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, along with a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas. We then summarize recent advances 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 propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform. The sampled signal coefficients form a new graph signal, whose corresponding graph structure preserves the first-order difference of the original graph signal. For general graphs, an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise; for graphs whose graph Fourier transforms are frames with maximal robustness to erasures as well as for Erd\H{o}s-R\'enyi graphs, random sampling leads to perfect recovery with high probability. We further establish the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs. To handle full-band graph signals, we propose a graph filter bank based on sampling theory on graphs. Finally, we apply the proposed sampling theory to semi-supervised classification on online blogs and digit images, where we achieve similar or better performance with fewer labeled samples compared to previous work.Comment: To appear in IEEE T-S

We present a study of optical Fe II emission in 302 AGNs selected from the SDSS. We group the strongest Fe II multiplets into three groups according to the lower term of the transition (b 4 F , a 6 S and a 4 G terms). These correspond approximately to the blue, central, and red part respectively of the "iron shelf" around Hβ. We calculate an Fe II template which takes into account transitions into these three terms and an additional group of lines, based on a reconstruction of the spectrum of I Zw 1. This Fe II template gives a more precise fit of the Fe II lines in broad-line AGNs than other templates. We extract Fe II, Hα, Hβ, [O III] and [N II] emission parameters and investigate correlations between them. We find that Fe II lines probably originate in an Intermediate Line Region. We notice that the blue, red, and central parts of the iron shelf have different relative intensities in different objects. Their ratios depend on continuum luminosity, FWHM Hβ, the velocity shift of Fe II, and the Hα/Hβ flux ratio. We examine the dependence of the well-known anti-correlation between the equivalent widths of Fe II and [O III] on continuum luminosity. We find that there is a Baldwin effect for [O III] but an inverse Baldwin effect for the Fe II emission. The [O III]/Fe II ratio thus decreases with L λ5100 . Since the ratio is a major component of the Boroson and Green eigenvector 1, this implies a connection between the Baldwin effect and eigenvector 1, and could be connected with AGN evolution. We find that spectra are different for Hβ FWHMs greater and less than ∼3000 kms −1 , and that there are different correlation coefficients between the parameters.

Frames have been used to capture significant signal characteristics, provide numerical stability of reconstruction, and enhance resilience to additive noise. This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of frames are sometimes themselves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet. With a simple model for quantization error, it is shown that a normalized frame minimizes mean-squared error if and only if it is tight. With one coefficient erased, a tight frame is again optimal among normalized frames, both in average and worst-case scenarios. For more erasures, a general analysis indicates some optimal designs. Being left with a tight frame after erasures minimizes distortion, but considering also the transmission rate and possible erasure events complicates optimizations greatly. 

This comprehensive and engaging textbook introduces the basic principles and techniques of signal processing, from the fundamental ideas of signals and systems theory to real-world applications. • Introduces students to the powerful foundations of modern signal processing, including the basic geometry of Hilbert space, the mathematics of Fourier transforms, and essentials of sampling, interpolation, approximation, and compression. • Discusses issues in real-world use of these tools such as effects of truncation and quantization, limitations on localization, and computational costs. • Includes over 160 homework problems and over 220 worked examples, specifically designed to test and expand students' understanding of the fundamentals of signal processing. • Accompanied by extensive online materials designed to aid learning, including Mathematica resources and interactive demonstrations.

No abstract

Abstract-We consider the problem of signal recovery on graphs. Graphs model data with complex structure as signals on a graph. Graph signal recovery recovers one or multiple smooth graph signals from noisy, corrupted, or incomplete measurements. We formulate graph signal recovery as an optimization problem, for which we provide a general solution through the alternating direction methods of multipliers. We show how signal inpainting, matrix completion, robust principal component analysis, and anomaly detection all relate to graph signal recovery and provide corresponding specific solutions and theoretical analysis. We validate the proposed methods on real-world recovery problems, including online blog classification, bridge condition identification, temperature estimation, recommender system for jokes, and expert opinion combination of online blog classification.

No abstract