Estimating graph topology from sparse graph signals with an application to image denoising

Graph signal processing is a framework that allows us to work with general unstructured discrete data that cannot be handled with classical Discrete signal processing. The underlying graph topology plays a crucial role in determining the definition of Fourier transform on graphs. Graph topology for a given graph signal is not always available and may also not be unique. In this paper we address the problem of estimating graph topology from signals that are sparse in the frequency domain. We estimate the graph Laplacian matrix in an optimization framework that minimizes errors in relations known to exist between the graph signals and their Fourier transforms. We also propose to use this algorithm for adapting an existing graph based non-local image denoising algorithm, which is known to perform well only for piece-wise smooth images. We provide results on natural, texture and smooth images that support our claim that with our topology estimation algorithm the denoising algorithm is able to adapt to different image structures. We compare our results with the graph based non-local method and the state-of-art BM3D algorithm, using different performance measures.