On the Convergence of PatchMatch and Its Variants

Many problems in image/video processing and computer vision require the computation of a dense k-nearest neighbor field (k-NNF) between two images. For each patch in a query image, the k-NNF determines the positions of the k most similar patches in a database image. With the introduction of the PatchMatch algorithm, Barnes et al. demonstrated that this large search problem can be approximated efficiently by collaborative search methods that exploit the local coherency of image patches. After its introduction, several variants of the original PatchMatch algorithm have been proposed, some of them reducing the computational time by two orders of magnitude. In this work we propose a theoretical framework for the analysis of PatchMatch and its variants, and apply it to derive bounds on their covergence rate. We consider a generic PatchMatch algorithm from which most specific instances found in the literature can be derived as particular cases. We also derive more specific bounds for two of these particular cases: the original PatchMatch and Coherency Sensitive Hashing. The proposed bounds are validated by contrasting them to the convergence observed in practice.