Abstract
AbstractOur previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives—were developed for analyzing data recorded on vertices of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally $$\kappa $$
κ
-dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of $$\kappa $$
κ
-dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.
Funder
Directorate for Mathematical and Physical Sciences
National Science Foundation
Office of Naval Research
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,Radiology, Nuclear Medicine and imaging,Signal Processing,Algebra and Number Theory,Analysis
Reference56 articles.
1. Wickerhauser, M.V.: Adapted Wavelet Analysis from Theory to Software. A K Peters Ltd, Wellesley (1994)
2. Jaffard, S., Meyer, Y., Ryan, R.D.: Wavelets: Tools for Science & Technology. SIAM, Philadelphia (2001)
3. Mallat, S.: A Wavelet Tour of Signal Processing, 3rd edn. Academic Press, Burlington (2009)
4. Sayood, K.: Introduction to Data Compression, 3rd edn. Morgan Kaufmann Publishers Inc, San Francisco (2006)
5. Saito, N., Coifman, R.R.: Local discriminant bases and their applications. J. Math. Imaging Vis. 5(4), 337–358 (1995). Invited paper