Computing Homology Group Generators of Images Using Irregular Graph Pyramids

Conference object English OPEN
Peltier , Samuel ; Ion , Adrian ; Haxhimusa , Yll ; Kropatsch , Walter ; Damiand , Guillaume (2007)
  • Publisher: Springer Berlin / Heidelberg
  • Related identifiers: doi: 10.1007/978-3-540-72903-7_26
  • Subject: [ INFO.INFO-DS ] Computer Science [cs]/Data Structures and Algorithms [cs.DS]

International audience; We introduce a method for computing homology groups and their generators of a 2D image, using a hierarchical structure i.e. irregular graph pyramid. Starting from an image, a hierarchy of the image is built, by two operations that preserve homology of each region. Instead of computing homology generators in the base where the number of entities (cells) is large, we first reduce the number of cells by a graph pyramid. Then homology generators are computed efficiently on the top level of the pyramid, since the number of cells is small, and a top down process is then used to deduce homology generators in any level of the pyramid, including the base level i.e. the initial image. We show that the new method produces valid homology generators and present some experimental results.
  • References (21)
    21 references, page 1 of 3

    1. Kovalevsky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics, and Image Processing 46 (1989) 141-161

    2. Kovalevsky, V.A.: Digital Geometry Based on the Topology of Abstract Cellular Complexes. In Chassery, J.M., Francon, J., Montanvert, A., RĀ“eveill`es, J.P., eds.: GĀ“eometrie Discr`ete en Imagery, Fondements et Applications, Strasbourg, France (1993) 259-284

    3. Agoston, M.K.: Algebraic Topology, a first course. Pure and applied mathematics. Marcel Dekker Ed. (1976)

    4. Allili, M., Mischaikow, K., Tannenbaum, A.: Cubical homology and the topological classification of 2d and 3d imagery. In: Proceedings of International Conference Image Processing. Volume 2. (2001) 173-176

    5. Sonka, M., Hlavac, V., Boyle, R.: Image Processing, Analysis and Machine Vision. Brooks/Cole Publishing Company (1999)

    6. Kaczynksi, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer (2004)

    7. Niethammer, M., Stein, A.N., Kalies, W.D., Pilarczyk, P., Mischaikow, K., Tannenbaum, A.: Analysis of blood vessels topology by cubical homology. In: Proceedings of International Conference Image Processing. Volume 2. (2002) 969-972

    8. Damiand, G., Peltier, S., Fuchs, L.: Computing homology for surfaces with generalized maps: Application to 3d images. In: Procedings of 2nd International Symposium on Visual Computing. Volume 4292 of LNCS., Lake Tahoe, Nevada, USA, Springer-verlag (2006) 1151-1160

    9. Munkres, J.R.: Elements of algebraic topology. Perseus Books (1984)

    10. Jolion, J.M., Rosenfeld, A.: A Pyramid Framework for Early Vision. Kluwer (1994)

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