Computation and application issues in multidimensional shape description
by
Daniela Giorgi
Institute of Applied Mathematics and Information Technology, National Research Council - Genova (Italy)
Coauthors: S. Biasotti (IMATI-CNR, Genova) A. Cerri (University of Bologna and ARCES) P. Frosini (University of Bologna and ARCES) C. Landi (University of Modena and Reggio Emilia)

In Computer Graphics and Vision, computational-topology offers a theoretical framework for the formalization and solution of problems related to shape analysis, description and comparison. Methods that make use of the properties of real functions defined on the shape [1] are often well suited to describe objects that are non-rigidly related to each other. The role of these functions, that we may call measuring functions, is to measure quantitative geometric properties of the shape, while taking into account its topology.

Recent advances in Size Theory have shown that it is possible to derive a concise and informative geometrical-topological shape descriptor also in the case of multi-variate measuring functions, that is functions taking values in R^k [2]. The idea of using k-dimensional measuring functions arises from the observation that the shape of an object can be better characterized by means of a set of functions, each investigating specific features of the shape under study. For instance, scientific simulations of real phenomena typically require the analysis of a huge and composite amount of data. At the same time, there are properties that are naturally k-dimensional: a first example is colour, which lives in the 3-dimensional RGB space. The possibility of working from the beginning with k-dimensional functions, instead of merging the information of k separate functions a posteriori, allows one to produce a single descriptor containing the information of the k functions at the same time. In other words, k different functions concur to produce a single descriptor.

In this talk, we will deal with the main issues related to the application in a discrete setting of the concepts introduced in Multidimensional Size Theory, in particular multidimensional size functions and multidimensional matching distances. A computational scheme coherent with the mathematical model will be given, highlighting that the technique proposed is able to deal with different model representations, such as simplicial complexes and digital spaces. Experimental results will be provided so as to illustrate the feasibility of the approach, in terms of storage space, computation time and efficacy of description. Finally, different families of measuring functions will be analyzed, in the light of their capability to capture salient shape features.

[1] S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, M. Spagnuolo: Describing shapes by geometrical-topological properties of real functions. ACM Computing Surveys, in press.

[2] S. Biasotti, A. Cerri, P. Frosini, D. Giorgi, C. Landi: Multidimensional size functions for shape comparison. Journal of Mathematical Imaging and Vision, in press (DOI 10.1007/s10851-008-0096-z).

Date received: May 22, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxd-21.