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Computing Homology within Image Context

Object class invariants play a key role in computer imagery, and more specifically in image analysis and geometric modeling. Computing and representing topological information (neighborhood, connectedness, orientation, etc.) form an important part in applications such as image classification, indexing, shape description, shape recognition. Geometric modeling applications also take topological criteria into account to ensure the reliability of construction or to control the result of construction operations. Homology is an algorithmically computable topological invariant that characterizes an object by its "holes". The notion of "hole" is defined in any dimension. Informally “holes” of a 3D-object are its connected components in dimension 0, its tunnels in dimension 1, its cavities in dimension 2. This project deals with the computation of homological information (homology groups and their generators) of objects contained in images, and its use for image applications.

We plan to develop a theoretical and practical framework for efficiently extracting “meaningful” homology information in the context of computer imagery. To achieve this goal, we intend to combine known techniques in algebraic topology, discrete geometry and computational geometry in order to develop new homology based algorithms for computer imagery. One challenge, and originality, of the project will be to acquire a better understanding of the behavior of homology information on structures and under operators used in computer imagery. The results of this study will be used both to reduce the complexity of computing homology groups of image objects and to determine the relevance of homology elements depending on the application context. Our research will be led along the following topics: stability of generators under image operations, homological classification of images, specificity of different combinatorial structures, and efficient computation of homology information, and to succeed we will address the following questions:

  • How stable are homology generators under different kinds of perturbations (noise, data distortion…), or transformations (fusion of objects, cutting “holes”…)?

  • Is it possible to deduce homological information of an object from some of its projections or cuts?

  • Which combinatorial structures are well suited for efficient homology computation?

  • Is there a notion of “adjacency” for classes of objects, defined by their generators?

  • Given an application (video tracking, object categorization…), is it possible to determine a well-suited set of homology generators?

This research is based on the complementary scientific expertise of the partners. PRIP (Vienna, Austria), SIC (Poitiers, France), and LAIC (Clermont, France) have already shown their interest and competences through publications dealing with the computation of topological invariants in digital imagery. Moreover the advanced theoretical background that is needed in this project belongs to the area of expertise of LMA (Poitiers, France). A joint project between the groups PRIP, SIC, LAIC and LMA is planned.