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Topology plays an important role in computer vision by capturing the structure of the objects. Nevertheless, its potential applications have not been sufficiently developed yet. In this paper, we combine the topological properties of an image with hierarchical approaches to build a topology preserving irregular image pyramid (TIIP). The TIIP algorithm uses combinatorial maps as data structure which implicitly capture the structure of the image in terms of the critical points. Thus, we can achieve a compact representation of an image, preserving the structure and topology of its critical points (maxima, the minima and the saddles). The parallel algorithmic complexity of building the pyramid is O(log d) where d is the diameter of the largest object. We achieve promising results for image reconstruction using only a few color values and the structure of the image, although preserving fine details including the texture of the image.

In this paper we provide a new characterization of cell decomposition (called slope complex) of a given 2-dimensional continuous surface. Each patch (cell) in the decomposition must satisfy that there exists a monotonic path for any two points in the cell. We prove that any triangulation of such surface is a slope complex and explain how to obtain new slope complexes with a smaller number of slope regions decomposing the surface. We give the minimal number of slope regions by counting certain bounding edges of a triangulation of the surface obtained from its critical points.