Convex optimization for multi-class image labeling with a novel family of total variation based regularizers

We introduce a linearly weighted variant of the total variation for vector fields in order to formulate regularizers for multi-class labeling problems with non-trivial interclass distances. We characterize the possible distances, show that Euclidean distances can be exactly represented, and review some methods to approximate non-Euclidean distances in order to define novel total variation based regularizers. We show that the convex relaxed problem can be efficiently optimized to a prescribed accuracy with optimality certificates using Nesterov's method, and evaluate and compare our approach on several synthetical and real-world examples.

[1]  Olga Veksler,et al.  Fast approximate energy minimization via graph cuts , 2001, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[2]  Jing Yuan,et al.  Convex Multi-class Image Labeling by Simplex-Constrained Total Variation , 2009, SSVM.

[3]  Gilbert Strang,et al.  Maximal flow through a domain , 1983, Math. Program..

[4]  C. Michelot.  A finite algorithm for finding the projection of a point onto the canonical simplex of ∝n , 1986 .

[5]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[6]  Guillermo Sapiro,et al.  Anisotropic diffusion of multivalued images with applications to color filtering , 1996, IEEE Trans. Image Process..

[7]  Richard Szeliski,et al.  A Comparative Study of Energy Minimization Methods for Markov Random Fields , 2006, ECCV.

[8]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[9]  J. Aujol,et al.  Some algorithms for total variation based image restoration , 2008 .

[10]  A. Fiacco.  A Finite Algorithm for Finding the Projection of a Point onto the Canonical Simplex of R " , 2009 .

[11]  Mila Nikolova,et al.  Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models , 2006, SIAM J. Appl. Math..

[12]  Gilles Aubert,et al.  Efficient Schemes for Total Variation Minimization Under Constraints in Image Processing , 2009, SIAM J. Sci. Comput..

[13]  P. Groenen,et al.  Modern multidimensional scaling , 1996 .

[14]  Jan-Michael Frahm,et al.  Fast Global Labeling for Real-Time Stereo Using Multiple Plane Sweeps , 2008, VMV.

[15]  J. Gower.  Properties of Euclidean and non-Euclidean distance matrices , 1985 .

[16]  Hiroshi Ishikawa,et al.  Exact Optimization for Markov Random Fields with Convex Priors , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  L. Vese,et al.  A projected gradient algorithm for color image decomposition , 2008 .

[18]  Daniel Cremers,et al.  A convex approach for computing minimal partitions , 2008 .

[19]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[20]  Éva Tardos,et al.  Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields , 2002, JACM.

[21]  Vladimir Kolmogorov,et al.  What metrics can be approximated by geo-cuts, or global optimization of length/area and flux , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[22]  Nikos Komodakis,et al.  Approximate Labeling via Graph Cuts Based on Linear Programming , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[23]  Vladimir Kolmogorov,et al.  An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision , 2001, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[24]  Charles R. Johnson,et al.  Connections between the real positive semidefinite and distance matrix completion problems , 1995 .