Topological analysis and characterization of discrete scalar fields

In this paper, we address the problem of analyzing the topology of discrete scalar fields defined on triangulated domains. To this aim, we introduce the notions of discrete gradient vector field and of Smalelike decomposition for the domain of a d-dimensional scalar field. We use such notions to extract the most relevant features representing the topology of the field. We describe a decomposition algorithm, which is independent of the dimension of the scalar field, and, based on it, methods for extracting the critical net of a scalar field. A complete classification of the critical points of a 2-dimensional field that corresponds to a piecewise differentiable field is also presented.