Spectral Domain Spline Graph Filter Bank

In this paper, we present a structure for two-channel spline graph filter bank with spectral sampling (SGFBSS) on arbitrary undirected graphs. Our proposed structure has many desirable properties; namely, perfect reconstruction, critical sampling in spectral domain, flexibility in choice of shape and cut-off frequency of the filters, and low complexity implementation of the synthesis section, thanks to our closed-form derivation of the synthesis filter and its sparse structure. These properties play a pivotal role in multi-scale transforms of graph signals. Additionally, this framework can use both normalized and non-normalized Laplacian of any undirected graph. We evaluate the performance of our proposed SGFBSS structure in nonlinear approximation and denoising applications through simulations. We also compare our method with the existing graph filter bank structures and show its superior performance.Comment: 5 pages, 6 figures, and one tabl

Graph Signal Processing: A Signal Representation Approach to Convolution and Sampling

The paper presents sampling in GSP as 1) linear operations (change of bases) between signal representations and 2) downsampling as linear shift invariant filtering and reconstruction (interpolation) as filtering, both in the spectral domain. To achieve this, it considers a spectral shift $M$ that leads to a spectral graph signal processing theory, $\text{GSP}_{\textrm{sp}}$, dual to GSP but that starts from the spectral domain and $M$. The paper introduces alternative signal representations, convolution of graph signals for these alternative representations, presenting a $\textit{fast}$ GSP convolution that uses the DSP FFT algorithm, and sampling as solutions of algebraic linear systems of equations.Comment: Added missing space in arXiv titl

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE