Honors Project On-Campus Access Only
Multiscale analysis of signals on graphs often involves the downsampling of a graph. In this paper, we provide a thorough survey of existing sampling and reconstruction theory for signals on graphs, and then empirically analyze the performance of different sampling and interpolation methods on various classes of graph signals. We present a new result that if a graph signal supported on a specific region of the graph Fourier spectrum that can be perfectly reconstructed from its values on a subset of the graph’s vertices, then a graph signal supported on the complement of the spectrum can be perfectly reconstructed from its values on the complement of the subset of vertices. We use this result to implement a critically sampled filter bank, where a fullband signal is broken into low and high pass components, which are subsampled on complementary sets of vertices. Through illustrative numerical examples, we show that the resulting dictionary atoms of this transform are jointly localized in the vertex and spectral domains, and the transform can sparsely represent piecewise-smooth signals.
Jin, Yan and Shuman, David I., "Sampling Theories for Graph Signals, with Applications to Critically Sampled Filter Banks" (2016). Mathematics, Statistics, and Computer Science Honors Projects. 48.
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