Rocio Gonzalez-Diaz

Strong Euler wellcomposedness

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

2021-11-23

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 $n\geq 4$.

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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.

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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)

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. This result is surprising since we know that CWCness and DWCness are equivalent in 2D and 3D. To reach our goal, we use local homology.

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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)

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. its complementary, $\bar{I}$). Then, a cell decomposition of a picture $I$ is self-dual Euler well-composed if both $K(I)$ and $K(\bar{I})$ are Euler well-composed. We prove in this paper that, first, self-dual Euler well-composedness is equivalent to digital well-composedness in dimension 2 and 3, and second, in dimension 4, self-dual Euler well-composedness implies digital well-composedness, though the converse is not true.

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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. In this paper we give a combinatorial procedure to compute a simplicial complex $P_S(\bar{I})$ which decomposes the complement space of $|P_S(I)|$ and prove that $P_S(\bar{I})$ is also wWC. This paper means one more step on the way to our ultimate goal: to prove that the $n$-D repaired complex is continuously well-composed (CWC), that is, the boundary of its continuous analog is an $(n-1)$-manifold.

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

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

2018-07-04

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.

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