Cech homology for shape recognition in the presence of occlusions

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.