A Random Walk on Image Patches

作者：Taylor Kye M^*; Meyer Francois G

来源：SIAM Journal on Imaging Sciences, 2012, 5(2): 688-725.

DOI：10.1137/110839370

• image patches
• diffusion maps
• Laplacian eigenmaps
• graph Laplacian
• commute time

In this paper we attempt to understand the success of algorithms that organize patches according to graph-based metrics. Algorithms that analyze patches extracted from images or time series have led to state-of-the art techniques for classification, denoising, and the study of nonlinear dynamics. The main contribution of this work is to provide a theoretical explanation for the above experimental observations. Our approach relies on a detailed analysis of the commute time metric on prototypical graph models that epitomize the geometry observed in general patch-graphs. We prove that a parametrization of the graph based on commute times shrinks the mutual distances between patches that correspond to rapid local changes in the signal, while the distances between patches that correspond to slow local changes expand. In effect, our results explain why the parametrization of the set of patches based on the eigenfunctions of the Laplacian can concentrate patches that correspond to rapid local changes, which would otherwise be scattered in the space of patches. While our results are based on a large sample analysis, numerical experiments on synthetic and real data indicate that the results hold for datasets that are very small in practice.

• 出版日期2012