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Finding important edges in a graph is a crucial problem for various research fields, such as network epidemics, signal processing, machine learning, and sensor networks. In this paper, we tackle the problem based on sampling theory on graphs. We convert the original graph to a line graph where its nodes and edges, respectively, represent the original edges and the connections between the edges. We then perform node sampling of the line graph based on the edge smoothness assumption: This process selects the most critical edges in the original graph. We present a general framework of edge sampling based on graph sampling theory and reveal a theoretical relationship between the degree of the original graph and the line graph. We also propose an acceleration method for edge sampling in the proposed framework by using the relationship between two types of Laplacian of the node and edge domains. Experimental results in synthetic and real-world graphs validate the effectiveness of our approach against some alternative edge selection methods., Comment: 9 pages, 8 figures

This paper introduces the physics-inspired synthesized underwater image dataset (PHISWID), a dataset tailored for enhancing underwater image processing through physics-inspired image synthesis. Deep learning approaches to underwater image enhancement typically demand extensive datasets, yet acquiring paired clean and degraded underwater ones poses significant challenges. While several underwater image datasets have been proposed using physics-based synthesis, a publicly accessible collection has been lacking. Additionally, most underwater image synthesis approaches do not intend to reproduce atmospheric scenes, resulting in incomplete enhancement. PHISWID addresses this gap by offering a set of paired ground-truth (atmospheric) and synthetically degraded underwater images, showcasing not only color degradation but also the often-neglected effects of marine snow, a composite of organic matter and sand particles that considerably impairs underwater image clarity. The dataset applies these degradations to atmospheric RGB-D images, enhancing the dataset's realism and applicability. PHISWID is particularly valuable for training deep neural networks in a supervised learning setting and for objectively assessing image quality in benchmark analyses. Our results reveal that even a basic U-Net architecture, when trained with PHISWID, substantially outperforms existing methods in underwater image enhancement. We intend to release PHISWID publicly, contributing a significant resource to the advancement of underwater imaging technology.

This paper proposes a compression framework for adjacency matrices of weighted graphs based on graph filter banks. Adjacency matrices are widely used mathematical representations of graphs and are used in various applications in signal processing, machine learning, and data mining. In many problems of interest, these adjacency matrices can be large, so efficient compression methods are crucial. In this paper, we propose a lossy compression of weighted adjacency matrices, where the binary adjacency information is encoded losslessly (so the topological information of the graph is preserved) while the edge weights are compressed lossily. For the edge weight compression, the target graph is converted into a line graph, whose nodes correspond to the edges of the original graph, and where the original edge weights are regarded as a graph signal on the line graph. We then transform the edge weights on the line graph with a graph filter bank for sparse representation. Experiments on synthetic data validate the effectiveness of the proposed method by comparing it with existing lossy matrix compression methods.

We propose a node clustering method for time-varying graphs based on the assumption that the cluster labels are changed smoothly over time. Clustering is one of the fundamental tasks in many science and engineering fields including signal processing, machine learning, and data mining. Although most existing studies focus on the clustering of nodes in static graphs, we often encounter time-varying graphs for time-series data, e.g., social networks, brain functional connectivity, and point clouds. In this paper, we formulate a node clustering of time-varying graphs as an optimization problem based on spectral clustering, with a smoothness constraint of the node labels. We solve the problem with a primal-dual splitting algorithm. Experiments on synthetic and real-world time-varying graphs are performed to validate the effectiveness of the proposed approach.

We present an attention-based spatial graph convolution (AGC) for graph neural networks (GNNs). Existing AGCs focus on only using node-wise features and utilizing one type of attention function when calculating attention weights. Instead, we propose two methods to improve the representational power of AGCs by utilizing 1) structural information in a high-dimensional space and 2) multiple attention functions when calculating their weights. The first method computes a local structure representation of a graph in a high-dimensional space. The second method utilizes multiple attention functions simultaneously in one AGC. Both approaches can be combined. We also propose a GNN for the classification of point clouds and that for the prediction of point labels in a point cloud based on the proposed AGC. According to experiments, the proposed GNNs perform better than existing methods. Our codes open at https://github.com/liyang-tuat/SFAGC.

In this paper, we consider a sensor placement problem where sensors can move within a network over time. Sensor placement problem aims to select K sensor positions from N candidates where K < N. Most existing methods assume that sensor positions are static, i.e., they do not move, however, many mobile sensors like drones, robots, and vehicles can change their positions over time. Moreover, underlying measurement conditions could also be changed, which are difficult to cover with statically placed sensors. We tackle the problem by allowing the sensors to change their positions in their neighbors on the network. We sequentially learn the dictionary from a pool of observed signals on the network based on graph sampling theory. Using the learned dictionary, we dynamically determine the sensor positions such that the non-observed signals on the network can be best recovered from the observations. In experiments, we validate the effectiveness of the proposed method via the mean squared error (MSE) of the reconstructed signals. The proposed dynamic sensor placement outperforms the existing static ones for both synthetic and real data.

In this paper, we consider multi-channel sampling (MCS) for graph signals. We generally encounter full-band graph signals beyond the bandlimited one in many applications, such as piecewise constant/smooth and union of bandlimited graph signals. Full-band graph signals can be represented by a mixture of multiple signals conforming to different generation models. This requires the analysis of graph signals via multiple sampling systems, i.e., MCS, while existing approaches only consider single-channel sampling. We develop a MCS framework based on generalized sampling. We also present a sampling set selection (SSS) method for the proposed MCS so that the graph signal is best recovered. Furthermore, we reveal that existing graph filter banks can be viewed as a special case of the proposed MCS. In signal recovery experiments, the proposed method exhibits the effectiveness of recovery for full-band graph signals.

This letter proposes a fast implementation of the regularity-constrained discrete sine transform (R-DST). The original DST \textit{leaks} the lowest frequency (DC: direct current) components of signals into high frequency (AC: alternating current) subbands. This property is not desired in many applications, particularly image processing, since most of the frequency components in natural images concentrate in DC subband. The characteristic of filter banks whereby they do not leak DC components into the AC subbands is called \textit{regularity}. While an R-DST has been proposed, it has no fast implementation because of the singular value decomposition (SVD) in its internal algorithm. In contrast, the proposed regularity-constrained fast sine transform (R-FST) is obtained by just appending a regularity constraint matrix as a postprocessing of the original DST. When the DST size is $M\times M$ ($M=2^\ell$, $\ell\in\mathbb{N}_{\geq 1}$), the regularity constraint matrix is constructed from only $M/2-1$ rotation matrices with the angles derived from the output of the DST for the constant-valued signal (i.e., the DC signal). Since it does not require SVD, the computation is simpler and faster than the R-DST while keeping all of its beneficial properties. An image processing example shows that the R-FST has fine frequency selectivity with no DC leakage and higher coding gain than the original DST. Also, in the case of $M=8$, the R-FST saved approximately $0.126$ seconds in a 2-D transformation of $512\times 512$ signals compared with the R-DST because of fewer extra operations.

We propose a generalized sampling framework for stochastic graph signals. Stochastic graph signals are characterized by graph wide sense stationarity (GWSS) which is an extension of wide sense stationarity (WSS) for standard time-domain signals. In this paper, graph signals are assumed to satisfy the GWSS conditions and we study their sampling as well as recovery procedures. In generalized sampling, a correction filter is inserted between sampling and reconstruction operators to compensate for non-ideal measurements. We propose a design method for the correction filters to reduce the mean-squared error (MSE) between original and reconstructed graph signals. We derive the correction filters for two cases: The reconstruction filter is arbitrarily chosen or predefined. The proposed framework allows for arbitrary sampling methods, i.e., sampling in the vertex or graph frequency domain. We also show that the graph spectral response of the resulting correction filter parallels that for generalized sampling for WSS signals if sampling is performed in the graph frequency domain. Furthermore, we reveal the theoretical connection between the proposed and existing correction filters. The effectiveness of our approach is validated via experiments by comparing its MSE with existing approaches.

Block frames called directional analytic discrete cosine frames (DADCFs) are proposed for sparse image representation. In contrast to conventional overlapped frames, the proposed DADCFs require a reduced amount of 1) computational complexity, 2) memory usage, and 3) global memory access. These characteristics are strongly required for current high-resolution image processing. Specifically, we propose two DADCFs based on discrete cosine transform (DCT) and discrete sine transform (DST). The first DADCF is constructed from parallel separable transforms of DCT and DST, where the DST is permuted by row. The second DADCF is also designed based on DCT and DST, while the DST is customized to have no DC leakage property which is a desirable property for image processing. Both DADCFs have rich directional selectivity with slightly different characteristics each other and they can be implemented as non-overlapping block-based transforms, especially they form Parseval block frames with low transform redundancy. We perform experiments to evaluate our DADCFs and compare them with conventional directional block transforms in image recovery.

In this paper, we propose a spatial graph convolution (GC) to classify signals on a graph. Existing GC methods are limited to using the structural information in the feature space. Additionally, the single step of GCs only uses features on the one-hop neighboring nodes from the target node. In this paper, we propose two methods to improve the performance of GCs: 1) Utilizing structural information in the feature space, and 2) exploiting the multi-hop information in one GC step. In the first method, we define three structural features in the feature space: feature angle, feature distance, and relational embedding. The second method aggregates the node-wise features of multi-hop neighbors in a GC. Both methods can be simultaneously used. We also propose graph neural networks (GNNs) integrating the proposed GC for classifying nodes in 3D point clouds and citation networks. In experiments, the proposed GNNs exhibited a higher classification accuracy than existing methods.

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.

This paper introduces a new benchmarking dataset for marine snow removal of underwater images. Marine snow is one of the main degradation sources of underwater images that are caused by small particles, e.g., organic matter and sand, between the underwater scene and photosensors. We mathematically model two typical types of marine snow from the observations of real underwater images. The modeled artifacts are synthesized with underwater images to construct large-scale pairs of ground truth and degraded images to calculate objective qualities for marine snow removal and to train a deep neural network. We propose two marine snow removal tasks using the dataset and show the first benchmarking results of marine snow removal. The Marine Snow Removal Benchmarking Dataset is publicly available online., Comment: APSIPA ASC 2023, Taipei, Taiwan, Nov. 2023

We propose a blind deconvolution method for signals on graphs, with the exact sparseness constraint for the original signal. Graph blind deconvolution is an algorithm for estimating the original signal on a graph from a set of blurred and noisy measurements. Imposing a constraint on the number of nonzero elements is desirable for many different applications. This paper deals with the problem with constraints placed on the exact number of original sources, which is given by an optimization problem with an $\ell_0$ norm constraint. We solve this non-convex optimization problem using the ADMM iterative solver. Numerical experiments using synthetic signals demonstrate the effectiveness of the proposed method.

The world is suffering from a pandemic called COVID-19, caused by the SARS-CoV-2 virus. National governments have problems evaluating the reach of the epidemic, due to having limited resources and tests at their disposal. This problem is especially acute in low and middle-income countries (LMICs). Hence, any simple, cheap and flexible means of evaluating the incidence and evolution of the epidemic in a given country with a reasonable level of accuracy is useful. In this paper, we propose a technique based on (anonymous) surveys in which participants report on the health status of their contacts. This indirect reporting technique, known in the literature as network scale-up method, preserves the privacy of the participants and their contacts, and collects information from a larger fraction of the population (as compared to individual surveys). This technique has been deployed in the CoronaSurveys project, which has been collecting reports for the COVID-19 pandemic for more than two months. Results obtained by CoronaSurveys show the power and flexibility of the approach, suggesting that it could be an inexpensive and powerful tool for LMICs., Comment: Presented at The KDD Workshop on Humanitarian Mapping, San Diego, California USA, August 24, 2020

The study of sampling signals on graphs, with the goal of building an analog of sampling for standard signals in the time and spatial domains, has attracted considerable attention recently. Beyond adding to the growing theory on graph signal processing (GSP), sampling on graphs has various promising applications. In this article, we review current progress on sampling over graphs focusing on theory and potential applications. Although most methodologies used in graph signal sampling are designed to parallel those used in sampling for standard signals, sampling theory for graph signals significantly differs from the theory of Shannon--Nyquist and shift-invariant sampling. This is due in part to the fact that the definitions of several important properties, such as shift invariance and bandlimitedness, are different in GSP systems. Throughout this review, we discuss similarities and differences between standard and graph signal sampling and highlight open problems and challenges.

We propose a novel framework for learning time-varying graphs from spatiotemporal measurements. Given an appropriate prior on the temporal behavior of signals, our proposed method can estimate time-varying graphs from a small number of available measurements. To achieve this, we introduce two regularization terms in convex optimization problems that constrain sparseness of temporal variations of the time-varying networks. Moreover, a computationally-scalable algorithm is introduced to efficiently solve the optimization problem. The experimental results with synthetic and real datasets (point cloud and temperature data) demonstrate our proposed method outperforms the existing state-of-the-art methods., Comment: 13 pages, submitted to IEEE Transactions on Signal Processing

We propose a framework for generalized sampling of graph signals that parallels sampling in shift-invariant (SI) subspaces. This framework allows for arbitrary input signals, which are not constrained to be bandlimited. Furthermore, the sampling and reconstruction filters may be different. We present design methods of the correction filter that compensate for these differences and lead to closed form expressions in the graph frequency domain. In this study, we consider two priors on graph signals: The first is a subspace prior, where the signal is assumed to lie in a periodic graph spectrum (PGS) subspace. The PGS subspace is proposed as a counterpart of the SI subspace used in standard sampling theory. The second is a smoothness prior that imposes a smoothness requirement on the graph signal. We suggest the use of recovery techniques for when the recovery filter can be optimized and under a setting in which a predefined filter must be used. Sampling is performed in the graph frequency domain, which is a counterpart of "sampling by modulation" used in SI subspaces. We compare our approach with existing sampling techniques on graph signal processing. The effectiveness of the proposed generalized sampling approach is validated numerically through several experiments.

This paper proposes a class of $M$-channel spectral graph filter banks with a symmetric structure, that is, the transform has sampling operations and spectral graph filters on both the analysis and synthesis sides. The filter banks achieve maximum decimation, perfect recovery, and orthogonality. Conventional spectral graph transforms with decimation have significant limitations with regard to the number of channels, the structures of the underlying graph, and their filter design. The proposed transform uses sampling in the graph frequency domain. This enables us to use any variation operators and apply the transforms to arbitrary graphs even when the filter banks have symmetric structures. We clarify the perfect reconstruction conditions and show design examples. An experiment on graph signal denoising conducted to examine the performance of the proposed filter bank is described.

This paper addresses the problem of selecting an optimal sampling set for signals on graphs. The proposed sampling set selection (SSS) is based on a localization operator that can consider both vertex domain and spectral domain localization. We clarify the relationships among the proposed method, sensor position selection methods in machine learning, and conventional SSS methods based on graph frequency. In contrast to the conventional graph signal processing-based approaches, the proposed method does not need to compute the eigendecomposition of a variation operator, while still considering (graph) frequency information. We evaluate the performance of our approach through comparisons of prediction errors and execution time.

We propose two-channel critically-sampled filter banks for signals on undirected graphs that utilize spectral domain sampling. Unlike conventional approaches based on vertex domain sampling, our transforms have the following desirable properties: 1) perfect reconstruction regardless of the characteristics of the underlying graphs and graph variation operators and 2) a symmetric structure; i.e., both analysis and synthesis filter banks are built using similar building blocks. Along with the structure of the filter banks, this paper also proves the general criterion for perfect reconstruction and theoretically shows that the vertex and spectral domain sampling coincide for a special case. The effectiveness of our approach is evaluated by comparing its performance in nonlinear approximation and denoising with various conventional graph transforms.

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation.

Sampling methods for graph signals in the graph spectral domain are presented. Though conventional sampling of graph signals can be regarded as sampling in the graph vertex domain, it does not have the desired characteristics in regard to the graph spectral domain. With the proposed methods, the down- and upsampled graph signals inherit the frequency domain characteristics of the sampled signals defined in the time/spatial domain. The properties of the sampling effects were evaluated theoretically in comparison with those obtained with the conventional sampling method in the vertex domain. Various examples of signals on simple graphs enable precise understanding of the problem considered. Fractional sampling and Laplacian pyramid representation of graph signals are potential applications of these methods., Comment: accepted to IEEE Transactions on Signal Processing

We propose an approximation method for thresholding of singular values using Chebyshev polynomial approximation (CPA). Many signal processing problems require iterative application of singular value decomposition (SVD) for minimizing the rank of a given data matrix with other cost functions and/or constraints, which is called matrix rank minimization. In matrix rank minimization, singular values of a matrix are shrunk by hard-thresholding, soft-thresholding, or weighted soft-thresholding. However, the computational cost of SVD is generally too expensive to handle high dimensional signals such as images; hence, in this case, matrix rank minimization requires enormous computation time. In this paper, we leverage CPA to (approximately) manipulate singular values without computing singular values and vectors. The thresholding of singular values is expressed by a multiplication of certain matrices, which is derived from a characteristic of CPA. The multiplication is also efficiently computed using the sparsity of signals. As a result, the computational cost is significantly reduced. Experimental results suggest the effectiveness of our method through several image processing applications based on matrix rank minimization with nuclear norm relaxation in terms of computation time and approximation precision., Comment: This is a journal paper