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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 ArticleDOI

A graph convolutional neural network for classification of building patterns using spatial vector data

TL;DR: A novel graph convolution is introduced by converting it from the vertex domain into a point-wise product in the Fourier domain using the graph Fourier transform and convolution theorem, which achieves a significant improvement over existing methods.
Journal ArticleDOI

Advances in Distributed Graph Filtering

TL;DR: This paper generalizes state-of-the-art distributed graph filters to filters where every node weights the signal of its neighbors with different values while keeping the aggregation operation linear, and characterize a subset of shift-invariant graph filters that can be described with edge-variant recursions.
Journal ArticleDOI

A multiresolution descriptor for deformable 3D shape retrieval

TL;DR: This work presents a spectral graph wavelet framework for the analysis and design of efficient shape signatures for nonrigid 3D shape retrieval and proposes a multiresolution signature via a cubic spline wavelet generating kernel.
Posted Content

RGCNN: Regularized Graph CNN for Point Cloud Segmentation

TL;DR: A regularized graph convolutional neural network (RGCNN) that directly consumes point clouds is proposed that significantly reduces the computational complexity while achieving competitive performance with the state of the art.
Journal ArticleDOI

On the Graph Fourier Transform for Directed Graphs

TL;DR: This paper addresses the general case of directed graphs and proposes an alternative approach that builds the graph Fourier basis as the set of orthonormal vectors that minimize a continuous extension of the graph cut size, known as the Lovász extension.
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.
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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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