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Title: Wavelets from the loop scheme
Authors: Han, B.
Shen, Z. 
Cohen, A.
Keywords: Box splines
High-dimensional Riesz wavelet bases
Wavelets from the Loop scheme
Issue Date: Dec-2005
Citation: Han, B., Shen, Z., Cohen, A. (2005-12). Wavelets from the loop scheme. Journal of Fourier Analysis and Applications 11 (6) : 615-637. ScholarBank@NUS Repository.
Abstract: Anewwavelet-based geometric mesh compression algorithm was developed recently in the area of computer graphics by Khodakovsky, Schröder, and Sweldens in their interesting article [23]. The new wavelets used in [23] were designed from the Loop scheme by using ideas and methods of [26, 27], where orthogonal wavelets with exponential decay and pre-wavelets with compact support were constructed. The wavelets have the same smoothness order as that of the basis function of the Loop scheme around the regular vertices which has a continuous second derivative; the wavelets also have smaller supports than those wavelets obtained by constructions in [26, 27] or any other compactly supported biorthogonal wavelets derived from the Loop scheme (e.g., [11, 12]). Hence, the wavelets used in [23] have a good time frequency localization. This leads to a very efficient geometric mesh compression algorithm as proposed in [23]. As a result, the algorithm in [23] outperforms several available geometric mesh compression schemes used in the area of computer graphics. However, it remains open whether the shifts and dilations of the wavelets form a Riesz basis of L2(ℝ2). Riesz property plays an important role in any wavelet-based compression algorithm and is critical for the stability of any wavelet-based numerical algorithms. We confirm here that the shifts and dilations of the wavelets used in [23] for the regular mesh, as expected, do indeed form a Riesz basis of L2(ℝ2) by applying the more general theory established in this article. © 2005 Birkhäuser Boston. All rights reserved.
Source Title: Journal of Fourier Analysis and Applications
ISSN: 10695869
DOI: 10.1007/s00041-005-5013-x
Appears in Collections:Staff Publications

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