Graph Filter Research Articles

Wide and Deep Graph Neural Network With Distributed Online Learning

Graph neural networks (GNNs) are naturally distributed architectures for learning representations from network data. This renders them suitable candidates for decentralized tasks. In these scenarios, the underlying graph often changes with time due to link failures or topology variations, creating a mismatch between the graphs on which GNNs were trained and the ones on which they are tested. Online learning can be leveraged to retrain GNNs at testing time to overcome this issue. However, most online algorithms are centralized and usually offer guarantees only on convex problems, which GNNs rarely lead to. This paper develops the Wide and Deep GNN (WD-GNN), a novel architecture that can be updated with distributed online learning mechanisms. The WD-GNN consists of two components: the wide part is a linear graph filter and the deep part is a nonlinear GNN. At training time, the joint wide and deep architecture learns nonlinear representations from data. At testing time, the wide, linear part is retrained, while the deep, nonlinear one remains fixed. This often leads to a convex formulation. We further propose a distributed online learning algorithm that can be implemented in a decentralized setting. We also show the stability of the WD-GNN to changes of the underlying graph and analyze the convergence of the proposed online learning procedure. Experiments on movie recommendation, source localization and robot swarm control corroborate theoretical findings and show the potential of the WD-GNN for distributed online learning.

Kernel graph filtering-A new method for dynamic sinogram denoising.

Low count PET (positron emission tomography) imaging is often desirable in clinical diagnosis and biomedical research, but its images are generally very noisy, due to the very weak signals in the sinograms used in image reconstruction. To address this issue, this paper presents a novel kernel graph filtering method for dynamic PET sinogram denoising. This method is derived from treating the dynamic sinograms as the signals on a graph, and learning the graph adaptively from the kernel principal components of the sinograms to construct a lowpass kernel graph spectrum filter. The kernel graph filter thus obtained is then used to filter the original sinogram time frames to obtain the denoised sinograms for PET image reconstruction. Extensive tests and comparisons on the simulated and real life in-vivo dynamic PET datasets show that the proposed method outperforms the existing methods in sinogram denoising and image enhancement of dynamic PET at all count levels, especially at low count, with a great potential in real life applications of dynamic PET imaging.

Learning spatial filters from EEG signals with Graph Signal Processing methods.

In this paper, we propose to learn a spatial filter directly from Electroencephalography (EEG) signals using graph signal processing tools. We combine a graph learning algorithm with a high-pass graph filter to remove spatially large signals from the raw data. This approach increases topographical localization, and attenuates volume-conducted features. We empirically show that our method gives similar results that the surface Laplacian in the noiseless case while being more robust to noise or defective electrodes.Clinical relevance- The proposed method is an alternative to the surface Laplacian filter that is commonly used for processing EEG signals. It could be used in cases where this standard approach does not provide satisfying results (low signal-to-noise ratios due to a low number of epochs, defective electrodes). This could be particularly interesting in case of an electrode defect, as it can happen in clinical practice.

A term-based and citation network-based search system for COVID-19.

The COVID-19 pandemic resulted in an unprecedented production of scientific literature spanning several fields. To facilitate navigation of the scientific literature related to various aspects of the pandemic, we developed an exploratory search system. The system is based on automatically identified technical terms, document citations, and their visualization, accelerating identification of relevant documents. It offers a multi-view interactive search and navigation interface, bringing together unsupervised approaches of term extraction and citation analysis. We conducted a user evaluation with domain experts, including epidemiologists, biochemists, medicinal chemists, and medicine students. In general, most users were satisfied with the relevance and speed of the search results. More interestingly, participants mostly agreed on the capacity of the system to enable exploration and discovery of the search space using the graph visualization and filters. The system is updated on a weekly basis and it is publicly available at http://www.nactem.ac.uk/cord/.

Improved stereo matching algorithm based on multi-scale fusion

Aiming at the low matching accuracy of local stereo matching algorithm in weak texture or discontinuous disparity areas, a stereo matching algorithm combining multi-scale fusion of convolutional neural network (CNN) and feature pyramid structure (FPN) is proposed. The feature pyramid is applied on the basis of the convolutional neural network to realize the multi-scale feature extraction and fusion of the image, which improves the matching similarity of the image blocks. The guide graph filter is used to quickly and effectively complete the cost aggregation. The disparity selection stage adapts the improvement dynamic programming algorithm to obtain the initial disparity map. The initial disparity map is refined so as to obtain the final disparity map. The algorithm is trained and tested on the image provided by Middlebury data set, and the result shows that the disparity map obtained by the algorithm has good effect.

Anomaly detection method of packet loss node location in heterogeneous hash networks

When the current method is used to detect the location anomaly of packet loss nodes in heterogeneous hash networks, the detection takes a long time, and the detection results obtained have large errors, which have the problems of low detection efficiency and low accuracy. This paper presents an anomaly detection method for packet loss node location in heterogeneous hash networks, The node distribution model and heterogeneous hash network topology are constructed, It provides relevant information for location anomaly detection of packet loss nodes, The high-pass graph filter is used to process the network signal to obtain high-frequency components, divide heterogeneous hash networks, obtain specific frequency components corresponding to sub-graph output signals, judge sub-graph signals through thresholds, establish suspected abnormal node sets, and compare suspected abnormal node sets and sub-graph node sets to realize abnormal detection of packet loss node location. The experimental results show that the method has good positioning effect, high detection efficiency, and better detection effectiveness and stability than the comparative test method.

Large Scale Distributed Collaborative Unlabeled Motion Planning With Graph Policy Gradients

In this letter, we present a learning method to solve the unlabelled motion problem with motion constraints and space constraints in 2D space for a large number of robots. To solve the problem of arbitrary dynamics and constraints we propose formulating the problem as a multi-agent problem. We are able to demonstrate the scalability of our methods for a large number of robots by employing a graph neural network (GNN) to parameterize policies for the robots. The GNN reduces the dimensionality of the problem by learning filters that aggregate information among robots locally, similar to how a convolutional neural network is able to learn local features in an image. Additionally, by employing a GNN we are also able to overcome the computational overhead of training policies for a large number of robots by first training graph filters for a small number of robots followed by zero-shot policy transfer to a larger number of robots. We demonstrate the effectiveness of our framework through various simulations.

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).

DeepDT: Learning Geometry From Delaunay Triangulation for Surface Reconstruction

In this paper, a novel learning-based network, named DeepDT, is proposed to reconstruct the surface from Delaunay triangulation of point cloud. DeepDT learns to predict inside/outside labels of Delaunay tetrahedrons directly from a point cloud and corresponding Delaunay triangulation. The local geometry features are first extracted from the input point cloud and aggregated into a graph deriving from the Delaunay triangulation. Then a graph filtering is applied on the aggregated features in order to add structural regularization to the label prediction of tetrahedrons. Due to the complicated spatial relations between tetrahedrons and the triangles, it is impossible to directly generate ground truth labels of tetrahedrons from ground truth surface. Therefore, we propose a multi-label supervision strategy which votes for the label of a tetrahedron with labels of sampling locations inside it. The proposed DeepDT can maintain abundant geometry details without generating overly complex surfaces, especially for inner surfaces of open scenes. Meanwhile, the generalization ability and time consumption of the proposed method is acceptable and competitive compared with the state-of-the-art methods. Experiments demonstrate the superior performance of the proposed DeepDT.

The Dual Graph Shift Operator: Identifying the Support of the Frequency Domain

Contemporary data is often supported by an irregular structure, which can be conveniently captured by a graph. Accounting for this graph support is crucial to analyze the data, leading to an area known as graph signal processing (GSP). The two most important tools in GSP are the graph shift operator (GSO), which is a sparse matrix accounting for the topology of the graph, and the graph Fourier transform (GFT), which maps graph signals into a frequency domain spanned by a number of graph-related Fourier-like basis vectors. This alternative representation of a graph signal is denominated the graph frequency signal. Several attempts have been undertaken in order to interpret the support of this graph frequency signal, but they all resulted in a one-dimensional interpretation. However, if the support of the original signal is captured by a graph, why would the graph frequency signal have a simple one-dimensional support? Departing from existing work, we propose an irregular support for the graph frequency signal, which we coin dual graph. A dual GSO leads to a better interpretation of the graph frequency signal and its domain, helps to understand how the different graph frequencies are related and clustered, enables the development of better graph filters and filter banks, and facilitates the generalization of classical SP results to the graph domain.

Adaptive Graph Filtering with Intra-Patch Pixel Smoothing for Image Denoising

Graph filtering has recently attracted significant attention in signal processing. This study proposes an adaptive graph filtering method with intra-patch pixel smoothing (AWGF-PS) to improve the quality of denoised images. A detailed coefficient shrinkage algorithm is provided for the graph filter, which makes the noisy component in each graph frequency band adaptively shrink with its component significance. It replaces the traditional ideal low-pass graph filter and is better at removing image noise in the entire band. As for the design of the graph filters whose performance depends heavily on the topological structure among signals, we infer the graph from the super-pixels of similar patches by considering the smoothing attribute of intra-patch pixels. This graph fully exploits the relationship among the pixels in local areas. Moreover, it can achieve an efficient calculation of graph Fourier bases on a small-scale graph, which effectively enhances the graph filtering’s practicality. The experiments demonstrate that our AWGF-PS method suitably restores the denoised images. It outperforms several state-of-the-art model-based methods and is competitive with certain discriminative learning methods. In particular, it has more advantages in tackling images with significant noise.

Bayesian Topology Learning and noise removal from network data

Learning the topology of a graph from available data is of great interest in many emerging applications. Some examples are social networks, internet of things networks (intelligent IoT and industrial IoT), biological connection networks, sensor networks and traffic network patterns. In this paper, a graph topology inference approach is proposed to learn the underlying graph structure from a given set of noisy multi-variate observations, which are modeled as graph signals generated from a Gaussian Markov Random Field (GMRF) process. A factor analysis model is applied to represent the graph signals in a latent space where the basis is related to the underlying graph structure. An optimal graph filter is also developed to recover the graph signals from noisy observations. In the final step, an optimization problem is proposed to learn the underlying graph topology from the recovered signals. Moreover, a fast algorithm employing the proximal point method has been proposed to solve the problem efficiently. Experimental results employing both synthetic and real data show the effectiveness of the proposed method in recovering the signals and inferring the underlying graph.

Minimax Design of Graph Filter Using Chebyshev Polynomial Approximation

In this brief, the minimax design problem of graph filter using Chebyshev polynomial approximation (CPA) is studied. First, conventional CPA graph filter design is investigated to show that it does not provide the minimax design, so the peak error of the spectral response of the designed filter is not minimized. Then, a spectral transformation is used to convert the minimax design problem of CPA graph filter into the one of the type-I linear-phase FIR digital filter such that the Parks-McClellan method can be employed to obtain the optimal filter coefficients of minimax design. Next, using the recurrence relation of Chebyshev polynomials, an implementation structure of CPA graph filter with low computational complexity is presented. Finally, a graph signal denoising application example is illustrated to show the usefulness of the proposed CPA graph filter.

Anomalous IoT Sensor Data Detection: An Efficient Approach Enabled by Nonlinear Frequency-Domain Graph Analysis

The detection of anomalous Internet-of-Things (IoT) sensor data is extremely important in many industrial applications due to the catastrophic consequences of the faulty or unreliable sensor data. Good anomalous data detection performance with high detection efficiency is indeed a dilemma since it is difficult to derive an explicit detection function to characterize the relationships between the anomalous values and the detection indicator. To overcome this difficulty, the location information of the IoT sensors is exploited in this study to characterize and reconstruct the relationships among the sensor data based on a second-order nonlinear polynomial graph filter (NPGF). The analysis of the sensor data reconstruction model is then conducted in the frequency domain based on the 2-D inverse graph Fourier transform (GFT), and the reconstruction error function for the sensor data is analytically derived based on the second-order GFT coefficients. It is shown that the detection efficiency can be greatly improved if the input graph signal is designed to be bandlimited. The anomalous sensor data detection is then conducted in the frequency domain as high-frequency components are more sensitive to the deviation values. An NPGF-based frequency-domain algorithm is proposed for the anomalous sensor data detection, which is illustrated and validated with a real-world data set for temperature monitoring. The simulation results demonstrate the detection performance and efficiency improvement of the proposed algorithm in anomaly detection.

Robust and label efficient bi-filtering graph convolutional networks for node classification

Due to their success at node classification, Graph Convolutional Networks (GCN) have raised a research upsurge of deep learning on graph-structured data. For the semi-supervised classification, graph convolution essentially acts as a low-pass filter on graph spectral domain. According to Graph Signal Processing theory, the low-pass filter in GCN is a finite impulse response (FIR) graph filter. However, compared with FIR graph filters, infinite impulse response (IIR) graph filters exhibit more powerful representation ability and flexibility. Intuitively, it is feasible to replace FIR filter in GCN with IIR graph filter to improve GCN. Therefore, inspired by the direct implementation of IIR graph filters, we propose a Bi-filtering Graph Convolutional Network (BGCN) which can be realized by simply cascading two sub filtering modules. Experimental results demonstrate that BGCN works well in node classification task and achieves comparable performance to GCN and its variants. The improvement of BGCN, however, is at the expense of a time-complexity increase. To simplify the proposed BGCN, we construct a Simple Bi-filtering Graph Convolution framework (SBGC) from the perspective of Graph Signal Processing. Furthermore, for the implementations of BGCN and SBGC, we design a novel low-pass graph filter to capture the low-frequency components that are beneficial to data representation for the task of node classification. Extensive experiments show that SBGC not only outperforms other baseline methods in performance, but also keeps a high level in computational efficiency. Moreover, it is particularly worth noting that both BGCN and SBGC are robust to feature noise and exhibit high label efficiency.

Fractional Order Graph Filters: Design and Implementation

Existing graph filters, polynomial or rational, are mainly of integer order forms. However, there are some frequency responses which are not easily achieved by integer order approximation. It will substantially increase the flexibility of the filters if we relax the integer order to fractional ones. Motivated by fractional order models, we introduce the fractional order graph filters (FOGF), and propose to design the filter coefficients by genetic algorithm. In order to implement distributed computation on a graph, an FOGF can be approximated by the continued fraction expansion and transformed to an infinite impulse response graph filter.

Improved Visual Localization via Graph Filtering.

Vision-based localization is the problem of inferring the pose of the camera given a single image. One commonly used approach relies on image retrieval where the query input is compared against a database of localized support examples and its pose is inferred with the help of the retrieved items. This assumes that images taken from the same places consist of the same landmarks and thus would have similar feature representations. These representations can learn to be robust to different variations in capture conditions like time of the day or weather. In this work, we introduce a framework which aims at enhancing the performance of such retrieval-based localization methods. It consists in taking into account additional information available, such as GPS coordinates or temporal proximity in the acquisition of the images. More precisely, our method consists in constructing a graph based on this additional information that is later used to improve reliability of the retrieval process by filtering the feature representations of support and/or query images. We show that the proposed method is able to significantly improve the localization accuracy on two large scale datasets, as well as the mean average precision in classical image retrieval scenarios.

Medical image fusion using segment graph filter and sparse representation

This study proposes a novel medical image fusion approach based on the segment graph filter (SGF) and sparse representation (SR). Specifically, using the SGF, source images are decomposed into base and detail images, based on which the edge information is integrated into the fused image as much as possible. The base images are then fused applying a fusion rule based on the normalized Shannon entropy, whereas the detail images are fused using an SR-based fusion method. Finally, the resultant fused image is computed by combining the fused base and detail images. For quantitative performance evaluations, five metrics are adopted: the feature-based metric, structure-based metric, normalized mutual information, nonlinear correlation information entropy, and phase congruency metric. Experimental results indicate that the fusion performance of the proposed method is comparable to those of state-of-the-art methods with respect to both subjective visual performance and objective quantification.

Minimum-Degree Distributed Graph Filter Design

Here we consider the problem of designing finite-impulse-response (FIR) graph filter (GF) in a fully distributed way. For a directed graph with $\mathsf {N}$ nodes, each node designs filter coefficients in a distributed manner, when the knowledge of the graph structure, recognized as global information, is unavailable to each node. By modeling graph signal shifting with observations at a node as a linear dynamical system, we establish fundamental connections between local response of shifting at a node, concerned in the graph signal processing (GSP) field, and the observability of the system, investigated in control theory. The observability, as a measure of how well internal states of a dynamical system can be learned from node's observations, is reflected by the minimal polynomial of a matrix pair related to the system. Specifically, by introducing a notion of observable graph frequencies to a node, we show that the output signals (observations) at a node only contain the spectral components of its so-called observable graph frequencies. Furthermore, we unveil that the observability of a node to the spectral components of a GS is related to the rank of its observability matrix from the perspective of control theory. Our work reveals that partial signal outputs at a node are sufficient to design the FIR GF locally in terms of node-variant (NV) GFs. These findings further enable us to characterize the minimum-degree NV GF, where a minimum number of its shifted GSs are involved in the filter's output, and devise a distributed GF design algorithm for it.

Graphon Signal Processing

Graphons are infinite-dimensional objects that represent the limit of convergent sequences of graphs as their number of nodes goes to infinity. This paper derives a theory of graphon signal processing centered on the notions of graphon Fourier transform and linear shift invariant graphon filters, the graphon counterparts of the graph Fourier transform and graph filters. It is shown that for convergent sequences of graphs and associated graph signals: (i) the graph Fourier transform converges to the graphon Fourier transform when the graphon signal is bandlimited; (ii) the spectral and vertex responses of graph filters converge to the spectral and vertex responses of graphon filters with the same coefficients. These theorems imply that for graphs that belong to certain families, i.e., that are part of sequences that converge to a certain graphon, graph Fourier analysis and graph filter design have well defined limits. In turn, these facts extend applicability of graph signal processing to graphs with large number of nodes -- since signal processing pipelines designed for limit graphons can be applied to finite graphs -- and to dynamic graphs -- since we can relate the result of SP pipelines designed for different graphs from the same convergent graph sequence.