Reconstructing shapes with guarantees by unions of convex sets

A simple way to reconstruct a shape <i>A</i> from a sample <i>P</i> is to output an <i>r</i>-offset <i>P</i> + <i>r B</i>, where <i>B</i> = {<i>x</i> ∈ <b>R</b><sup><i>N</i></sup> <i>x</i> ≤ 1} designates the unit Euclidean ball centered at the origin. Recently, it has been proved that the output <i>P</i> + <i>r B</i> is homotopy equivalent to the shape <i>A</i>, for a dense enough sample <i>P</i> of <i>A</i> and for a suitable value of the parameter <i>r</i>. In this paper, we extend this result and find convex sets <i>C</i> ⊂ <b>R</b><sup><i>N</i></sup>, besides the unit Euclidean ball <i>B</i>, for which <i>P</i> + <i>rC</i> reconstructs the topology of <i>A</i>. This class of convex sets includes in particular <i>N</i>-dimensional cubes in <b>R</b><sup><i>N</i></sup>. We proceed in two steps. First, we establish the result when <i>P</i> is an ε-offset of <i>A</i>. Building on this first result, we then consider the case when <i>P</i> is a finite noisy sample of <i>A</i>.

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