Similarity Analysis from Limiting Quantum Walks

Similarity compression is a critical step to improve the efficiency of edge detection. In this paper, we compare two approaches for compressing/decompressing similarity matrices, being edge detection our application domain. In this regard, state-of-the-art contour detectors rely on spectral clustering where pixel or patch similarity is encoded in a symmetric weight matrix and the eigenvectors of the normalized Laplacian derived from this matrix are clustered in order to find contours (normalized cuts and its variants). Despite significant interest in learning the similarity measure for providing well localized boundaries, the underlying spectral analysis has played a subsidiary role, and has mostly been based on classical random walks and the heat kernel. However, recent findings based on continuous-time quantum walks suggest that under the complex wave equation there are long-range interactions not present in the classical case. In the case of the edge map this opens up a means of controlling texture in the edge map by a simple thresholding. In this paper, we use the long-time averages of quantum walks for edge detection, and show that texture is a consequence of short-rangedness of these interactions. This is due to the local-to-global property of limiting quantum walks. In addition, when analyzing the role of limiting quantum walks as intermediate/indirect similarity decompression, we find that quantum walks are able of recovering the original edge structure when a factorization compressor is used, whereas this is not the case when compression relies on the Szemeéredi Regularity Lemma, despite this latter method is by far more efficient.