Title :
Multiscale geometric wavelets for the analysis of point clouds
Author :
Chen, Guangliang ; Maggioni, Mauro
Author_Institution :
Dept. of Math., Duke Univ., Durham, NC, USA
Abstract :
Data sets are often modeled as point clouds in ¿D, for D large. It is often assumed that the data has some interesting low-dimensional structure, for example that of a d-dimensional manifold M, with d much smaller than D. When M is simply a linear subspace, one may exploit this assumption for encoding efficiently the data by projecting onto a dictionary of d vectors in ¿D (for example found by SVD), at a cost (d + n)D for n data points. When M is nonlinear, there are no "explicit" constructions of dictionaries that achieve a similar efficiency: typically one uses either random dictionaries, or dictionaries obtained by black-box optimization. In this paper we construct data-dependent multiscale dictionaries that aim at efficient encoding and manipulating of the data. Their construction is fast, and so are the algorithms to map data points to dictionary coefficients and vice versa. In addition, data points are guaranteed to have a sparse representation in terms of the dictionary. We think of dictionaries as the analogue of wavelets, but for approximating point clouds rather than functions.
Keywords :
data handling; database management systems; optimisation; wavelet transforms; black box optimization; data dependent multiscale dictionaries; data encoding; data manipulation; data sets; multiscale geometric wavelets; point clouds analysis; Clouds; Dictionaries; Encoding; Machine learning; Manifolds; Mathematics; Optimization methods; Signal processing; Signal processing algorithms; Wavelet analysis; Data Sets; Frames; Multiscale Analysis; Point Clouds; Sparse Approximation; Wavelets;
Conference_Titel :
Information Sciences and Systems (CISS), 2010 44th Annual Conference on
Conference_Location :
Princeton, NJ
Print_ISBN :
978-1-4244-7416-5
Electronic_ISBN :
978-1-4244-7417-2
DOI :
10.1109/CISS.2010.5464843