Topology-based Digital Image Processing (I): Homological Spanning Forest representation
Author: Pedro Real (Univ. Sevilla)
In this talk, we introduce a non-unique graph-based representation for a 2D digital object O, called Homological Spanning Forest (or HSF, for short) for O, which provides in a simple way advanced topological and geometrical information about it. In order to calculate a HSF representation for a ROI O inside a digital image I, we have two options: (a) Object-based computation, that is, to directly compute a HSF from O; (b) Ambiance-based computation, starting with the computation of a HSF of the whole image, passing through an adaptation process to another HSF which "fits well with" the shape of $O$ (creating a kind of "crack" between the object and the ambiance, using trees whose nodes are 1-cells and 2-cells), and finally isolating the ROI from the background. The final graph structure obtained for O in this way is a new HSF for it. The advantage of the approach (a) is that we only deal with the cells of the object O. The advantage of the approach (b) is that it is susceptible to employ parallelism in a natural way. Several methods belonging to one of the previous approaches for computing HSF representations are specified. Simple examples showing the different notions or concepts we will exhaustively use in this theory are showed: nodes, edges, arrows, vectors, vector fields, trees, CW-complexes, boundary operator, chain homotopy operators, flow, strong deformation retracts, integral-chain complexes, computational homological algebra, homology, cohomology, relative homology,... We use pedagogical software for topogeometric analysis of 2D and 3D digital images based on HSF to make the talk more understandable.