Homological Tree-Based Strategies for Image Analysis

Abstract

Homological characteristics of digital objects can be obtained in a straightforward manner computing an algebraic map φ over a finite cell complex K (with coefficients in the finite field $\textbf{F}_2=\{0,1\}$) which represents the digital object [9]. Computable homological information includes the Euler characteristic, homology generators and representative cycles, higher (co)homology operations, etc. This algebraic map φ is described in combinatorial terms using a mixed three-level forest. Different strategies changing only two parameters of this algorithm for computing φ are presented. Each one of those strategies gives rise to different maps, although all of them provides the same homological information for K. For example, tree-based structures useful in image analysis like topological skeletons and pyramids can be obtained as subgraphs of this forest.

This work has been partially supported by PAICYT research project FQM-296, “Andalusian research project” PO6-TIC-02268, Spanish MEC project MTM2006-03722 and the Austrian Science Fund under grant P20134-N13.