Entropic metric alignment for correspondence problems

Many shape and image processing tools rely on computation of correspondences between geometric domains. Efficient methods that stably extract "soft" matches in the presence of diverse geometric structures have proven to be valuable for shape retrieval and transfer of labels or semantic information. With these applications in mind, we present an algorithm for probabilistic correspondence that optimizes an entropy-regularized Gromov-Wasserstein (GW) objective. Built upon recent developments in numerical optimal transportation, our algorithm is compact, provably convergent, and applicable to any geometric domain expressible as a metric measure matrix. We provide comprehensive experiments illustrating the convergence and applicability of our algorithm to a variety of graphics tasks. Furthermore, we expand entropic GW correspondence to a framework for other matching problems, incorporating partial distance matrices, user guidance, shape exploration, symmetry detection, and joint analysis of more than two domains. These applications expand the scope of entropic GW correspondence to major shape analysis problems and are stable to distortion and noise.

[1]  Leonidas J. Guibas,et al.  Near-Optimal Joint Object Matching via Convex Relaxation , 2014, ICML.

[2]  Ron Kimmel,et al.  Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Teofilo F. Gonzalez,et al.  P-Complete Approximation Problems , 1976, J. ACM.

[4]  Quentin Mérigot,et al.  A Multiscale Approach to Optimal Transport , 2011, Comput. Graph. Forum.

[5]  Facundo Mémoli,et al.  Spectral Gromov-Wasserstein distances for shape matching , 2009, 2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops.

[6]  Facundo Mémoli,et al.  Gromov-Hausdorff distances in Euclidean spaces , 2008, 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops.

[7]  Guillermo Sapiro,et al.  Graph Matching: Relax at Your Own Risk , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  C. Villani Topics in Optimal Transportation , 2003 .

[9]  Qi-Xing Huang,et al.  Dense Human Body Correspondences Using Convolutional Networks , 2015, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[10]  Le Song,et al.  Kernelized Sorting , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[12]  Karl-Theodor Sturm The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces , 2012, 1208.0434.

[13]  Le Song,et al.  A Hilbert Space Embedding for Distributions , 2007, Discovery Science.

[14]  Leonidas J. Guibas,et al.  Soft Maps Between Surfaces , 2012, Comput. Graph. Forum.

[15]  Éric Gaussier,et al.  Relation between PLSA and NMF and implications , 2005, SIGIR '05.

[16]  Ronen Basri,et al.  Tight relaxation of quadratic matching , 2015, SGP '15.

[17]  Remco C. Veltkamp,et al.  A Survey of Content Based 3D Shape Retrieval Methods , 2004, SMI.

[18]  Y. Aflalo,et al.  On convex relaxation of graph isomorphism , 2015, Proceedings of the National Academy of Sciences.

[19]  Andrea Tagliasacchi,et al.  High-contrast computational caustic design , 2014, ACM Trans. Graph..

[20]  J. Tropp An Alternating Minimization Algorithm for Non-negative Matrix Approximation , 2003 .

[21]  Leonidas J. Guibas,et al.  One Point Isometric Matching with the Heat Kernel , 2010, Comput. Graph. Forum.

[22]  Martial Hebert,et al.  A spectral technique for correspondence problems using pairwise constraints , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[23]  H. Urakawa Calculus of variations and harmonic maps , 1993 .

[24]  Pierre Alliez,et al.  An Optimal Transport Approach to Robust Reconstruction and Simplification of 2d Shapes , 2022 .

[25]  Mathieu Desbrun,et al.  Blue noise through optimal transport , 2012, ACM Trans. Graph..

[26]  Stephen DiVerdi,et al.  IsoMatch: Creating Informative Grid Layouts , 2015, Comput. Graph. Forum.

[27]  Ron Kimmel,et al.  Spectral multidimensional scaling , 2013, Proceedings of the National Academy of Sciences.

[28]  Leonidas J. Guibas,et al.  Shape google: Geometric words and expressions for invariant shape retrieval , 2011, TOGS.

[29]  Eranda Çela,et al.  The quadratic assignment problem : theory and algorithms , 1999 .

[30]  Vladlen Koltun,et al.  Robust Nonrigid Registration by Convex Optimization , 2015, 2015 IEEE International Conference on Computer Vision (ICCV).

[31]  Jorge Nocedal,et al.  An interior algorithm for nonlinear optimization that combines line search and trust region steps , 2006, Math. Program..

[32]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[33]  K. Kurdyka On gradients of functions definable in o-minimal structures , 1998 .

[34]  Guillermo Sapiro,et al.  A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching , 2010, International Journal of Computer Vision.

[35]  Vladimir G. Kim,et al.  Blended intrinsic maps , 2011, ACM Trans. Graph..

[36]  Remco C. Veltkamp,et al.  A survey of content based 3D shape retrieval methods , 2004, Proceedings Shape Modeling Applications, 2004..

[37]  John Wright,et al.  When Are Nonconvex Problems Not Scary? , 2015, ArXiv.

[38]  Stephen DiVerdi,et al.  Exploring collections of 3D models using fuzzy correspondences , 2012, ACM Trans. Graph..

[39]  Facundo Mémoli,et al.  Gromov–Wasserstein Distances and the Metric Approach to Object Matching , 2011, Found. Comput. Math..

[40]  Daniela Giorgi,et al.  SHape REtrieval Contest 2007: Watertight Models Track , 2007 .

[41]  Alexander M. Bronstein,et al.  Graph matching: relax or not? , 2014, ArXiv.

[42]  Vikas Singh,et al.  Solving the multi-way matching problem by permutation synchronization , 2013, NIPS.

[43]  G. Burton TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

[44]  Gert R. G. Lanckriet,et al.  On the Convergence of the Concave-Convex Procedure , 2009, NIPS.

[45]  Thomas Hofmann,et al.  Probabilistic Latent Semantic Indexing , 1999, SIGIR Forum.

[46]  Sartaj Sahni,et al.  Computationally Related Problems , 1974, SIAM J. Comput..

[47]  Facundo Mémoli,et al.  Eurographics Symposium on Point-based Graphics (2007) on the Use of Gromov-hausdorff Distances for Shape Comparison , 2022 .

[48]  Leonidas J. Guibas,et al.  Dirichlet Energy for Analysis and Synthesis of Soft Maps , 2013 .

[49]  Hédy Attouch,et al.  Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality , 2008, Math. Oper. Res..

[50]  S. Łojasiewicz Sur la géométrie semi- et sous- analytique , 1993 .

[51]  Gabriel Peyré,et al.  Iterative Bregman Projections for Regularized Transportation Problems , 2014, SIAM J. Sci. Comput..

[52]  Maks Ovsjanikov,et al.  Functional maps , 2012, ACM Trans. Graph..

[53]  M. Coste AN INTRODUCTION TO O-MINIMAL GEOMETRY , 2002 .

[54]  V. Deineko,et al.  The Quadratic Assignment Problem: Theory and Algorithms , 1998 .

[55]  Radu Ioan Bot,et al.  An inertial forward–backward algorithm for the minimization of the sum of two nonconvex functions , 2014, EURO J. Comput. Optim..

[56]  Leonidas J. Guibas,et al.  Map-based exploration of intrinsic shape differences and variability , 2013, ACM Trans. Graph..

[57]  Leonidas J. Guibas,et al.  Earth mover's distances on discrete surfaces , 2014, ACM Trans. Graph..

[58]  Abdel Nasser,et al.  A Survey of the Quadratic Assignment Problem , 2014 .

[59]  M. Gromov Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[60]  L. Bregman The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[61]  Eric Mjolsness,et al.  A Novel Optimizing Network Architecture with Applications , 1996, Neural Computation.

[62]  D. Hunter,et al.  Quantile Regression via an MM Algorithm , 2000 .

[63]  Facundo Mémoli,et al.  The Gromov-Wasserstein Distance: A Brief Overview , 2014, Axioms.

[64]  Panos M. Pardalos,et al.  Quadratic Assignment and Related Problems , 1994 .

[65]  Heinz H. Bauschke,et al.  Joint minimization with alternating Bregman proximity operators , 2005 .

[66]  Leonidas J. Guibas,et al.  Consistent Shape Maps via Semidefinite Programming , 2013, SGP '13.

[67]  Leonidas J. Guibas,et al.  The Earth Mover's Distance as a Metric for Image Retrieval , 2000, International Journal of Computer Vision.

[68]  Jitendra Malik,et al.  Shape matching and object recognition using low distortion correspondences , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[69]  Robert R. Meyer,et al.  Sufficient Conditions for the Convergence of Monotonic Mathematical Programming Algorithms , 1976, J. Comput. Syst. Sci..

[70]  Ghassan Hamarneh,et al.  A Survey on Shape Correspondence , 2011, Comput. Graph. Forum.

[71]  M. V. D. Panne,et al.  Displacement Interpolation Using Lagrangian Mass Transport , 2011 .

[72]  Alan L. Yuille,et al.  A Convergence Proof for the Softassign Quadratic Assignment Algorithm , 1996, NIPS.

[73]  Alan L. Yuille,et al.  Convergence Properties of the Softassign Quadratic Assignment Algorithm , 1999, Neural Computation.

[74]  Steven Gold,et al.  A Graduated Assignment Algorithm for Graph Matching , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[75]  Ninad Thakoor,et al.  Hidden Markov Model-Based Weighted Likelihood Discriminant for 2-D Shape Classification , 2007, IEEE Transactions on Image Processing.

[76]  Marco Cuturi,et al.  Sinkhorn Distances: Lightspeed Computation of Optimal Transportation , 2013, NIPS 2013.