Adaptive Online Learning of Separable Path Graph Transforms for Intra-prediction

Current video coding standards, including H.264/AVC, HEVC, and VVC, employ discrete cosine transform (DCT), discrete sine transform (DST), and secondary to Karhunen-Loeve transforms (KLTs) decorrelate the intra-prediction residuals. However, the efficiency of these transforms in decorrelation can be limited when the signal has a non-smooth and non-periodic structure, such as those occurring in textures with intricate patterns. This paper introduces a novel adaptive separable path graph-based transform (GBT) that can provide better decorrelation than the DCT for intra-predicted texture data. The proposed GBT is learned in an online scenario with sequential K-means clustering, which groups similar blocks during encoding and decoding to adaptively learn the GBT for the current block from previously reconstructed areas with similar characteristics. A signaling overhead is added to the bitstream of each coding block to indicate the usage of the proposed graph-based transform. We assess the performance of this method combined with H.264/AVC intra-coding tools and demonstrate that it can significantly outperform H.264/AVC DCT for intra-predicted texture data.

Lossy Compression of Adjacency Matrices by Graph Filter Banks

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.

Fast graph-based denoising for point cloud color information

Point clouds are utilized in various 3D applications such as cross-reality (XR) and realistic 3D displays. In some applications, e.g., for live streaming using a 3D point cloud, real-time point cloud denoising methods are required to enhance the visual quality. However, conventional high-precision denoising methods cannot be executed in real time for large-scale point clouds owing to the complexity of graph constructions with K nearest neighbors and noise level estimation. This paper proposes a fast graph-based denoising (FGBD) for a large-scale point cloud. First, high-speed graph construction is achieved by scanning a point cloud in various directions and searching adjacent neighborhoods on the scanning lines. Second, we propose a fast noise level estimation method using eigenvalues of the covariance matrix on a graph. Finally, we also propose a new low-cost filter selection method to enhance denoising accuracy to compensate for the degradation caused by the acceleration algorithms. In our experiments, we succeeded in reducing the processing time dramatically while maintaining accuracy relative to conventional denoising methods. Denoising was performed at 30fps, with frames containing approximately 1 million points.

Optimizing $k$ in $k$NN Graphs with Graph Learning Perspective

In this paper, we propose a method, based on graph signal processing, to optimize the choice of $k$ in $k$-nearest neighbor graphs ($k$NNGs). $k$NN is one of the most popular approaches and is widely used in machine learning and signal processing. The parameter $k$ represents the number of neighbors that are connected to the target node; however, its appropriate selection is still a challenging problem. Therefore, most $k$NNGs use ad hoc selection methods for $k$. In the proposed method, we assume that a different $k$ can be chosen for each node. We formulate a discrete optimization problem to seek the best $k$ with a constraint on the sum of distances of the connected nodes. The optimal $k$ values are efficiently obtained without solving a complex optimization. Furthermore, we reveal that the proposed method is closely related to existing graph learning methods. In experiments on real datasets, we demonstrate that the $k$NNGs obtained with our method are sparse and can determine an appropriate variable number of edges per node. We validate the effectiveness of the proposed method for point cloud denoising, comparing our denoising performance with achievable graph construction methods that can be scaled to typical point cloud sizes (e.g., thousands of nodes).

Frequency analysis and filter design for directed graphs with polar decomposition

In this study, we challenge the traditional approach of frequency analysis on directed graphs, which typically relies on a single measure of signal variation such as total variation. We argue that the inherent directionality in directed graphs necessitates a multifaceted analytical approach, one that incorporates multiple definitions of signal variations. Our methodology leverages the polar decomposition to define two distinct variations, each associated with different matrices derived from this decomposition. This approach not only provides a novel interpretation in the node domain but also reveals aspects of graph signals that may be overlooked with a singular measure of variation. Additionally, we develop graph filters specifically designed to smooth graph signals in accordance with our proposed variations. These filters allow for the bypassing of costly filtering operations associated with the original graph through effective cascading. We demonstrate the efficacy of our methodology using an M-block cyclic graph example, thereby validating our claims and showcasing the advantages of our multifaceted approach in analyzing signals on directed graphs.

Irregularity-Aware Bandlimited Approximation for Graph Signal Iinterpolation

In most work to date, graph signal sampling and reconstruction algorithms are intrinsically tied to graph properties, assuming bandlimitedness and optimal sampling set choices. However, practical scenarios often defy these assumptions, leading to suboptimal performance. In the context of sampling and reconstruction, graph irregularities lead to varying contributions from sampled nodes for interpolation and differing levels of reliability for interpolated nodes. The existing GFT-based methods in the literature make bandlimited signal approximations without taking into account graph irregularities and the relative significance of nodes, resulting in suboptimal reconstruction performance under various mismatch conditions. In this paper, we leverage the GFT equipped with a specific inner product to address graph irregularities and account for the relative importance of nodes during the bandlimited signal approximation and interpolation process. Empirical evidence demonstrates that the proposed method outperforms other GFT-based approaches for bandlimited signal interpolation in challenging scenarios, such as sampling sets selected independently of the underlying graph, low sampling rates, and high noise levels.

Towards a geometric understanding of Spatio Temporal Graph Convolution Networks

Spatiotemporal graph convolutional networks (STGCNs) have emerged as a desirable model for skeleton-based human action recognition. Despite achieving state-of-the-art performance, there is a limited understanding of the representations learned by these models, which hinders their application in critical and real-world settings. While layerwise analysis of CNN models has been studied in the literature, to the best of our knowledge, there exists no study on the layerwise explainability of the embeddings learned on spatiotemporal data using STGCNs. In this paper, we first propose to use a local Dataset Graph (DS-Graph) obtained from the feature representation of input data at each layer to develop an understanding of the layer-wise embedding geometry of the STGCN. To do so, we develop a window-based dynamic time warping (DTW) method to compute the distance between data sequences with varying temporal lengths. To validate our findings, we have developed a layer-specific Spatiotemporal Graph Gradient-weighted Class Activation Mapping (L-STG-GradCAM) technique tailored for spatiotemporal data. This approach enables us to visually analyze and interpret each layer within the STGCN network. We characterize the functions learned by each layer of the STGCN using the label smoothness of the representation and visualize them using our L-STG-GradCAM approach. Our proposed method is generic and can yield valuable insights for STGCN architectures in different applications. However, this paper focuses on the human activity recognition task as a representative application. Our experiments show that STGCN models learn representations that capture general human motion in their initial layers while discriminating different actions only in later layers. This justifies experimental observations showing that fine-tuning deeper layers works well for transfer between related tasks.

The faces of Convolution: from the Fourier theory to algebraic signal processing

In this expository article, we provide a self-contained overview of the notion of convolution embedded in different theories: from the classical Fourier theory to the theory of algebraic signal processing. We discuss their relations and differences. Toward the end, we provide an opinion on whether there is a consistent approach to convolution that unifies seemingly different approaches by different theories.

Signal Variation Metrics and Graph Fourier Transforms for Directed Graphs

In this paper we consider the problem of constructing graph Fourier transforms (GFTs) for directed graphs (digraphs), with a focus on developing multiple GFT designs that can capture different types of variation over the digraph node-domain. Specifically, for any given digraph we propose three GFT designs based on the polar decomposition. Our method is closely related to existing polar decomposition based GFT designs, but with added interpretability in the digraph node-domain. Each of our proposed digraph GFTs has a clear node domain variation interpretation, so that one or more of the GFTs can be used to extract different insights from available graph signals. We demonstrate the benefits of our approach experimentally using M-block cyclic graphs, showing that the diffusion of signals on the graph leads to changes in the spectrum obtained from our proposed GFTs, but cannot be observed with the conventional GFT definition.

Graph Signal Processing: History, Development, Impact, and Outlook

Graph signal processing (GSP) generalizes signal processing (SP) tasks to signals living on non-Euclidean domains whose structure can be captured by a weighted graph. Graphs are versatile, able to model irregular interactions, easy to interpret, and endowed with a corpus of mathematical results, rendering them natural candidates to serve as the basis for a theory of processing signals in more irregular domains. In this article, we provide an overview of the evolution of GSP, from its origins to the challenges ahead. The first half is devoted to reviewing the history of GSP and explaining how it gave rise to an encompassing framework that shares multiple similarities with SP. A key message is that GSP has been critical to develop novel and technically sound tools, theory, and algorithms that, by leveraging analogies with and the insights of digital SP, provide new ways to analyze, process, and learn from graph signals. In the second half, we shift focus to review the impact of GSP on other disciplines. First, we look at the use of GSP in data science problems, including graph learning and graph-based deep learning. Second, we discuss the impact of GSP on applications, including neuroscience and image and video processing. We conclude with a brief discussion of the emerging and future directions of GSP.