Rocio Gonzalez-Diaz

In Journal of Combinatorial Optimization

Abstract In this paper, we define a new flavour of well-composedness, called strong Euler well-composedness. In the general setting of regular cell complexes, a regular cell complex of dimension $n$ is strongly Euler well-composed if the Euler characteristic of the link of each boundary cell is $1$, which is the Euler characteristic of an $(n-1)$-dimensional ball. Working in the particular setting of cubical complexes canonically associated with $n$-D pictures, we formally prove in this paper that strong Euler well-composedness implies digital well-composedness in any dimension $n\geq 2$ and that the converse is not true when $n\geq 4$.

Continuous well-composedness implies digital well-composedness in $n$-D

By Nicolas Boutry, Rocio Gonzalez-Diaz, Laurent Najman, Thierry Géraud

2021-11-09

In Journal of Mathematical Imaging and Vision

Abstract In this paper, we prove that when a $n$-D cubical set is continuously well-composed (CWC), that is, when the boundary of its continuous analog is a topological $(n-1)$-manifold, then it is digitally well-composed (DWC), which means that it does not contain any critical configuration. We prove this result thanks to local homology. This paper is the sequel of a previous paper where we proved that DWCness does not imply CWCness in 4D.

A 4D counter-example showing that DWCness does not imply CWCness in $n$-D

By Nicolas Boutry, Rocio Gonzalez-Diaz, Laurent Najman, Thierry Géraud

2020-07-21

In Combinatorial image analysis: Proceedings of the 20th international workshop, IWCIA 2020, novi sad, serbia, july 16–18, 2020

Abstract In this paper, we prove that the two flavours of well-composedness called Continuous Well-Composedness (shortly CWCness), stating that the boundary of the continuous analog of a discrete set is a manifold, and Digital Well-Composedness (shortly DWCness), stating that a discrete set does not contain any critical configuration, are not equivalent in dimension 4. To prove this, we exhibit the example of a configuration of 8 tesseracts (4D cubes) sharing a common corner (vertex), which is DWC but not CWC.

Euler well-composedness

By Nicolas Boutry, Rocio Gonzalez-Diaz, Maria-Jose Jimenez, Eduardo Paluzo-Hildago

2020-07-21

In Combinatorial image analysis: Proceedings of the 20th international workshop, IWCIA 2020, novi sad, serbia, july 16–18, 2020

Abstract In this paper, we define a new flavour of well-composedness, called Euler well-composedness, in the general setting of regular cell complexes: A regular cell complex is Euler well-composed if the Euler characteristic of the link of each boundary vertex is $1$. A cell decomposition of a picture $I$ is a pair of regular cell complexes $\big(K(I),K(\bar{I})\big)$ such that $K(I)$ (resp. $K(\bar{I})$) is a topological and geometrical model representing $I$ (resp.

One more step towards well-composedness of cell complexes over $n$-D pictures

By Nicolas Boutry, Rocio Gonzalez-Diaz, Maria-Jose Jimenez

2019-06-18

In Proceedings of the 21st international conference on discrete geometry for computer imagery (DGCI)

Abstract An $n$-D pure regular cell complex $K$ is weakly well-composed (wWC) if, for each vertex $v$ of $K$, the set of $n$-cells incident to $v$ is face-connected. In previous work we proved that if an $n$-D picture $I$ is digitally well composed (DWC) then the cubical complex $Q(I)$ associated to $I$ is wWC. If $I$ is not DWC, we proposed a combinatorial algorithm to locally repair $Q(I)$ obtaining an $n$-D pure simplicial complex $P_S(I)$ homotopy equivalent to $Q(I)$ which is always wWC.

Weakly well-composed cell complexes over $n$D pictures

By Nicolas Boutry, Rocio Gonzalez-Diaz, Maria-Jose Jimenez

0001-01-01

In Information Sciences

Abstract In previous work we proposed a combinatorial algorithm to “locally repair” the cubical complex $Q(I)$ that is canonically associated with a given 3D picture I. The algorithm constructs a 3D polyhedral complex $P(I)$ which is homotopy equivalent to $Q(I)$ and whose boundary surface is a 2D manifold. A polyhedral complex satisfying these properties is called well-composed. In the present paper we extend these results to higher dimensions. We prove that for a given $n$-dimensional picture the obtained cell complex is well-composed in a weaker sense but is still homotopy equivalent to the initial cubical complex.

Strong Euler wellcomposedness

Continuous well-composedness implies digital well-composedness in $n$-D

A 4D counter-example showing that DWCness does not imply CWCness in $n$-D

Euler well-composedness

One more step towards well-composedness of cell complexes over $n$-D pictures

Weakly well-composed cell complexes over $n$D pictures

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