Graph Filter Banks Research Articles

A new design scheme of two-channel graph filter bank

The existing designs of a two-channel graph filter bank (2c-graph-FB) are polynomial-based and have the numerical calculation problem when designing high performance/order 2c-graph-FB. In this study, a new design scheme is proposed to overcome the numerical calculation problem. The basic idea is adopting the expression form of the traditional symmetric finite impulse response (FIR) filter's frequency response as that of the frequency-domain graph filter of a 2c-graph-FB. Further, for solving the contradiction between the order and the vertex-domain localization in the existing designs as well as the proposed design, the idea of approximate vertex-domain localization is proposed. The designed 2c-graph-FB via our proposed scheme can have strict or approximate vertex-domain localization property. The simulation results reveal that the proposed scheme has no numerical calculation problem. Moreover, compared with the existing designs on the condition of all being able to achieve exceedingly small reconstruction error, the proposed design with the strict localization property has superior frequency selectivity. Among the existing designs (also including the proposed design with strict or approximate localization property) which can achieve exceedingly small reconstruction error, the proposed design scheme with the approximate localization property performs best in terms of the frequency selectivity while having certain localization property.

Minimax design of two-channel critically sampled graph QMF banks

We address here the issue of designing high spectral selectivity graph filter banks, which require high-degree polynomial approximation. To achieve sharp transition band roll required for high-selectivity, the minimax criterion is used in the design formulation. A constraint optimization method is used for the design process, and a spectral transformation is employed to circumvent the numerical problems that arise with high-degree polynomials. Using the transformation, approximation problem can be recast as an approximation problem with trigonometric polynomials. The transformed constraint optimization problem can be solved using the readily available sequential quadratic programming (SQP) method. High order filters, with low maximum approximation error (MAE) and low maximum reconstruction error (MRE), can be designed with this method. An algorithm is also developed that allows the user to choose design parameters that will give the lowest possible MRE. The method can also be adapted to trade-off between reconstruction quality and spectral selectivity of the resulting filter bank. The suite of techniques presented allow the user to easily shape the spectral response of the filters, to suit the application at hand. Numerous examples, with comparisons, will be presented to demonstrate the versatility of the proposed techniques. Applications to nonlinear approximation and denoising of graph signal will also be considered.

Iterative alternating optimization of bi-orthogonal two-channel graph filter bank

Iterative alternating optimization of bi-orthogonal two-channel graph filter bank

Multi-Channel Sampling on Graphs and Its Relationship to Graph Filter Banks

In this paper, we consider multi-channel sampling (MCS) for graph signals. We generally encounter full-band graph signals beyond the bandlimited ones in many applications, such as piecewise constant/smooth graph signals 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.

Preface

The 2022 3rd International Conference on Electrical, Electronic Information and Communication Engineering (EEICE 2022) was successfully held on April 22-24, 2022. Due to the COVID-19 pandemic around the world, and with the strict travelling rules in China, it is still difficult to take international travel for our attendees abroad. Therefore, EEICE 2022 was held both in physical (Guilin, China) and online (Zoom).EEICE 2022 is to bring together innovative academics and industrial experts in the field of “Electrical”, “Electronic Information”, “Communication Engineering” and “Communication and Information System”, etc. to a common forum. The primary goal of the conference is to promote research and developmental activities in “Electronic Information Technology”, “Communication Engineering” and another goal is to promote scientific information interchange between researchers, developers, engineers, students, and practitioners working all around the world.During the conference, the conference model was divided into three sessions, including oral presentations, keynote speeches, and online Q&A discussion. In the first part, some scholars, whose submissions were selected as the excellent papers, were given about 5-10 minutes to perform their oral presentations one by one. Then in the second part, keynote speakers were each allocated 30-45 minutes to hold their speeches.There were over 160 experts and scholars in the area of Electrical, Electronic Information and Communication Engineering representing different famous universities and institutes around the globe to attend this hybrid conference. During the conference, we were pleased to invite five distinguished experts to present their insightful speeches. Prof. Jizhong Zhu, South China University of Technology, China. His research interest is in the analysis, operation, planning and control of power systems, smart grid, power markets as well as applications of renewable energy. And then we had Prof. Kai Yang, Huazhong University of Science and Technology, China. He performed a wonderful speech: Research on novel hybrid flux permanent magnet motor and its control system. Prof. Ting Yang, School of Electrical and Information Engineering, Tianjin University, China. His main research interests include energy & power, Internet of things, artificial intelligence and intelligent manufacturing. Prof. Xiaohuan Li, Guilin University of Electronic Te c h n o l o g y, China. His current research interests include wireless sensor networks, vehicular networks, U AV networks, and cognitive radios. Our finale keynote speaker, Prof. Junzheng Jiang, Guilin University of Electronic Technology, China. His research interests include graph filter bank, distributed signal processing on graphs, and time-vertex signal processing on graphs. Their insightful speeches had triggered heated discussion. Every participant praised this conference for disseminating useful and insightful knowledge.List of Committee member, Conference Chairman, Organizing Committees, Technical Program Committees are available in this pdf.

Spline Graph Filter Bank with Spectral Sampling

Spline Graph Filter Bank with Spectral Sampling

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.

Spectral Domain Spline Graph Filter Bank

In this paper, we present a structure for two-channel spline graph filter bank with spectral sampling (SGFBSS) on arbitrary undirected graphs. Our proposed structure has many desirable properties; namely, perfect reconstruction, critical sampling in spectral domain, flexibility in choice of shape and cut-off frequency of the filters, and low complexity implementation of the synthesis section, thanks to our closed-form derivation of the synthesis filter and its sparse structure. These properties play a pivotal role in multi-scale transforms of graph signals. Additionally, this framework can use both normalized and non-normalized Laplacian of any undirected graph. We evaluate the performance of our proposed SGFBSS structure in nonlinear approximation and denoising applications through simulations. We also compare our method with the existing graph filter bank structures and show its superior performance.

Theory and Design of Joint Time-Vertex Nonsubsampled Filter Banks

Graph signal processing (GSP) is a field that deals with data residing on irregular domains, i.e. graph signals. In this field, the graph filter bank is one of the most important developments, owing to its ability to provide multiresolution analysis of graph signals. However, most of the current research on graph filter bank focuses on static graph signals. The research does not exploit the temporal correlations of time-varying signals in real-world applications, such as in wireless sensor networks. In this paper, the theory and design of joint time-vertex nonsubsampled filter bank are developed, using a generalized product graph framework. Several methods are proposed to design the filter bank with perfect reconstruction, while still achieving filters with good spectral characteristics. A notable feature of the designed filter bank is that it can be completely realized in a distributed manner. The subband filters are either of polynomial type or defined implicitly via iterative equations. In either case, implementing the subband filters requires only the exchange of information between neighboring nodes. The filter banks are therefore of low implementation complexity and suitable for processing large time-varying datasets. Numerical examples will demonstrate the effectiveness of the proposed designed methods. Application in time-varying graph signal denoising will show the superiority of joint time-vertex filter bank over other methods.

Ht-index for empirical evaluation of the sampled graph-based Discrete Pulse Transform

The Discrete Pulse Transform decomposes a signal into pulses, with the most recent and effective implementation being a graph-base algorithm called the Roadmaker’s Pavage. Even though an efficient implementation, the theoretical structure results in a slow, deterministic algorithm. This paper examines the use of the spectral domain of graphs and designs graph filter banks to downsample the algorithm, investigating the extent to which this speeds up the algorithm. Converting graph signals to the spectral domain is costly, thus estimation for filter banks is examined, as well as the design of a reusable filter bank. The sampled version requires hyperparameters to reconstruct the same textures of the image as the original algorithm, preventing a large scale study. Here an objective and efficient way of deriving similar results between the original and our proposed Filtered Roadmaker’s Pavage is provided. The method makes use of the Ht-index, separating the distribution of information at scale intervals. Empirical research using benchmark datasets provides improved results, showing that using the proposed algorithm consistently runs faster, uses less computational resources, while having a positive SSIM with low variance. This provides an informative and faster approximation to the nonlinear DPT, a property not standardly achievable.

Random Node-Asynchronous Graph Computations: Novel Opportunities for Discrete-Time State-Space Recursions

For the analysis of data defined over irregular domains, the recent area of graph signal processing extends classical signal processing techniques to the case of graphs. A prominent idea in this context is the notion of graph shift, which provides a foundation for distributed signal processing over networks. Despite its distributed nature, successive implementations of graph shift require a synchronization over the network, which can be a restriction in practical applications. The main goal of this article is to review the recent advancements in random and asynchronous computations over graphs with a specific emphasis on asynchronous graph-filtering applications. This article describes the behavior of the randomized asynchronous graph shift from a linear system theory viewpoint while making connections with asynchronous fixed-point iterations and Markov jumpswitching systems. In the context of graph signal processing, two specific applications are considered. The first one combines polynomial filtering and asynchronous updates to compute the Fiedler vector for spectral clustering. The second one considers the asynchronous implementation of a given rational graph filter. Asynchronous graph filtering is also extended to implement polynomial graph filter banks and graph neural networks in an asynchronous manner. The considered applications have proven mean-squared convergence guarantees.

Localized Spectral Graph Filter Frames: A Unifying Framework, Survey of Design Considerations, and Numerical Comparison

Representing data residing on a graph as a linear combination of building block signals can enable efficient and insightful visual or statistical analysis of the data, and such representations prove useful as regularizers in signal processing and machine learning tasks. Designing collections of building block signals -- or more formally, dictionaries of atoms -- that specifically account for the underlying graph structure as well as any available representative training signals has been an active area of research over the last decade. In this article, we survey a particular class of dictionaries called localized spectral graph filter frames, whose atoms are created by localizing spectral patterns to different regions of the graph. After showing how this class encompasses a variety of approaches from spectral graph wavelets to graph filter banks, we focus on the two main questions of how to design the spectral filters and how to select the center vertices to which the patterns are localized. Throughout, we emphasize computationally efficient methods that ensure the resulting transforms and their inverses can be applied to data residing on large, sparse graphs. We demonstrate how this class of transform methods can be used in signal processing tasks such as denoising and non-linear approximation, and provide code for readers to experiment with these methods in new application domains.

A Modified Spline Graph Filter Bank

Two-channel critically sampled spline graph filter bank (SGFB) is one of the existing solutions for signal processing on arbitrary graphs. This paper addresses an approach to modify the filter design for SGFB. Our method improves the shape of analysis filters in the spectral domain and also reduces the computational complexity for the synthesis section. Additionally, the proposed method exploits polynomial filters and does not need to compute eigendecomposition for the Laplacian matrix of the underlying graph. Numerical results show the efficacy of our approach by comparing its performance in graph signal decomposition and denoising with the existing solutions.

Infrared and visible image fusion based on oversampled graph filter banks

The infrared image (RI) and visible image (VI) fusion method merges complementary information from the infrared and visible imaging sensors to provide an effective way for understanding the scene. The graph filter bank-based graph wavelet transform possesses the advantages of the classic wavelet filter bank and graph representation of a signal. Therefore, we propose an RI and VI fusion method based on oversampled graph filter banks. Specifically, we consider the source images as signals on the regular graph and decompose them into the multiscale representations with M-channel oversampled graph filter banks. Then, the fusion rule for the low-frequency subband is constructed using the modified local coefficient of variation and the bilateral filter. The fusion maps of detail subbands are formed using the standard deviation-based local properties. Finally, the fusion image is obtained by applying the inverse transform on the fusion subband coefficients. The experimental results on benchmark images show the potential of the proposed method in the image fusion applications.

Design of Nonsubsampled Graph Filter Banks via Lifting Schemes

Graph filter banks play a crucial role in the vertex and spectral representation of graph signals. The notion of two-channel nonsubsampled graph filter banks (NSGFBs) on an undirected graph was introduced recently. The absence of downsampling/upsampling operators allows greater flexibility in the design of NSGFBs that achieve perfect reconstruction. However the design of NSGFBs that take the spectral response into account has not been adequately addressed yet. Based on the polynomial/rational lifting scheme, this letter presents a simple method to design NSGFBs with good spectral response and perfect reconstruction. Experimental results will demonstrate the effectiveness of the proposed method in tailoring the spectral responses of the lifted NSGFBs. Application of the NSGFB to denoising will also be considered.

Iterative technique for optimizing two-channel: Quadrature mirror, bi-orthogonal and graph filter bank

An iterative technique is proposed to optimally design the two-channel linear phase (LP) FIR quadrature mirror filter bank (2c-LP-FIR-QMFB). In each iteration, first, the frequency-domain reconstruction error function of the 2c-LP-FIR-QMFB is approximately converted into a convex function, by approximating all its sub-functions using first order Taylor series expansion. Then, the approximated reconstruction error function is minimized as the optimization objective function while the frequency selectivity performance metric function of the sub-filter is taken as the constraint function. Further, the optimal step-size optimization problem for each iteration is solved using a similar iterative strategy. Additionally, the proposed technique can also be utilized to optimize the coefficients of a class of structurally perfect reconstruction two-channel LP FIR bi-orthogonal filter bank, as well as a class of two-channel graph filter bank. Simulation results reveal that the proposed technique outperforms the previously existing techniques in terms of the frequency-domain reconstruction error and/or the frequency selectivity performance metric of the sub-filter. Particularly, for the graph filter bank, the reconstruction error of the proposed technique is only (3.1e-11)% of its previous counterpart. The proposed technique might also be useful to optimize the coefficients of other classes of two-channel traditional signal or graph signal filter banks.

Compression of Hyperspectral Scenes through Integer-to-Integer Spectral Graph Transforms

Hyperspectral images are depictions of scenes represented across many bands of the electromagnetic spectrum. The large size of these images as well as their unique structure requires the need for specialized data compression algorithms. The redundancies found between consecutive spectral components and within components themselves favor algorithms that exploit their particular structure. One novel technique with applications to hyperspectral compression is the use of spectral graph filterbanks such as the GraphBior transform, that leads to competitive results. Such existing graph based filterbank transforms do not yield integer coefficients, making them appropriate only for lossy image compression schemes. We propose here two integer-to-integer transforms that are used in the biorthogonal graph filterbanks for the purpose of the lossless compression of hyperspectral scenes. Firstly, by applying a Triangular Elementary Rectangular Matrix decomposition on GraphBior filters and secondly by adding rounding operations to the spectral graph lifting filters. We examine the merit of our contribution by testing its performance as a spatial transform on a corpus of hyperspectral images; and share our findings through a report and analysis of our results.

Nonsubsampled Graph Filter Banks: Theory and Distributed Algorithms

In this paper, we consider nonsubsampled graph filter banks (NSGFBs) to process data on a sparse graph. The analysis filter banks of NSGFBs have small bandwidth, pass/block the normalized constant signal, and have stability on $\ell ^2$ . Given an analysis filter bank with small bandwidth, we introduce algebraic and optimization methods to construct well-localized synthesis filter banks such that the corresponding NSGFBs provide a perfect signal reconstruction in the noiseless setting. We also prove that the proposed NSGFBs can control the resonance effect in the presence of bounded noise and they can limit the influence of shot noise primarily to a small neighborhood near its location on the graph. We later introduce an iterative algorithm to implement the proposed NSGFBs in a distributed manner, and develop an NSGFB-based denoising technique which is demonstrated to have satisfactory performance on noise suppression.

Design of orthogonal graph filter bank with known eigenvalues of Laplacian matrix

In the conventional designs of orthogonal graph filter bank, both eigenvalues and eigenvectors of Laplacian matrix are assumed to be unknown such that graph filter bank can be efficiently implemented by using polynomial graph filters; that is, the output signal at each vertex is computed only from the signal at that vertex and its K -hop neighbourhoods. However, the disadvantage of these methods is that the designed orthogonal filter bank does not have the exact perfect reconstruction (PR). In this study, the authors first assume that the eigenvalues of Laplacian matrix are known but eigenvectors are still unknown. Then, the orthogonal graph filter bank is designed by solving the simultaneous equation or Lagrange multiplier method such that the PR condition at all eigenvalues can be achieved for a given graph structure. Although the proposed filter bank is a graph-structure-dependent design, it can be efficiently implemented by using polynomial graph filters. Finally, experimental results are shown to demonstrate the effectiveness of the proposed design method.

$M$-Channel Critically Sampled Spectral Graph Filter Banks With Symmetric Structure

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.