A Convex Relaxation of the Ambrosio-Tortorelli Elliptic Functionals for the Mumford-Shah Functional

In this paper, we revisit the phase-field approximation of Ambrosio and Tortorelli for the Mumford -- Shah functional. We then propose a convex relaxation for it to attempt to compute globally optimal solutions rather than solving the nonconvex functional directly, which is the main contribution of this paper. Inspired by McCormick's seminal work on factorable nonconvex problems, we split a nonconvex product term that appears in the Ambrosio -- Tortorelli elliptic functionals in a way that a typical alternating gradient method guarantees a globally optimal solution without completely removing coupling effects. Furthermore, not only do we provide a fruitful analysis of the proposed relaxation but also demonstrate the capacity of our relaxation in numerous experiments that show convincing results compared to a naive extension of the McCormick relaxation and its quadratic variant. Indeed, we believe the proposed relaxation and the idea behind would open up a possibility for convexifying a new class of functions in the context of energy minimization for computer vision.

[1]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[2]  Jean-Michel Morel,et al.  A variational method in image segmentation: Existence and approximation results , 1992 .

[3]  Massimo Fornasier,et al.  Theoretical Foundations and Numerical Methods for Sparse Recovery , 2010, Radon Series on Computational and Applied Mathematics.

[4]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[5]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[6]  G. David,et al.  Singular Sets of Minimizers for the Mumford-Shah Functional , 2005 .

[7]  T. Chan,et al.  Fast dual minimization of the vectorial total variation norm and applications to color image processing , 2008 .

[8]  Daniel Cremers,et al.  An algorithm for minimizing the Mumford-Shah functional , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[9]  Massimo Gobbino.  Finite Difference Approximation of the Mumford-Shah Functional , 1998 .

[10]  T. O’Neil.  Geometric Measure Theory , 2002 .

[11]  L. Ambrosio,et al.  Approximation of functional depending on jumps by elliptic functional via t-convergence , 1990 .

[12]  G. Bouchitté,et al.  The calibration method for the Mumford-Shah functional and free-discontinuity problems , 2001, math/0105013.

[13]  Tony F. Chan,et al.  Color TV: total variation methods for restoration of vector-valued images , 1998, IEEE Trans. Image Process..

[14]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[15]  Antonin Chambolle,et al.  Image Segmentation by Variational Methods: Mumford and Shah Functional and the Discrete Approximations , 1995, SIAM J. Appl. Math..

[16]  A. Chambolle,et al.  An introduction to Total Variation for Image Analysis , 2009 .

[17]  E. D. Giorgi,et al.  Existence theorem for a minimum problem with free discontinuity set , 1989 .

[18]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[19]  Andrea Braides.  Approximation of Free-Discontinuity Problems , 1998 .

[20]  Jean-Michel Morel,et al.  Variational methods in image segmentation , 1995 .

[21]  Daniel Cremers,et al.  The Natural Vectorial Total Variation Which Arises from Geometric Measure Theory , 2012, SIAM J. Imaging Sci..

[22]  Lawrence Evans.  Four important linear partial differential equations , 2010 .

[23]  Christopher V. Alvino,et al.  The Piecewise Smooth Mumford–Shah Functional on an Arbitrary Graph , 2009, IEEE Transactions on Image Processing.

[24]  Daniel Cremers,et al.  A convex representation for the vectorial Mumford-Shah functional , 2012, 2012 IEEE Conference on Computer Vision and Pattern Recognition.

[25]  Paul I. Barton,et al.  McCormick-Based Relaxations of Algorithms , 2009, SIAM J. Optim..

[26]  Andrea Braides,et al.  Non-local approximation of the Mumford-Shah functional , 1997 .