Reconstruction Of Graph Signals Research Articles

LRPS‐GCN: A millimeter wave sparse imaging algorithm based on graph signal

AbstractAiming at the problems of slow speed and poor accuracy of traditional millimeter wave sparse imaging, a sparse imaging algorithm based on graph convolution model is proposed from the perspective of sparse signal recovery. The graph signal model is constructed by combining the low‐rank and piecewise smoothing(LRPS) regular terms, based on which the proximal operator is replaced by the denoising graph convolution network, and the graph convolution sparse reconstruction network LRPS‐GCN is constructed, and the recovered target image is obtained by iterating with the optimal non‐linear sparse variation. For the proposed algorithm, simulation experiments are carried out using synthetic datasets under different target densities, iteration times and noise environments, and compared with the traditional graph signal reconstruction algorithm and the deep compressed sensing reconstruction algorithm, and then use the measured data with varying degrees of sparsity to validate. The experimental results show that the reconstructed images by this algorithm have better performance in terms of normalised mean square error, target to background ratio, reconstruction time and memory usage.

Bayesian reconstruction of Cartesian product graph signals with general patterns of missing data

In this paper, we address the challenge of signal reconstruction on Cartesian product graphs, which frequently arises in applications where node-level data may be corrupted by equipment failure or difficult to reliably collect. A recent approach to Graph Signal Reconstruction (GSR) is to formulate it as a Bayesian inference problem, where the underlying signal is presumed to be smooth with respect to the graph topology. However, the resulting linear system derived from this model specification is often intractable to solve directly for general Cartesian product graphs. To overcome this limitation, we introduce two robust and efficient iterative procedures for computing the posterior mean. The first method is a stationary iterative technique based on matrix splitting, while the second employs a conjugate gradient approach with a symmetric graph-spectral preconditioner. We provide a theoretical and numerical comparison of the convergence behaviour of these algorithms, demonstrating that the optimal choice depends on the model hyperparameters. Finally, we propose a novel supervised learning-based technique, enhanced by an active learning strategy, for estimating the posterior marginal variance. This circumvents the need to instantiate, invert or decompose large matrices directly in memory and results in an R2 statistic of over 0.95 on a synthetic test data set. This paper contributes to the development of advanced signal reconstruction techniques applicable in various domains, including large-scale sensor networks and social network analysis.

An efficient algorithm with fast convergence rate for sparse graph signal reconstruction

In this paper, we consider the graph signals are sparse in the graph Fourier domain and propose an iterative threshold compressed sensing reconstruction (ITCSR) algorithm to reconstruct sparse graph signals in the graph Fourier domain. The proposed ITCSR algorithm derives from the well-known compressed sensing by considering a threshold for sparsity-promoting reconstruction of the underlying graph signals. The proposed ITCSR algorithm enhances the performance of sparse graph signal reconstruction by introducing a threshold function to determine a suitable threshold. Furthermore, we demonstrate that the suitable parameters for the threshold can be automatically determined by leveraging the sparrow search algorithm. Moreover, we analytically prove the convergence property of the proposed ITCSR algorithm. In the experimental, numerical tests with synthetic as well as 3D point cloud data demonstrate the merits of the proposed ITCSR algorithm relative to the baseline algorithms.

Graph signal reconstruction based on spatio-temporal features learning

This paper presents a new algorithm for reconstructing time-varying graph signals using spatiotemporal feature learning. We introduce a time series analysis method to capture the temporal stationarity of graph signals and propose a reconstruction model based on spatiotemporal coupling features. However, prior knowledge of the temporal stationarity of graph signals is required for this task. To address the issue, we analyze the spatio-temporal coupling feature in a graph-spectral view and design a learning framework to capture only the temporal feature based on the sampled signals. We reconstruct the graph signals using the proposed reconstruction model and feature-learning framework by solving an unconstrained optimization problem consisting of data fidelity and two regularization terms. Numerical results using both synthetic and real-world datasets demonstrate the superiority of the proposed reconstruction algorithm over existing methods.

Graph Signal Reconstruction Under Heterogeneous Noise via Adaptive Uncertainty-Aware Sampling and Soft Classification

Graph Signal Reconstruction Under Heterogeneous Noise via Adaptive Uncertainty-Aware Sampling and Soft Classification

Optimal sensor placement for leak location in water distribution networks: A feature selection method combined with graph signal processing

Nowadays, real-time monitoring of smart water networks is essentially performed by Supervisory Control and Data Acquisition (SCADA) systems and through the use of sensors strategically positioned in the water distribution network (WDN). The purpose of sensor placement is to reasonably collect signals and thus improve the efficiency of subsequent leak diagnosis, while also limiting their deployment and operational costs. Most current sensor placement methods rely on well-calibrated hydraulic models for node selection and subsequent leak location evaluation. In this study, an efficient Optimal Sensor Placement (OSP) method for WDN is proposed, which avoids the reliance on the hydraulic model in the node selection step. It can be generalized as an optimization process, and the algorithm to solve the problem is a heuristic algorithm. We consider the user nodes in the WDN as features and propose a feature selection algorithm combined with graph signal processing. By this method, the importance of different nodes in the WDN is analyzed and the redundancy between different nodes is considered in the iterative selection process. A graph signal reconstruction method is applied to recover pressure data of all nodes using the data monitored by the selected nodes, and the relationship between the number of sensors and the reconstruction error is analyzed. The proposed method is tested on two benchmark networks of different sizes and types. A comparative analysis with other methods shows that the proposed OSP method can be easily extended to other WDNs. In addition, the proposed method achieves better monitoring results while selecting nodes quickly.

Rating prediction based on the graph Fourier basis and PSD estimation from the perspective of graph signal reconstruction

Accurately and quickly predicting unknown ratings from fewer known ratings has always been a challenging problem for recommender systems. Inspired by emerging graph signal processing techniques, this paper proposes a rating prediction method via a graph signal reconstruction technique. The most novel point is that the proposed method does not need to construct (or learn) the graph structures in advance and then estimate the power spectral density (PSD), but directly estimates the graph Fourier basis and the PSD from data. Furthermore, we present an approximate solution strategy to significantly reduce the computational complexity of the reconstructed model. Finally, we conduct experiments with two public databases to test the proposed method and compare it with existing methods. The experiments give an encouraging result in both prediction accuracy and efficiency.

Salt and pepper noise removal method based on graph signal reconstruction

Salt and Pepper (SAP) noise is a very common type of impulse noise which frequently arises in image acquisition and transmission. Removing SAP noise is an important task in image restoration. In this paper, we model the SAP removal problem as a graph signal reconstruction problem, so that the information of noisy pixels can also participate in the estimation of their gray values. In the proposed model, the regularization term depends on the Power Spectral Density (PSD) of original signals, which makes the reconstructed signal follows the frequency content of the potential original signal well. Further, an algorithm for estimating the local PSD of an image block is designed and corresponding theoretical analysis is given. Finally, a block SAP noise removal algorithm is proposed. The computational complexity is also discussed. Experimental results on ten commonly used test images demonstrate that the proposed algorithm outperforms the state-of-the-art methods.

Joint Sampling and Reconstruction of Time-Varying Signals Over Directed Graphs

Vertex-domain and temporal-domain smoothness of time-varying graph signals are cardinal properties that can be exploited for effective graph signal reconstruction from limited samples. However, existing approaches are not directly applicable when the signal's frequency occupancy changes with time. Moreover, while e.g., sensor network applications can benefit from directed graph models, the non-orthogonality of the graph eigenvectors can challenge spectral-based signal reconstruction algorithms. In this context, here we consider <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> -sparse time-varying signals with unknown frequency supports. By exploiting the smoothness of the varying graph frequency supports and employing shift operations over directed graphs, we study joint sampling of multiple varying signals based on Schur decomposition to reconstruct each signal by orthogonal frequency components. Firstly, joint frequency support of the multiple signals is identified by proposing a two-stage Individual-Joint sampling scheme. Based on the estimated frequency support, the GFT coefficients of each signal can be recovered using data collected in individual sampling stage. Greedy algorithms are proposed for vertex set selection and graph shift order selection, which enable a robust signal reconstruction against additive noise. Considering the signals in applications may be approximately <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> -sparse, we further exploit the samples in both individual and joint sampling stages and investigate the optimal signal reconstruction as a convex optimization problem with adaptive frequency support selection. The proposed optimal sampling and reconstruction algorithms outperform several existing schemes in random network and sensor network data gathering.

Graph Signal Reconstruction Techniques for IoT Air Pollution Monitoring Platforms

Air pollution monitoring platforms play a very important role in preventing and mitigating the effects of pollution. Recent advances in the field of graph signal processing have made it possible to describe and analyze air pollution monitoring networks using graphs. One of the main applications is the reconstruction of the measured signal in a graph using a subset of sensors. Reconstructing the signal using information from sensor neighbors can help improve the quality of network data, examples are filling in missing data with correlated neighboring nodes, or correcting a drifting sensor with neighboring sensors that are more accurate. This paper compares the use of various types of graph signal reconstruction methods applied to real data sets of Spanish air pollution reference stations. The methods considered are Laplacian interpolation, graph signal processing low-pass based graph signal reconstruction, and kernel-based graph signal reconstruction, and are compared on actual air pollution data sets measuring O3, NO2, and PM10. The ability of the methods to reconstruct the signal of a pollutant is shown, as well as the computational cost of this reconstruction. The results indicate the superiority of methods based on kernel-based graph signal reconstruction, as well as the difficulties of the methods to scale in an air pollution monitoring network with a large number of low-cost sensors. However, we show that scalability can be overcome with simple methods, such as partitioning the network using a clustering algorithm.

Revisiting graph neural networks from hybrid regularized graph signal reconstruction

Graph neural networks (GNNs) have shown strong graph-structured data processing capabilities. However, most of them are generated based on the message-passing mechanism and lack of the systematic approach to guide their developments. Meanwhile, a unified point of view is hard to explain the design concepts of different GNN models. This paper presents a unified optimization framework from hybrid regularized graph signal reconstruction to establish the connection between the aggregation operations of different GNNs, showing that exploring the optimal solution is the process of GNN information aggregation. We use this new framework to mathematically explain several classic GNN models and summarizes their commonalities and differences from a macro perspective. The proposed framework not only provides convenience to understand GNNs, but also has a guiding significance for the proposal of new GNNs. Moreover, we design a model-driven fixed-point iteration method and a data-driven dictionary learning network according to the corresponding optimization objective and sparse representation. Then the new model, GNN based on model-driven and data-driven (GNN-MD), is established by using alternating iteration methods. We also theoretically analyze its convergence. Numerous node classification experiments on multiple datasets illustrate that the proposed GNN-MD has excellent performance and outperforms all baselines on high-feature-dimension datasets.

Spectral Graph Theory-Based Recovery Method for Missing Harmonic Data

In large-scale applications, parts of harmonic data are inevitably lost during transmission. This study presents an approach for the recovery of missing harmonic data based on the spectral graph theory. The proposed methodology involves graph theory for constructing a Laplacian matrix and a graph signal reconstruction function, the merging <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -means algorithm for building a priori information model, and the accelerated segmentation Bregman iterative algorithm for solving the reconstruction function. Compared with existing methods on data recovery in power systems, the method maintains a good recovery accuracy when the data correlation of measurement units is low and the prior information is little. The proposed method has good anti-noise performances and low computational complexity. The feasibility and accuracy of the proposed method are verified through simulation and field recorded data.

Volterra Graph-Based Outlier Detection for Air Pollution Sensor Networks

Today's air pollution sensor networks pose new challenges given their heterogeneity of low-cost sensors and high-cost instrumentation. Recently, with the advent of graph signal processing, sensor network measurements have been successfully represented by graphs depicting the relationships between sensors. However, one of the main problems of these sensor networks is their reliability, especially due to the inclusion of low-cost sensors, so the detection and identification of outliers is extremely important for maintaining the quality of the network data. In order to better identify the outliers of the sensors composing a network, we propose the Volterra graph-based outlier detection (VGOD) mechanism, which uses a graph learned from data and a Volterra-like graph signal reconstruction model to detect and localize abnormal measurements in air pollution sensor networks. The proposed unsupervised decision process is compared with other outlier detection methods, state-of-the-art graph-based methods and non-graph-based methods, showing improvements in both detection and localization of anomalous measurements, so that anomalous measurements can be corrected and malfunctioning sensors can be replaced.

A recommendation prediction method based on the estimation of PSD of sampled signals on graph

In recommendation system, the rating prediction is a very challenging problem. In this paper, we model users (items) and their relationships as a weighted undirected graph, and propose a rating prediction model based on graph signal reconstruction. Different from the existing reconstruction model, the proposed model takes into account the frequency structure of the signal itself, and tends to protect the frequency contents with higher power spectral density (PSD) and impose large penalties on those with lower PSD when reconstructing a signal. Furthermore, based on the assumption of the stationarity of the graph signal, we propose a method to estimate the PSD of a stationarity stochastic graph signal directly from some sampled signals without its any other complete implementation. Related theoretical analysis is also given. Finally, a rating prediction algorithm based on the graph signal reconstruction is derived. Experimental results show that the proposed model leads a significant improvement of the predictive accuracy.

Robust Adaptive Estimation of Graph Signals Based on Welsch Loss

This paper considers the problem of adaptive estimation of graph signals under the impulsive noise environment. The existing least mean squares (LMS) approach suffers from severe performance degradation under an impulsive environment that widely occurs in various practical applications. We present a novel adaptive estimation over graphs based on Welsch loss (WL-G) to handle the problems related to impulsive interference. The proposed WL-G algorithm can efficiently reconstruct graph signals from the observations with impulsive noises by formulating the reconstruction problem as an optimization based on Welsch loss. An analysis on the performance of the WL-G is presented to develop effective sampling strategies for graph signals. A novel graph sampling approach is also proposed and used in conjunction with the WL-G to tackle the time-varying case. The performance advantages of the proposed WL-G over the existing LMS regarding graph signal reconstruction under impulsive noise environment are demonstrated.

Salt and Pepper Noise Removal Method Based on Graph Signal Reconstruction

Salt and Pepper Noise Removal Method Based on Graph Signal Reconstruction

Reconstruction of Time-Varying Graph Signals via Sobolev Smoothness

Graph Signal Processing (GSP) is an emerging research field that extends the concepts of digital signal processing to graphs. GSP has numerous applications in different areas such as sensor networks, machine learning, and image processing. The sampling and reconstruction of static graph signals have played a central role in GSP. However, many real-world graph signals are inherently time-varying and the smoothness of the temporal differences of such graph signals may be used as a prior assumption. In the current work, we assume that the temporal differences of graph signals are smooth, and we introduce a novel algorithm based on the extension of a Sobolev smoothness function for the reconstruction of time-varying graph signals from discrete samples. We explore some theoretical aspects of the convergence rate of our Time-varying Graph signal Reconstruction via Sobolev Smoothness (GraphTRSS) algorithm by studying the condition number of the Hessian associated with our optimization problem. Our algorithm has the advantage of converging faster than other methods that are based on Laplacian operators without requiring expensive eigenvalue decomposition or matrix inversions. The proposed GraphTRSS is evaluated on several datasets including two COVID-19 datasets and it has outperformed many existing state-of-the-art methods for time-varying graph signal reconstruction. GraphTRSS has also shown excellent performance on two environmental datasets for the recovery of particulate matter and sea surface temperature signals.

A reconstruction method for graph signals based on the power spectral density estimation

Graph signal reconstruction is a classic problem of graph signal processing. The ultimate goal of signal reconstruction is to obtain an estimate as close as possible to the original signals. Most of existing reconstruction models are based on assumption of signal smoothness. This paper proposes a new reconstruction model based on signal power spectral density estimation. The basic idea is to suppress those graph frequency components with low power spectral density when reconstructing a signal, while preserving those graph frequency contents with high power spectral density. Therefore, the reconstructed signals can well maintain the frequency structures of the original signals. Under the stationarity assumption of stochastic graph signals, we further propose a method to estimate the power spectral density directly from the sampled signals. Experimental results show that the proposed power spectral density estimation algorithm works well even when the proportion of labeled vertices is extremely low, and is robust to the PSD of signals itself and the various structures of graphs. Compared with traditional models based on smoothness assumption, the proposed reconstruction model can significantly improve the reconstruction performance.

Graph-signal Reconstruction and Blind Deconvolution for Structured Inputs

Key to successfully deal with complex contemporary datasets is the development of tractable models that account for the irregular structure of the information at hand. This paper provides a comprehensive and unifying view of several sampling, reconstruction, and recovery problems for signals defined on irregular domains that can be accurately represented by a graph. The workhorse assumption is that the (partially) observed signals can be modeled as the output of a graph filter to a structured (parsimonious) input graph signal. When either the input or the filter coefficients are known, this is tantamount to assuming that the signals of interest live on a subspace defined by the supporting graph. When neither is known, the model becomes bilinear. Upon imposing different priors and additional structure on either the input or the filter coefficients, a broad range of relevant problem formulations arise. The goal is then to leverage those priors, the shift operator of the supporting graph, and the samples of the signal of interest to recover: the signal at the non-sampled nodes (graph-signal interpolation), the input (deconvolution), the filter coefficients (system identification), or any combination thereof (blind deconvolution).

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.