Spectral Graph Wavelet Transform Research Articles

Network time series forecasting using spectral graph wavelet transform

We propose a novel method for forecasting network time series that occur in graphs or networks. Our approach is based on a spectral graph wavelet transform (SGWT) that provides the localized behavior of graph signals around each node. The proposed method improves forecasting performance over other existing methods. In particular, the advantages of the proposed method stand out when signals observed on a graph are inhomogeneous or non-stationary. We demonstrate the strength of the proposed approach through real-world data analysis. This analysis uses two network time series datasets: the daily number of people getting on and off the Seoul Metropolitan Subway, and daily Covid-19 confirmed cases reported in South Korea. We further conduct a simulation study to evaluate the effectiveness of the proposed method.

Multi-dimensional spectral graph wavelet transform

In this paper, we introduce the two-dimensional spectral graph wavelet transform (SGWT) of discrete functions defined on weighted Cartesian product graphs. The graphs we consider are undirected, non-planar, and can be cyclic with no multiple loops and no multiple edges. We build this transform with the help of spectral decomposition of $$N_1N_2 \times N_1N_2$$ Laplacian matrix $$\mathcal {L}$$ of the Cartesian product graph $$G=G_1\square G_2$$ , where $$N_1N_2$$ is the number of vertices of the Cartesian product graph. We have established the inversion formula for SGWT for continuous scale parameters $$s_1$$ and $$s_2$$ . By sampling the scale parameters at discrete values, we obtain discrete SGWT coefficients at different scales and vertices. We have obtained the frame bounds of the frame formed by the set of scaling and wavelet coefficients. Further, we have proved the localization result which holds for both normalized and non-normalized form of Laplacian. Towards the end, we have given the implementation details of SGWT through an example to show how to calculate the graph wavelet coefficients.

Undersampled MRI reconstruction based on spectral graph wavelet transform

Compressed sensing magnetic resonance imaging (CS-MRI) has exhibited great potential to accelerate magnetic resonance imaging if an image can be sparsely represented. How to sparsify the image significantly affects the reconstruction quality of images. In this paper, a spectral graph wavelet transform (SGWT) is introduced to sparsely represent magnetic resonance images in iterative image reconstructions. The SGWT is achieved by extending the traditional wavelets transform to the signal defined on the vertices of the weighted graph, i.e. the spectral graph domain. This SGWT uses only the connectivity information encoded in the edge weights, and does not rely on any other attributes of the vertices. Therefore, SGWT can be defined and calculated for any domain where the underlying relations between data locations can be represented by a weighted graph. Furthermore, we present a Chebyshev polynomial approximation algorithm for fast computing this SGWT transform. The l1 norm regularized CS-MRI reconstruction model is introduced and solved by the projected iterative soft-thresholding algorithm to verify its feasibility. Numerical experiment results demonstrate that our proposed method outperforms several state-of-the-art sparsify transforms in terms of suppressing artifacts and achieving lower reconstruction errors on the tested datasets.

Dynamic PET images denoising using spectral graph wavelet transform.

Positron emission tomography (PET) is a non-invasive molecular imaging method for quantitative observation of physiological and biochemical changes in living organisms. The quality of the reconstructed PET image is limited by many different physical degradation factors. Various denoising methods including Gaussian filtering (GF) and non-local mean (NLM) filtering have been proposed to improve the image quality. However, image denoising usually blurs edges, of which high frequency components are filtered as noises. On the other hand, it is well-known that edges in a PET image are important to detection and recognition of a lesion. Denoising while preserving the edges of PET images remains an important yet challenging problem in PET image processing. In this paper, we propose a novel denoising method with good edge-preserving performance based on spectral graph wavelet transform (SGWT) for dynamic PET images denoising. We firstly generate a composite image from the entire time series, then perform SGWT on the PET images, and finally reconstruct the low graph frequency content to get the denoised dynamic PET images. Experimental results on simulation and in vivo data show that the proposed approach significantly outperforms the GF, NLM and graph filtering methods. Compared with deep learning-based method, the proposed method has the similar denoising performance, but it does not need lots of training data and has low computational complexity.

Analysis of the Spatio-Temporal Dynamics of COVID-19 in Massachusetts via Spectral Graph Wavelet Theory

The rapid spread of COVID-19 disease has had a significant impact on the world. In this paper, we study COVID-19 data interpretation and visualization using open-data sources for 351 cities and towns in Massachusetts from December 6, 2020 to September 25, 2021. Because cities are embedded in rather complex transportation networks, we construct the spatio-temporal dynamic graph model, in which the graph attention neural network is utilized as a deep learning method to learn the pandemic transition probability among major cities in Massachusetts. Using the spectral graph wavelet transform (SGWT), we process the COVID-19 data on the dynamic graph, which enables us to design effective tools to analyze and detect spatio-temporal patterns in the pandemic spreading. We design a new node classification method, which effectively identifies the anomaly cities based on spectral graph wavelet coefficients. It can assist administrations or public health organizations in monitoring the spread of the pandemic and developing preventive measures. Unlike most work focusing on the evolution of confirmed cases over time, we focus on the spatio-temporal patterns of pandemic evolution among cities. Through the data analysis and visualization, a better understanding of the epidemiological development at the city level is obtained and can be helpful with city-specific surveillance.

Data-driven thresholding in denoising with Spectral Graph Wavelet Transform

This paper is devoted to adaptive signal denoising in the context of Graph Signal Processing (GSP) using Spectral Graph Wavelet Transform (SGWT). This issue is addressed via a data-driven thresholding process in the transformed domain by optimizing the parameters in the sense of the Mean Square Error (MSE) using the Stein’s Unbiased Risk Estimator (SURE). The SGWT considered is built upon a partition of unity making the transform semi-orthogonal so that the optimization can be performed in the transformed domain. However, since the SGWT is over-complete, the divergence term in the SURE needs to be computed in the context of correlated noise. Two thresholding strategies called coordinatewise and block thresholding process are investigated. For each of them, the SURE is derived for a whole family of elementary thresholding functions among which the soft threshold and the James–Stein threshold. This multi-scales analysis shows better performance than the most recent methods from the literature. That is illustrated numerically for a series of signals on different graphs.

Fractional Spectral Graph Wavelets and Their Applications

One of the key challenges in the area of signal processing on graphs is to design transforms and dictionary methods to identify and exploit structure in signals on weighted graphs. In this paper, we first generalize graph Fourier transform (GFT) to spectral graph fractional Fourier transform (SGFRFT), which is then used to define a novel transform named spectral graph fractional wavelet transform (SGFRWT), which is a generalized and extended version of spectral graph wavelet transform (SGWT). A fast algorithm for SGFRWT is also derived and implemented based on Fourier series approximation. Some potential applications of SGFRWT are also presented.

Analysis and Classification of Dermoscopic Images Using Spectral Graph Wavelet Transform

The abnormal growth of skin cells leads to skin cancer which occurs due to the unrepaired DNA impairment to the skin cells. Worldwide every year more than 1.23 lakhs of skin cancers are diagnosed, out of which Melanoma is the deadliest one. The aim of this research work is Recognition of Melanoma and skin lesion classification from the Dermoscopic images. Feature extraction of Dermoscopic images can be done by Shape, Color and Texture features. Texture features comprises of Gray Level Co-occurrence Matrix (GLCM) and statistical texture features calculated from the coefficients of the Multiresolution transforms such as Discrete Wavelet Transform (DWT), Curvelet, Tetrolet, and Spectral Graph Wavelet Transform (SGWT). The novelty in this work is using SGWT for extraction of texture features. The superiority of SGWT over conventional wavelet transform is its ability to work on irregular shaped images. Weighted graphs are the base for the SGWT which can be obtained in the form of meshes for irregular shapes. In the present work, skin lesions are obtained from the International Skin Imaging Collaboration (ISIC) 2016 archive. The features obtained from the Dermoscopic images are classified using Naïve Baye's, K-Nearest Neighbor (KNN) and Support Vector Machine (SVM) classifiers. The proposed method using Shape, Color and Texture based features for Melanoma Recognition with SGWT results in an Area Under the Curve (AUC) of 0.951 with Accuracy of 96.79 %, Sensitivity of 88 % and Specificity of 98.26 %. Further, the AUC of skin lesion classification such as Melanoma vs Nevus, Seborrheic Keratosis vs Squamous Cell Carcinoma, Melanoma vs Seborrheic Keratosis, Melanoma vs Basal Cell Carcinoma and Nevus vs Basal Cell Carcinoma using SGWT is 0.895, 0.945, 0.9645, 0.945 and 0.98 respectively.

Complex networks approach for depth of anesthesia assessment

Despite numerous attempts to develop a reliable depth of anesthesia (DoA) index to avoid patients’ intraoperative awareness during surgery, designing an accurate DoA index is a grand challenge in anesthesia research. In this paper, an attempt is made to design a new DoA index. We applied a statistical model and spectral graph wavelet transform (SGWT) to monitor the DoA. The de-noised electroencephalography (EEG) signals are partitioned into segments using a window technique. The window size is determined empirically, then each EEG segment is divided into sub-blocks to make the signal quasi stationary. 10 statistical characteristics are extracted from each sub-block. As a result, a vector of statistical characteristics is pulled out from each segment. Each vector of the features is then mapped as a weighted graph and spectral graph wavelet transform is performed. The total energy of wavelet coefficients at different scales is tested. The energy of wavelet coefficients at scale 3 is selected to form a SGWTDoA function. The SGWTDoA is evaluated using an anesthesia EEG recordings and the bispectral (BIS) from 22 subjects. The Bland-Altman, regression, Q-Q plot and Pearson correlation are used to verify the agreement between the SGWTDoA and the BIS. The experimental results demonstrate that the SGWTDoA has the ability to estimate the DoA accurately. The SGWTDoA is also compared and tested with the BIS in the case of poor signal quality. Our findings show that, the SGWTDoA can reflect the transition from unconsciousness to consciousness efficiently even for a poor signal while the BIS fails to display the DoA values on the monitor.

Sparse representation on graphs by tight wavelet frames and applications

In this paper, we introduce a new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start with defining the quasi-affine systems on a given manifold $\cM$ that is formed by generalized dilations and shifts of a finite collection of wavelet functions $\Psi:=\{\psi_j: 1\le j\le r\}\subset L_2(\R)$. We further require that $\psi_j$ is generated by some refinable function $\phi$ with mask $a_j$. We present the condition needed for the masks $\{a_j: 0\le j\le r\}$ so that the associated quasi-affine system generated by $\Psi$ is a tight frame for $L_2(\cM)$. Then, we discuss how the transition from the continuum (manifolds) to the discrete setting (graphs) can be naturally done. In order for the proposed discrete tight wavelet frame transforms to be useful in applications, we show how the transforms can be computed efficiently and accurately by proposing the fast tight wavelet frame transforms for graph data (WFTG). Finally, we consider two specific applications of the proposed WFTG: graph data denoising and semi-supervised clustering. Utilizing the sparse representation provided by the WFTG, we propose $\ell_1$-norm based optimization models on graphs for denoising and semi-supervised clustering. On one hand, our numerical results show significant advantage of the WFTG over the spectral graph wavelet transform (SGWT) by [1] for both applications. On the other hand, numerical experiments on two real data sets show that the proposed semi-supervised clustering model using the WFTG is overall competitive with the state-of-the-art methods developed in the literature of high-dimensional data classification, and is superior to some of these methods.

Infrared and visible image fusion with spectral graph wavelet transform.

Infrared and visible image fusion technique is a popular topic in image analysis because it can integrate complementary information and obtain reliable and accurate description of scenes. Multiscale transform theory as a signal representation method is widely used in image fusion. In this paper, a novel infrared and visible image fusion method is proposed based on spectral graph wavelet transform (SGWT) and bilateral filter. The main novelty of this study is that SGWT is used for image fusion. On the one hand, source images are decomposed by SGWT in its transform domain. The proposed approach not only effectively preserves the details of different source images, but also excellently represents the irregular areas of the source images. On the other hand, a novel weighted average method based on bilateral filter is proposed to fuse low- and high-frequency subbands by taking advantage of spatial consistency of natural images. Experimental results demonstrate that the proposed method outperforms seven recently proposed image fusion methods in terms of both visual effect and objective evaluation metrics.

Anatomically-adapted graph wavelets for improved group-level fMRI activation mapping

A graph based framework for fMRI brain activation mapping is presented. The approach exploits the spectral graph wavelet transform (SGWT) for the purpose of defining an advanced multi-resolutional spatial transformation for fMRI data. The framework extends wavelet based SPM (WSPM), which is an alternative to the conventional approach of statistical parametric mapping (SPM), and is developed specifically for group-level analysis. We present a novel procedure for constructing brain graphs, with subgraphs that separately encode the structural connectivity of the cerebral and cerebellar gray matter (GM), and address the inter-subject GM variability by the use of template GM representations. Graph wavelets tailored to the convoluted boundaries of GM are then constructed as a means to implement a GM-based spatial transformation on fMRI data. The proposed approach is evaluated using real as well as semi-synthetic multi-subject data. Compared to SPM and WSPM using classical wavelets, the proposed approach shows superior type-I error control. The results on real data suggest a higher detection sensitivity as well as the capability to capture subtle, connected patterns of brain activity.

Spectral Laplace-Beltrami wavelets with applications in medical images.

The spectral graph wavelet transform (SGWT) has recently been developed to compute wavelet transforms of functions defined on non-Euclidean spaces such as graphs. By capitalizing on the established framework of the SGWT, we adopt a fast and efficient computation of a discretized Laplace-Beltrami (LB) operator that allows its extension from arbitrary graphs to differentiable and closed 2-D manifolds (smooth surfaces embedded in the 3-D Euclidean space). This particular class of manifolds are widely used in bioimaging to characterize the morphology of cells, tissues, and organs. They are often discretized into triangular meshes, providing additional geometric information apart from simple nodes and weighted connections in graphs. In comparison with the SGWT, the wavelet bases constructed with the LB operator are spatially localized with a more uniform "spread" with respect to underlying curvature of the surface. In our experiments, we first use synthetic data to show that traditional applications of wavelets in smoothing and edge detectio can be done using the wavelet bases constructed with the LB operator. Second, we show that multi-resolutional capabilities of the proposed framework are applicable in the classification of Alzheimer's patients with normal subjects using hippocampal shapes. Wavelet transforms of the hippocampal shape deformations at finer resolutions registered higher sensitivity (96%) and specificity (90%) than the classification results obtained from the direct usage of hippocampal shape deformations. In addition, the Laplace-Beltrami method requires consistently a smaller number of principal components (to retain a fixed variance) at higher resolution as compared to the binary and weighted graph Laplacians, demonstrating the potential of the wavelet bases in adapting to the geometry of the underlying manifold.

Tight Wavelet Frames on Multislice Graphs

We present a framework for the design of wavelet transforms tailored to data defined on multislice graphs (i.e., multiplex or dynamic graphs). Graphs with multiple types of interactions are ubiquitous in real life, motivating the extension of wavelets to these complex domains. Our framework generalizes the recently proposed spectral graph wavelet transform (SGWT) [D. Hammond, P. Vandergheynst, and R. Gribonval, “Wavelets on Graphs via Spectral Graph Theory,” Appl. Comput. Harmon. Anal., vol. 30, pp. 129-150, Mar. 2011], which is designed in the spectral (frequency) domain of an arbitrary finite weighted graph. We extend the SGWT to form a tight frame, which conserves energy in the wavelet domain, and define the relationship between conventional and spectral graph wavelets. We then propose a design for multislice graphs that is based on the higher-order singular value decomposition (HOSVD), a powerful tool from multilinear algebra. In particular, the multiple adjacency matrices are stacked to form a tensor and the HOSVD decomposition provides information about its third-order structure, analogous to that provided by matrix factorizations. We obtain a set of “eigennetworks” and from these graph wavelets, which exploit the variability across the graphs. We demonstrate the feasibility of our method 1) by capturing different orientations of a gray-scale image and 2) by decomposing brain signals from functional magnetic resonance imaging. We show its effectiveness to identify variability across graph edges and provide meaningful decompositions.