On the Space Between Critical Points

González-Dı́az

Lecture Notes in Computer Science

2021

Self Cite

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.

Section: Motivation and Contributionmentioning

confidence: 99%

Image = Structure + Few Colors

González-Dı́az

Lecture Notes in Computer Science

2021

Self Cite

“…In [6,7] Edelsbrunner et al propose an algorithm of constructing a hierarchy of increasingly coarse Morse-Smale complexes to decompose a piecewise linear 2D-manifold with all its critical points being distinct. In our previous research work [8,9], we further generalize this concept beyond Morse-Smale complex and present a new hierarchy of increasingly coarse complexes decomposing 2D continuous surfaces denoted as slope complexes.…”

Section: Introductionmentioning

confidence: 99%

“…The main contribution of this paper is to provide orientation to the dual graph. A first step in this direction was done in [9] but here we provide it's interpretation in the image context and give properties of the oriented dual graphs to provide a reduction technique to meet our aims.…”

Section: Introductionmentioning

confidence: 99%

Congratulations! Dual Graphs Are Now Orientated!

Graph-Based Representations in Pattern Recognition

Casablanca

et al. 2019

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A digital image can be perceived as a 2.5D surface consisting of pixel coordinates and the intensity of pixel as height of the point in the surface. Such surfaces can be efficiently represented by the pair of dual plane graphs: neighborhood (primal) graph and its dual. By defining orientation of edges in the primal graph and use of Local Binary Patters (LBPs), we can categorize the vertices corresponding to the pixel into critical (maximum, minimum, saddle) or slope points. Basic operation of contraction and removal of edges in primal graph result in configuration of graphs with different combinations of critical and non-critical points. The faces of graph resemble a slope region after restoration of the continuous surface by successive monotone cubic interpolation. In this paper, we define orientation of edges in the dual graph such that it remains consistent with the primal graph. Further we deliver the necessary and sufficient conditions for merging of two adjacent slope regions.3 There is a topological and a combinatorial isomorphism between G and G and it is a unique pair of graphs embedded in a surface [5, p. 70-80]

Computing and Reducing Slope Complexes

Casablanca

Computational Topology in Image Context

et al. 2018

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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.