Abstract
This chapter presents an overview of the Mumford and Shah model for image segmentation. It discusses its various formulations, some of its properties, the mathematical framework, and several approximations. It also presents numerical algorithms and segmentation results using the Ambrosio-Tortorelli phase-field approximations on one hand and level set formulations on the other hand. Several applications of the Mumford-Shah problem to image restoration are also presented.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Adams, R.A.: Sobolev Spaces. Academic, New York (1975)
Alicandro, R., Braides, A., Shah, J.: Free-discontinuity problems via functionals involving the L1-norm of the gradient and their approximation. Interfaces Free Bound 1, 17–37 (1999)
Ambrosio, L.: A compactness theorem for a special class of functions of bounded variation. Boll. Un. Mat. Ital. 3(B), 857–881 (1989)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)
Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma \)-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990)
Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B7(6), 105–123 (1992)
Aubert, G., Blanc-Féraud, L., March, R.: An approximation of the Mumford-Shah energy by a family of discrete edge-preserving functionals. Nonlinear Anal. 64(9), 1908–1930 (2006)
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Springer, New York (2006)
Bar, L., Brook, A., Sochen, N., Kiryati, N.: Deblurring of color images corrupted by impulsive noise. IEEE Trans. Image Process. 16(4), 1101–1111 (2007)
Bar, L., Sochen, N., Kiryati, N.: Variational pairing of image segmentation and blind restoration. In: Proceedings of 8th European Conference on Computer Vision, Prague. Volume 3022 of LNCS, pp. 166–177 (2004)
Bar, L., Sochen, N., Kiryati, N.: Image deblurring in the presence of salt-and-pepper noise. In: Proceedings of 5th International Conference on Scale Space and PDE Methods in Computer Vision, Hofgeismar. Volume 3459 of LNCS, pp. 107–118 (2005)
Bar, L., Sochen, N., Kiryati, N.: Image deblurring in the presence of impulsive noise. Int. J. Comput. Vis. 70, 279–298 (2006)
Bar, L., Sochen, N., Kiryati, N.: Semi-blind image restoration via Mumford-Shah regularization. IEEE Trans. Image Process. 15(2), 483–493 (2006)
Bar, L., Sochen, N., Kiryati, N.: Convergence of an iterative method for variational deconvolution and impulsive noise removal. SIAM J. Multiscale Model Simul. 6, 983–994 (2007)
Bar, L., Sochen, N., Kiryati, N.: Restoration of images with piecewise space-variant blur. In: Proceedings of 1st International Conference on Scale Space and Variational Methods in Computer Vision, Ischia, pp. 533–544 (2007)
Blake, A., Zisserman, A.: Visual Reconstruction. MIT, Cambridge (1987)
Bourdin, B.: Image segmentation with a finite element method. M2AN Math. Model. Numer. Anal. 33(2), 229–244 (1999)
Bourdin, B., Chambolle, A.: Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85(4), 609–646 (2000)
Braides, A.: Approximation of Free-Discontinuity Problems. Volume 1694 of Lecture Notes in Mathematics. Springer, Berlin (1998)
Braides, A., Dal Maso, G.: Nonlocal approximation of the Mumford-Shah functional. Calc Var 5, 293–322 (1997)
Bregman, L.M.: The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Phys. 7, 200–217 (1967)
Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. SIAM MMS 4(2), 490–530 (2005)
Chambolle, A.: Un théorème de γ-convergence pour la segmentation des signaux. C R Acad Sci Paris Sér. I Math 314(3), 191–196 (1992)
Chambolle, A.: Image segmentation by variational methods: Mumford and Shah functional, and the discrete approximation. SIAM J. Appl. Math. 55, 827–863 (1995)
Chambolle, A.: Finite-differences discretizations of the Mumford-Shah functional. M2AN Math. Model. Numer. Anal. 33(2), 261–288 (1999)
Chambolle, A.: Inverse problems in image processing and image segmentation: some mathematical and numerical aspects. In: Chidume, C.E. (ed.) Mathematical Problems in Image Processing. ICTP Lecture Notes Series, vol. 2. ICTP, Trieste (2000). http://publications.ictp.it/lns/vol2.html
Chambolle, A., Dal Maso, G.: Discrete approximation of the Mumford-Shah functional in dimension two. M2AN Math. Model. Numer. Anal. 33(4), 651–672 (1999)
Chan, T.F., Shen, J.: Image Processing and Analysis. Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005)
Chan, T., Vese, L.: An active contour model without edges. Lect. Notes Comput. Sci. 1682, 141–151 (1999)
Chan, T., Vese, L.: An efficient variational multiphase motion for the Mumford-Shah segmentation model. In: 34th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, vol. 1, pp. 490–494 (2000)
Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10, 266–277 (2001)
Chan, T., Vese, L.: A level set algorithm for minimizing the Mumford-Shah functional in image processing. In: IEEE/Computer Society Proceedings of the 1st IEEE Workshop on Variational and Level Set Methods in Computer Vision, Vancouver, pp. 161–168 (2001)
Chan, T.F., Wong, C.K.: Total variation blind deconvolution. IEEE Trans. Image Process. 7, 370–375 (1998)
Chung, G., Vese, L.A.: Energy minimization based segmentation and denoising using a multilayer level set approach. Lect. Notes Comput. Sci. 3757, 439–455 (2005)
Chung, G., Vese, L.A.: Image segmentation using a multilayer level-set approach. Comput. Vis. Sci. 12(6), 267–285 (2009)
Cohen, L.D.: Avoiding local minima for deformable curves in image analysis. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Curves and Surfaces with Applications in CAGD, pp. 77–84. Vanderbilt University Press, Nashville (1997)
Cohen, L., Bardinet, E., Ayache, N.: Surface reconstruction using active contour models. In: SPIE ‘93 Conference on Geometric Methods in Computer Vision, San Diego, July 1993
David, G.: Singular Sets of Minimizers for the Mumford-Shah Functional. Birkhäuser, Basel (2005)
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC, Boca Raton (1992)
Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE TPAMI 6, 721–741 (1984)
Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. SIAM MMS 6(2), 595–630 (2007)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)
Huber, P.J.: Robust Statistics. Wiley, New York (1981)
Jung, M., Chung, G., Sundaramoorthi, G., Vese, L.A., Yuille, A.L.: Sobolev gradients and joint variational image segmentation, denoising and deblurring. In: IS&T/SPIE on Electronic Imaging. Volume 7246 of Computational Imaging VII, San Jose, pp. 72460I-1–72460I-13 (2009)
Jung, M., Vese, L.A.: Nonlocal variational image deblurring models in the presence of gaussian or impulse noise. In: International Conference on Scale Space and Variational Methods in Computer Vision (SSVM’ 09), Voss. Volume 5567 of LNCS, pp. 402–413 (2009)
Kim, J., Tsai, A., Cetin, M., Willsky, A.S.: A curve evolution-based variational approach to simultaneous image restoration and segmentation. In: Proceedings of IEEE International Conference on Image Processing, Rochester, vol. 1, pp. 109–112 (2002)
Koepfler, G., Lopez, C., Morel, J.M.: A multiscale algorithm for image segmentation by variational methods. SIAM J. Numer. Anal. 31(1), 282–299 (1994)
Kundur, D., Hatzinakos, D.: Blind image deconvolution. Signal Process. Mag. 13, 43–64 (1996)
Kundur, D., Hatzinakos, D.: Blind image deconvolution revisited. Signal Process. Mag. 13, 61–63 (1996)
Larsen, C.J.: A new proof of regularity for two-shaded image segmentations. Manuscr. Math. 96, 247–262 (1998)
Leonardi, G.P., Tamanini, I.: On minimizing partitions with infinitely many components. Ann. Univ. Ferrara Sez. VII Sc. Mat. XLIV, 41–57 (1998)
Li, C., Kao, C.-Y., Gore, J.C., Ding, Z.: Implicit active contours driven by local binary fitting energy. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), CVPR’07, Minneapolis (2007)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1993)
Dal Maso, G., Morel, J.M., Solimini, S.: Variational approach in image processing – existence and approximation properties. C. R. Acad. Sci. Paris Sér. I Math. 308(19), 549–554 (1989)
Dal Maso, G., Morel, J.M., Solimini, S.: A variational method in image segmentation – existence and approximation properties. Acta Math. 168(1–2), 89–151 (1992)
Massari, U., Tamanini, I.: On the finiteness of optimal partitions. Ann. Univ. Ferrara Sez VII Sc. Mat. XXXIX, 167–185 (1993)
Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142 (1987)
Modica, L., Mortola, S.: Un esempio di γ-convergenza. Boll. Un. Mat. Ital. B5(14), 285–299 (1977)
Morel, J.-M., Solimini, S.: Variational Methods in Image Segmentation. Birkhäuser, Boston (1995)
Mumford, D., Shah, J.: Boundary detection by minimizing functionals. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, San Francisco, pp. 22–26 (1985)
Mumford, D., Shah, J.: Boundary detection by minimizing functionals. In: Ullman, S., Richards, W. (eds.) Image Understanding, pp. 19–43. Springer, Berlin (1989)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)
Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms: application to the processing of outliers. SIAM J. Numer. Anal. 40, 965–994 (2002)
Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20, 99–120 (2004)
Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation based image restoration. SIAM MMS 4, 460–489 (2005)
Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2002)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. J. Comput. Phys. 79, 12–49 (1988)
Rudin, L., Osher, S.: Total variation based image restoration with free local constraints. In: Proceedings of IEEE International Conference on Image Processing, Austin, vol. 1, pp. 31–35 (1994)
Rudin, L.I., Osher, S., Fatemi, E.: Non linear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Samson, C., Blanc-Féraud, L., Aubert, G., Zerubia, J.: Multiphase evolution and variational image classification. Technical report 3662, INRIA Sophia Antipolis (1999)
Sethian, J.A.: Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge Monograph on Applied and Computational Mathematics, Cambridge, United Kingdom, University Press, Cambridge (1996)
Sethian, J.A.: Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999)
Shah, J.: A common framework for curve evolution, segmentation and anisotropic diffusion. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, San Francisco, pp. 136–142 (1996)
Tamanini, I.: Optimal approximation by piecewise constant functions. In: Serapioni R., Tomarelli F. (eds.) Variational Methods for Discontinuous Structures: Applications to Image Segmentation, Continuum Mechanics, Homogenization, Villa Olmo, Como, 8–10 September 1994. Progress in Nonlinear Differential Equations and Their Applications, vol. 25, pp. 73–85. Birkhäuser, Basel (1996)
Tamanini, I., Congedo, G.: Optimal segmentation of unbounded functions. Rend. Sem. Mat. Univ. Padova 95, 153–174 (1996)
Tikhonov, A.N., Arsenin, V.: Solutions of Ill-Posed Problems. Winston, Washington (1977)
Tsai, A., Yezzi, A., Willsky, A.: Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. Image Process. 10(8), 1169–1186 (2001)
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), 271–293 (2002)
Vogel, C.R., Oman, M.E.: Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7, 813–824 (1998)
Weisstein, E.W.: Minimal residual method. MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/MinimalResidualMethod.html
You, Y., Kaveh, M.: A regularization approach to joint blur identification and image restoration. IEEE Trans. Image Process. 5, 416–428 (1996)
Zhao, H.K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127, 179–195 (1996)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this entry
Cite this entry
Bar, L. et al. (2015). Mumford and Shah Model and Its Applications to Image Segmentation and Image Restoration. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0790-8_25
Download citation
DOI: https://doi.org/10.1007/978-1-4939-0790-8_25
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-0789-2
Online ISBN: 978-1-4939-0790-8
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering