Maria-Jose Jimenez

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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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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