Digital Imaging: A Unified Topological Framework

Abstract

In this article, a tractable modus operandi is proposed to model a (binary) digital image (i.e., an image defined on Z n and equipped with a standard pair of adjacencies) as an image defined in the space (F n ) of cubical complexes. In particular, it is shown that all the standard pairs of adjacencies (namely the (4, 8) and (8, 4)-adjacencies in Z 2 , the (6, 18), (18, 6), (6, 26), and (26, 6)-adjacencies in Z 3 , and more generally the (2n, 3 n − 1) and (3 n − 1, 2n)-adjacencies in Z n ) can then be correctly modelled in F n . Moreover, it is established that the digital fundamental group of a digital image in Z n is isomorphic to the fundamental group of its corresponding image in F n , thus proving the topological correctness of the proposed approach. From these results, it becomes possible to establish links between topology-oriented methods developed either in classical digital spaces (Z n ) or cubical complexes (F n ), and to potentially unify some of them.

How to Make nD Functions Digitally Well-Composed in a Self-dual Way

“…The following proposition, which could also be derived from [11], follows easily. Since we have ∀ V ⊂ V, span(V ) = min(V ), max(V ) , whatever an nD function u, the interval-valued map I span (u) is such as I span (u) = I min (u) and I span (u) = I max (u).…”

A Tutorial on Well-Composedness

“…Effectively, Mazo [117] developed a method able to interpolate any image in n-D into a DWC one [26], based on the connectivity functions [117] where ε = 1 corresponds to the max interpolation and ε = −1 corresponds to the min interpolation. Even if this method is initially developed for binary images defined on Khalimsky grids, its extension to Z n and to gray level images is well known and frequently used.…”

“…Initially, Géraud and Najman [63,127] proposed in 2013 two dual ways to obtain a continuous representation (see Figure 43) of a given signal (such as a binary or a gray-level image) on H n : they used first the DWC min/max interpolations u min and u max of Mazo [117], and then they computed their span-based immersions on H n , to obtain respectively U min and U max .…”

Geometric Preservation of 2D Digital Objects Under Rigid Motions

“…unit squares defined as the Voronoi cells of the points of X within R 2 . Based on these different models, the structure of X can be defined in various topological frameworks which are mainly equivalent [10] to that of digital topology [11]. However, this digital topology of X is often non-coherent with the continuous topology of X .…”