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Pré-Publication, Document De Travail Année : 2009

Reconstructing shapes with guarantees by unions of convex sets

Résumé

A simple way to reconstruct a shape $A$ from a sample $P$ is to output an $r$-offset $P + r B$, where $B$ designates the unit Euclidean ball centered at the origin. Recently, it has been proved that the output $P + r B$ is homotopy equivalent to the shape $A$, for a dense enough sample $P$ of $A$ and for a suitable value of the parameter $r$. In this paper, we extend this result and find convex sets $C$, besides the unit Euclidean ball $B$, for which $P + r C$ reconstructs the topology of $A$. This class of convex sets includes in particular $N$-dimensional cubes in the $N$-dimensional Euclidean space. We proceed in two steps. First, we establish the result when $P$ is an $\epsilon$-offset of $A$. Building on this first result, we then consider the case when $P$ is a finite noisy sample of $A$.
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Dates et versions

hal-00427035 , version 1 (28-10-2009)
hal-00427035 , version 2 (10-12-2009)
hal-00427035 , version 3 (31-03-2010)

Identifiants

  • HAL Id : hal-00427035 , version 2

Citer

Dominique Attali, André Lieutier. Reconstructing shapes with guarantees by unions of convex sets. 2009. ⟨hal-00427035v2⟩
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