Cech homology for shape recognition in the presence of occlusions

TL;DR

This paper explores how Cech homology and size functions can be used to recognize shapes despite occlusions by analyzing how size functions change with shape overlaps, using algebraic topology tools.

Contribution

It introduces a condition relating size functions of shapes and their intersections, enabling shape recognition under occlusion using algebraic topology methods.

Findings

Size functions can detect partial shape matches.

The Mayer-Vietoris sequence helps analyze shape overlaps.

Shape recognition is robust to occlusions using these topological tools.

Abstract

In Computer Vision the ability to recognize objects in the presence of occlusions is a necessary requirement for any shape representation method. In this paper we investigate how the size function of a shape changes when a portion of the shape is occluded by another shape. More precisely, considering a set $X=A\cup B$ and a measuring function $\phi$ on $X$, we establish a condition so that $\ell_{(X,\phi)=\ell_{(A,\phi|_A)}+\ell_{(B,\phi|_B)}-\ell_{(A\cap B,\phi|_{A\cap B})}$. The main tool we use is the Mayer-Vietoris sequence of \v{C}ech homology groups. This result allows us to prove that size functions are able to detect partial matching between shapes by showing a common subset of cornerpoints.