convex relaxation for image segmentation with any number of regions, Inverse Probl. Imag. 7 (2013), pp. 1099–1113. doi: 10.3934/ipi.2013.7.1099
   Web of Science ®Google Scholar
• J. Zhang, K. Chen and D.A. Gould, A fast algorithm for automatic segmentation and extraction of a single object by active surfaces, Int. J. Comput. Math. 92 (2015), pp. 1251–1274. doi: 10.1080/00207160.2014.931943
   Web of Science ®Google Scholar
• J. Zhang, K. Chen and B. Yu, A 3D multi-grid algorithm for the Chan–Vese model of variational image segmentation, Int. J. Comput. Math. 89 (2012), pp. 160–189. doi: 10.1080/00207160.2011.632410
   Web of Science ®Google Scholar
• J. Zhang, K. Chen, B. Yu and D. Gould, A local information based variational model for selective image segmentation, Inverse Probl. Imag. 8 (2014), pp. 293–320. doi: 10.3934/ipi.2014.8.321
   Web of Science ®Google Scholar
• H.K. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys. 127 (1996), pp. 179–195. doi: 10.1006/jcph.1996.0167
   Web of Science ®Google Scholar
• S.C. Zhu and A. Yuille, Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation, IEEE Trans. Pattern Anal. Mach. Intell. 18 (1996), pp. 884–900. doi: 10.1109/34.537343
   Web of Science ®Google Scholar