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Wavelets on Graphs via Spectral Graph Theory

TLDR
In this paper, the spectral graph wavelet operator is defined based on spectral decomposition of the discrete graph Laplacian, and a wavelet generating kernel and a scale parameter are used to localize this operator to an indicator function.
Abstract
We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $\L$. Given a wavelet generating kernel $g$ and a scale parameter $t$, we define the scaled wavelet operator $T_g^t = g(t\L)$. The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on $g$, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing $\L$. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.

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Citations
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Journal Article

Optimal Transport Graph Neural Networks

TL;DR: OT-GNN is introduced that compute graph embeddings from optimal transport distances between the set of GNN nodeembeddings and "prototype" point clouds as free parameters, which consistently exhibits better generalization performance on several molecular property prediction tasks, yielding also smoother representations.
Posted ContentDOI

Enhancing experimental signals in single-cell RNA-sequencing data using graph signal processing

TL;DR: A continuous measure of the effect of an experiment across the transcriptomic space is described, using the manifold assumption to model the cellular state space as a graph (or network) with cells as nodes and edges connecting cells with similar transcriptomic profiles.
Posted Content

Predictive Generalized Graph Fourier Transform for Attribute Compression of Dynamic Point Clouds.

TL;DR: This work proposes a complete compression framework for attributes of 3D dynamic point clouds, focusing on optimal inter-coding and refined motion estimation via efficient registration prior to inter-prediction, which searches the temporal correspondence between adjacent frames of irregular point clouds.
Proceedings ArticleDOI

Hierarchical in-network attribute compression via importance sampling

TL;DR: A lossy network compression scheme called Slice Tree (ST), which partitions a network into smooth regions with respect to node/edge values and compresses each value as the average of its region, and an importance sampling algorithm to efficiently prune the search space of candidate slices in the ST construction.
Proceedings ArticleDOI

Depth Map Super-Resolution Using Synthesized View Matching for Depth-Image-Based Rendering

TL;DR: A novel super-resolution (SR) algorithm is proposed to increase the resolution of the received depth map at decoder to match the corresponding received high resolution texture map for DIBR to enable transmission of depth maps at low resolution for bit saving.
References
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Book

A wavelet tour of signal processing

TL;DR: An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Proceedings ArticleDOI

Object recognition from local scale-invariant features

TL;DR: Experimental results show that robust object recognition can be achieved in cluttered partially occluded images with a computation time of under 2 seconds.
Book

Ten lectures on wavelets

TL;DR: This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
Book

Functional analysis

Walter Rudin
Journal ArticleDOI

Ten Lectures on Wavelets

TL;DR: In this article, the regularity of compactly supported wavelets and symmetry of wavelet bases are discussed. But the authors focus on the orthonormal bases of wavelets, rather than the continuous wavelet transform.
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