Similarity between Hypergraphs Based on Mathematical Morphology

In the framework of structural representations for applications in image understanding, we establish links between similarities, hypergraph theory and mathematical morphology. We propose new similarity measures and pseudo-metrics on lattices of hypergraphs based on morphological operators. New forms of these operators on hypergraphs are introduced as well. Some examples based on various dilations and openings on hypergraphs illustrate the relevance of our approach.

[1]  Henk J. A. M. Heijmans,et al.  The algebraic basis of mathematical morphology. I Dilations and erosions , 1990, Comput. Vis. Graph. Image Process..

[2]  C. Tappert,et al.  A Survey of Binary Similarity and Distance Measures , 2010 .

[3]  Andrew J. Davison,et al.  Active Matching , 2008, ECCV.

[4]  Alain Bretto,et al.  Hypergraph Theory: An Introduction , 2013 .

[5]  Michael H. F. Wilkinson,et al.  Mathematical Morphology and Its Application to Signal and Image Processing, 9th International Symposium, ISMM 2009, Groningen, The Netherlands, August 24-27, 2009 Proceedings , 2009, ISMM.

[6]  Jean Cousty,et al.  Some Morphological Operators on Simplicial Complex Spaces , 2011, DGCI.

[7]  Jean Cousty,et al.  Morphological filtering on graphs , 2013, Comput. Vis. Image Underst..

[8]  V. Voloshin.  Introduction to Graph and Hypergraph Theory , 2013 .

[9]  L. Vincent.  Graphs and mathematical morphology , 1989 .

[10]  Isabelle Bloch,et al.  Mathematical morphology on hypergraphs, application to similarity and positive kernel , 2013, Comput. Vis. Image Underst..

[11]  N. Loménie,et al.  Morphological Mesh Filtering and α-objects , 2013 .

[12]  Isabelle Bloch,et al.  Mathematical Morphology on Hypergraphs: Preliminary Definitions and Results , 2011, DGCI.

[13]  Henk J. A. M. Heijmans,et al.  The algebraic basis of mathematical morphology : II. Openings and closings , 1991, CVGIP Image Underst..

[14]  H. Heijmans.  Morphological image operators , 1994 .

[15]  Zheru Chi,et al.  Salient object detection using content-sensitive hypergraph representation and partitioning , 2012, Pattern Recognit..

[16]  Jean Ponce,et al.  A Tensor-Based Algorithm for High-Order Graph Matching , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  Yue Gao,et al.  3-D Object Retrieval and Recognition With Hypergraph Analysis , 2012, IEEE Transactions on Image Processing.

[18]  Trevor Coward,et al.  Nova Science Publishers , 2013 .

[19]  Abderrahim Elmoataz,et al.  Partial Difference Equations over Graphs: Morphological Processing of Arbitrary Discrete Data , 2008, ECCV.

[20]  Dan A. Simovici.  Betweenness, Metrics and Entropies in Lattices , 2008, 38th International Symposium on Multiple Valued Logic (ismvl 2008).

[21]  Salvatore Tabbone,et al.  Hypergraph-based image retrieval for graph-based representation , 2012, Pattern Recognit..

[22]  Isabelle Bloch,et al.  Mathematical Morphology , 2007, Handbook of Spatial Logics.

[23]  Edwin R. Hancock,et al.  Hypergraph based information-theoretic feature selection , 2012, Pattern Recognit. Lett..

[24]  Jean Stawiaski,et al.  Morphology on Graphs and Minimum Spanning Trees , 2009, ISMM.

[25]  Maya R. Gupta,et al.  Similarity-based Classification: Concepts and Algorithms , 2009, J. Mach. Learn. Res..

[26]  Hans-Dieter Liess,et al.  Mathematical engineering , 2009, International Conference on Instrumentation, Communication, Information Technology, and Biomedical Engineering 2009.