Spectral Domain Sampling of Graph Signals

Abstract

Sampling methods for graph signals in the graph spectral domain are presented. Though the conventional sampling of graph signals can be regarded as sampling in the graph vertex domain, it does not have the desired characteristics in regard to the graph spectral domain. With the proposed methods, the down-and upsampled graph signals inherit the frequency-domain characteristics of the sampled signals defined in the time/spatial domain. The properties of the sampling effects were evaluated theoretically in comparison with those obtained with the conventional sampling method in the vertex domain. Various examples of signals on simple graphs enable precise understanding of the problem considered. Fractional sampling and Laplacian pyramid representation of graph signals are potential applications of these methods.

Topics in Graph Signal Processing: Convolution and Modulation

“…While this signal contains 0s and is a sampled signal by a factor of K = 2, it has no clear relation with the original x = [−1.992, .93, −.314, −.577] T . This further confirms that what we show in (64) explains why the method in [46] given by (62) only works for ring graphs and DSP and does not generalize to arbitrary graphs.…”

“…Our work does provide an interpretation of the sampling method in (62). In our framework, the sampling method (without decimation) in (62) can be interpreted using (24) Thus, the sampling method proposed in [46] can be written as P(M) · x = DFT Λ * DFT H · x. Taking the GFT −1 yields…”

“…In (64), the GFT −1 and the DFT do not cancel except when the graph is the ring graph. This explains why the sampling method proposed in [46] does not generalize to arbitrary graphs, working only for ring graphs and DSP. There is no vertex domain interpretation of (62) for an arbitrary graph, only for DSP.…”

“…The third is the P(M) · x proposed in this paper. The fourth is based on the spectral domain sampling method (62) in [46]. Our proposed P(M) · x from (37) matches the sampled signal perfectly in both the vertex and spectral domains.…”

“…These references address down-and up-spectral and vertex sampling, perfect, robust, greedy reconstruction, vertex domain eigenvector free sampling, interpolation of graph signals, sampling set selection, a probabilistic interpretation or a distance-based formulation of sampling, use graph sampling to solve sensor position selection, critical sampling for wavelet filterbanks, sampling of graph signals through successive local aggregations, uncertainty principles, among many other topics. Of particular relevance to our sampling work are [31,37,46], arXiv:1912.06762v1 [eess.SP] 14 Dec 2019 and we will discuss relations to these in section VIII. We emphasize that our main contribution is to provide a GSP sampling framework that parallels that of DSP sampling and showing the exact duality between vertex domain and spectral graph domain sampling, addressing the precise meaning of statements like "vertex domain sampling cannot inherit the desired characteristics of the sampling in the graph frequency domain [.]"…”