A new graph-theoretic approach to clustering and segmentation

We develop a framework for the image segmentation problem based on a new graph-theoretic formulation of clustering. The approach is motivated by the analogies between the intuitive concept of a cluster and that of a dominant set of vertices, a notion that generalizes that of a maximal complete subgraph to edge-weighted graphs. We also establish a correspondence between dominant sets and the extrema of a quadratic form over the standard simplex, thereby allowing us the use of continuous optimization techniques such as replicator dynamics from evolutionary game theory. Such systems are attractive as they can be coded in a few lines of any high-level programming language, can easily be implemented in a parallel network of locally interacting units, and offer the advantage of biological plausibility. We present experimental results on real-world images which show the effectiveness of the proposed approach.

[1]  Calvin C. Gotlieb,et al.  Semantic Clustering of Index Terms , 1968, J. ACM.

[2]  Kim L. Boyer,et al.  Quantitative Measures of Change Based on Feature Organization: Eigenvalues and Eigenvectors , 1998, Comput. Vis. Image Underst..

[3]  Jörgen W. Weibull,et al.  Evolutionary Game Theory , 1996 .

[4]  Panos M. Pardalos,et al.  Continuous Characterizations of the Maximum Clique Problem , 1997, Math. Oper. Res..

[5]  Jean Ponce,et al.  Computer Vision: A Modern Approach , 2002 .

[6]  Steven W. Zucker,et al.  Two Stages of Curve Detection Suggest Two Styles of Visual Computation , 1989, Neural Computation.

[7]  R. Punnett,et al.  The Genetical Theory of Natural Selection , 1930, Nature.

[8]  Anil K. Jain,et al.  Algorithms for Clustering Data , 1988 .

[9]  Dana H. Ballard,et al.  Computer Vision , 1982 .

[10]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[11]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[12]  Michael Werman,et al.  Self-Organization in Vision: Stochastic Clustering for Image Segmentation, Perceptual Grouping, and Image Database Organization , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Jack Minker,et al.  An Analysis of Some Graph Theoretical Cluster Techniques , 1970, JACM.

[14]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[15]  Azriel Rosenfeld,et al.  Scene Labeling by Relaxation Operations , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[16]  Marcello Pelillo,et al.  The Dynamics of Nonlinear Relaxation Labeling Processes , 1997, Journal of Mathematical Imaging and Vision.

[17]  Daniel P. Huttenlocher,et al.  Efficient Graph-Based Image Segmentation , 2004, International Journal of Computer Vision.

[18]  David G. Luenberger,et al.  Linear and Nonlinear Programming: Second Edition , 2003 .

[19]  M. Nowak,et al.  Evolutionary game theory , 1995, Current Biology.

[20]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[21]  A. Jagota,et al.  Feasible and infeasible maxima in a quadratic program for maximum clique , 1996 .

[22]  Richard M. Leahy,et al.  An Optimal Graph Theoretic Approach to Data Clustering: Theory and Its Application to Image Segmentation , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  E. Akin,et al.  Dynamics of games and genes: Discrete versus continuous time , 1983 .

[24]  L. Baum,et al.  An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology , 1967 .

[25]  Pietro Perona,et al.  A Factorization Approach to Grouping , 1998, ECCV.

[26]  Charles T. Zahn,et al.  and Describing GestaltClusters , 1971 .

[27]  Vijay V. Raghavan,et al.  A Comparison of the Stability Characteristics of Some Graph Theoretic Clustering Methods , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.