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Topological numbers and their use to characterize simple points for 2D binary images

Christophe Lohou

TL;DR

The number of possible configurations corresponding to a simple point represents the maximum limit of local configurations that a thinning algorithm operating by parallel deletion of simple (individual) points may delete while preserving topology while preserving topology.

Abstract

In this paper, we adapt the two topological numbers, which have been proposed to efficiently characterize simple points in specific neighborhoods for 3D binary images, to the case of 2D binary images. Unlike the 3D case, we only use a single neighborhood to define these two topological numbers for the 2D case. Then, we characterize simple points either by using the two topological numbers or by a single topological number linked to another one condition. We compare the characterization of simple points by topological numbers with two other ones based on Hilditch crossing number and Yokoi number. We also highlight the number of possible configurations corresponding to a simple point, which also represents the maximum limit of local configurations that a thinning algorithm operating by parallel deletion of simple (individual) points may delete while preserving topology (limit usually not reachable, depending on the deletion strategy).

Topological numbers and their use to characterize simple points for 2D binary images

TL;DR

The number of possible configurations corresponding to a simple point represents the maximum limit of local configurations that a thinning algorithm operating by parallel deletion of simple (individual) points may delete while preserving topology while preserving topology.

Abstract

In this paper, we adapt the two topological numbers, which have been proposed to efficiently characterize simple points in specific neighborhoods for 3D binary images, to the case of 2D binary images. Unlike the 3D case, we only use a single neighborhood to define these two topological numbers for the 2D case. Then, we characterize simple points either by using the two topological numbers or by a single topological number linked to another one condition. We compare the characterization of simple points by topological numbers with two other ones based on Hilditch crossing number and Yokoi number. We also highlight the number of possible configurations corresponding to a simple point, which also represents the maximum limit of local configurations that a thinning algorithm operating by parallel deletion of simple (individual) points may delete while preserving topology (limit usually not reachable, depending on the deletion strategy).

Paper Structure

This paper contains 10 sections, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Topology preservation, simple points
  4. Topological numbers
  5. Local characterization of simple points by the computing of topological numbers
  6. More efficient characterization of simple points with topological numbers
  7. Comparisons
  8. Hilditch crossing number
  9. Yokoi number
  10. Conclusion

Figures (4)

  • Figure 1: (a) A $2$D binary image, (b) a corresponding mapping to $\mathcal{Z}^2$.
  • Figure 2: (a) $N_4(x)$, (b) $N_8(x)$, (c) $4$-neighbors of $x$, (d) $8$-neighbors of $x$.
  • Figure 3: (a) $x$ is $4$-simple for $X$ and is not $8$-simple for $X$, (b) $x$ is both $4$-simple for $X$ and $8$-simple for $X$, (c) $x$ is $8$-simple for $X$ and is not $4$-simple for $X$.
  • Figure 4: (a) $x$ is not $n$-simple for $X$. It is not possible to locally (in $N^*_8(x)$) determine the reason why $x$ is not $n$-simple for $X$: (b) the deletion of $x$ will break the object $X$ onto two $n$-connected components, or (c) will merge two $\overline{n}$-connected components of $\overline{X}$.