New Characterizations of Simple Points in 2D, 3D, and 4D Discrete Spaces

Abstract

Abstract

Morphological filtering on graphs

“…Simplicial (and cubical) complexes (see [4,5] for image operators defined in cubical complexes and [18] for examples of morphological operators in 2D simplicial complexes) allows topological properties of discrete objects to be better handled than with graphs. These topological structures extend graphs to higher dimensions in the sense that a graph is a 1-D structure made of points and edges considered as 0D and 1D elements.…”

“…On the other hand, there is a growing interest for considering digital objects not only composed of points but also composed of elements lying between them and carrying structural information about how the points are glued together (see [6,4,18,31,10,2] for recent examples). The simplest of these representations are the graphs.…”

Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

“…Transformations that pre- serve all topological characteristics, known as topologypreserving transformations, are used in many applications of image analysis. Homotopic skeletonization [31,19] is the best known and most used transformation of this kind, with many applications both in 2D and 3D. In particular, skeletons are often used as a simplification of the original data, which facilitates shape recognition, registration, or animation.…”

Paths, Homotopy and Reduction in Digital Images

“…All these topological models have found practical applications in the context of digital image analysis, especially for the definition of "topology-preserving" procedures (i.e., procedures enabling to modify a binary digital image without altering its homotopy type), including reduction ones (used for skeletonisation or segmentation), see e.g. [8].…”