Graph Signal Denoising Research Articles

Graph Signal Restoration Using Nested Deep Algorithm Unrolling

Graph signal processing is a ubiquitous task in many applications such as sensor, social, transportation and brain networks, point cloud processing, and graph neural networks. Often, graph signals are corrupted in the sensing process, thus requiring restoration. In this paper, we propose two graph signal restoration methods based on deep algorithm unrolling (DAU). First, we present a graph signal denoiser by unrolling iterations of the alternating direction method of multiplier (ADMM). We then suggest a general restoration method for linear degradation by unrolling iterations of Plug-and-Play ADMM (PnP-ADMM). In the second approach, the unrolled ADMM-based denoiser is incorporated as a submodule, leading to a nested DAU structure. The parameters in the proposed denoising/restoration methods are trainable in an end-to-end manner. Our approach is interpretable and keeps the number of parameters small since we only tune graph-independent regularization parameters. We overcome two main challenges in existing graph signal restoration methods: 1) limited performance of convex optimization algorithms due to fixed parameters which are often determined manually. 2) large number of parameters of graph neural networks that result in difficulty of training. Several experiments for graph signal denoising and interpolation are performed on synthetic and real-world data. The proposed methods show performance improvements over several existing techniques in terms of root mean squared error in both tasks.

Untrained Graph Neural Networks for Denoising

A fundamental problem in signal processing is to denoise a signal. While there are many well-performing methods for denoising signals defined on regular domains, including images defined on a two-dimensional pixel grid, many important classes of signals are defined over <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">irregular</i> domains that can be conveniently represented by a graph. This paper introduces two untrained graph neural network architectures for graph signal denoising, develops theoretical guarantees for their denoising capabilities in a simple setup, and provides empirical evidence in more general scenarios. The two architectures differ on how they incorporate the information encoded in the graph, with one relying on graph convolutions and the other employing graph upsampling operators based on hierarchical clustering. Each architecture implements a different prior over the targeted signals. Finally, we provide numerical experiments with synthetic and real datasets that i) asses the denoising behavior predicted by our theoretical results and ii) compare the denoising performance of our architectures with that of existing alternatives.

Node-Adaptive Regularization for Graph Signal Reconstruction

A critical task in graph signal processing is to estimate the true signal from noisy observations over a subset of nodes, also known as the reconstruction problem. In this paper, we propose a node-adaptive regularization for graph signal reconstruction, which surmounts the conventional Tikhonov regularization, giving rise to more degrees of freedom; hence, an improved performance. We formulate the node-adaptive graph signal denoising problem, study its bias-variance trade-off, and identify conditions under which a lower mean squared error and variance can be obtained with respect to Tikhonov regularization. Compared with existing approaches, the node-adaptive regularization enjoys more general priors on the local signal variation, which can be obtained by optimally designing the regularization weights based on Prony's method or semidefinite programming. As these approaches require additional prior knowledge, we also propose a minimax (worst-case) strategy to address instances where this extra information is unavailable. Numerical experiments with synthetic and real data corroborate the proposed regularization strategy for graph signal denoising and interpolation, and show its improved performance compared with competing alternatives.

Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising

We propose an interpretable graph neural network framework to denoise single or multiple noisy graph signals. The proposed graph unrolling networks expand algorithm unrolling to the graph domain and provide an interpretation of the architecture design from a signal processing perspective. We unroll an iterative denoising algorithm by mapping each iteration into a single network layer where the feed-forward process is equivalent to iteratively denoising graph signals. We train the graph unrolling networks through unsupervised learning, where the input noisy graph signals are used to supervise the networks. By leveraging the learning ability of neural networks, we adaptively capture appropriate priors from input noisy graph signals, instead of manually choosing signal priors. A core component of graph unrolling networks is the edge-weight-sharing graph convolution operation, which parameterizes each edge weight by a trainable kernel function where the trainable parameters are shared by all the edges. The proposed convolution is permutation-equivariant and can flexibly adjust the edge weights to various graph signals. We then consider two special cases of this class of networks, graph unrolling sparse coding (GUSC) and graph unrolling trend filtering (GUTF), by unrolling sparse coding and trend filtering, respectively. To validate the proposed methods, we conduct extensive experiments on both real-world datasets and simulated datasets, and demonstrate that our methods have smaller denoising errors than conventional denoising algorithms and state-of-the-art graph neural networks. For denoising a single smooth graph signal, the normalized mean square error of the proposed networks is around 40% and 60% lower than that of graph Laplacian denoising and graph wavelets, respectively.

Spectral Graph Based Vertex-Frequency Wiener Filtering for Image and Graph Signal Denoising

In this article, we propose and develop a spectral graph based vertex varying Wiener filtering framework in the joint vertex-frequency domain for denoising of graph signals defined on weighted, undirected and connected graphs. To this end, we first extend the Zadeh time-frequency filter concept to graph signals and obtain vertex-frequency transfer function of the proposed Wiener filter by transforming its vertex varying impulse response that minimizes the mean square error between original and recovered signals. To facilitate the derived Wiener filter, we present a detailed derivation of a recently proposed graph Rihaczek vertex-frequency signal distribution (GRD) so as to match the structure of the proposed graph Zadeh filter, based on a graph translation operator defined by generalized convolution with a delta signal. We express the filter transfer function in terms of this graph transform of the original, noiseless signal. The form of the obtained Wiener filter is, interestingly, different than those of time-frequency Wiener filters prevalent in the classical signal processing. We also investigate the invertibility of the employed GRD. Assuming that the original graph signal of interest can be viewed as deterministic, we propose two algorithms to implement the proposed vertex-frequency Wiener filter from a single realization of a noisy, input signal. We derive mean and variance of the GRD of the noisy signal, since they are required in one of these algorithms. We apply proposed Wiener filter algorithms to denoise a standard set of images and three irregularly structured graph signals, and demonstrate their competitiveness with compared high performance denoising methods.

Autoregressive Moving Average Graph Filtering

One of the cornerstones of the field of signal processing on graphs are graph filters, direct analogues of classical filters, but intended for signals defined on graphs. This work brings forth new insights on the distributed graph filtering problem. We design a family of autoregressive moving average (ARMA) recursions, which (i) are able to approximate any desired graph frequency response, and (ii) give exact solutions for tasks such as graph signal denoising and interpolation. The design philosophy, which allows us to design the ARMA coefficients independently from the underlying graph, renders the ARMA graph filters suitable in static and, particularly, time-varying settings. The latter occur when the graph signal and/or graph are changing over time. We show that in case of a time-varying graph signal our approach extends naturally to a two-dimensional filter, operating concurrently in the graph and regular time domains. We also derive sufficient conditions for filter stability when the graph and signal are time-varying. The analytical and numerical results presented in this paper illustrate that ARMA graph filters are practically appealing for static and time-varying settings, as predicted by theoretical derivations.