Multiresolution for algebraic curves and surfaces using wavelets

This paper describes a multiresolution method for implicit curves and surfaces. The method is based on wavelets, and is able to simplify the topology. The implicit curves and surfaces are defined as the zero-valued algebraic isosurface of a tensor-product uniform cubic Bspline. A wavelet multiresolution method that deals with uniform cubic Bsplines on bounded domains has been constructed. Further, the report explains how to set the unknown coefficients to produce the most compact object, how to recover the initial object, a suitable data structure and, finally, points out several improvements that might produce better results.Postprint (published version

Multiresolution analysis as an approach for tool path planning in NC machining

Wavelets permit multiresolution analysis of curves and surfaces. A complex curve can be decomposed using wavelet theory into lower resolution curves. The low-resolution (coarse) curves are similar to rough-cuts and high-resolution (fine) curves to finish-cuts in numerical controlled (NC) machining.;In this project, we investigate the applicability of multiresolution analysis using B-spline wavelets to NC machining of contoured 2D objects. High-resolution curves are used close to the object boundary similar to conventional offsetting, while lower resolution curves, straight lines and circular arcs are used farther away from the object boundary.;Experimental results indicate that wavelet-based multiresolution tool path planning improves machining efficiency. Tool path length is reduced, sharp corners are smoothed out thereby reducing uncut areas and larger tools can be selected for rough-cuts

Reversing subdivision rules: local linear conditions and observations on inner products

AbstractIn a previous work (Samavati and Bartels, Comput. Graphics Forum 18 (1998) 97–119) we investigated how to reverse subdivision rules using global least-squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finite-dimensional Cartesian vector space. We produced simple and sparse reconstruction filters, but had to appeal to matrix factorization to obtain an efficient, exact decomposition. We also made some observations on how the inner product that defines the semiorthogonality influences the sparsity of the reconstruction filters. In this work we carry the investigation further by studying biorthogonal systems based upon subdivision rules and local least-squares fitting problems that reverse the subdivision. We are able to produce multiresolution structures for some common univariate subdivision rules that have both sparse reconstruction and decomposition filters. Three will be presented here – for quadratic and cubic B-spline subdivision and for the four-point interpolatory subdivision of Dyn et al. We observe that each biorthogonal system we produce can be interpreted as a semiorthogonal system with an inner product induced on the multiresolution that is quite different from that normally used. Some examples of the use of this approach on images, curves, and surfaces are given

Similarity Analysis of Nonlinear Equations and Bases of Finite Wavelength Solitons

We introduce a generalized similarity analysis which grants a qualitative description of the localised solutions of any nonlinear differential equation. This procedure provides relations between amplitude, width, and velocity of the solutions, and it is shown to be useful in analysing nonlinear structures like solitons, dublets, triplets, compact supported solitons and other patterns. We also introduce kink-antikink compact solutions for a nonlinear-nonlinear dispersion equation, and we construct a basis of finite wavelength functions having self-similar properties.Comment: 18 pages Latex, 6 figures ep

Multiresolution analysis of electronic structure: semicardinal and wavelet bases

This article reviews recent developments in multiresolution analysis which make it a powerful tool for the systematic treatment of the multiple length-scales inherent in the electronic structure of matter. Although the article focuses on electronic structure, the advances described are useful for non-linear problems in the physical sciences in general. The new language and notations introduced are well- suited for both formal manipulations and the development of computer software using higher-level languages such as C++. The discussion is self-contained, and all needed algorithms are specified explicitly in terms of simple operators and illustrated with straightforward diagrams which show the flow of data. Among the reviewed developments is the construction of_exact_ multiresolution representations from extremely limited samples of physical fields in real space. This new and profound result is the critical advance in finally allowing systematic, all electron calculations to compete in efficiency with state-of-the-art electronic structure calculations which depend for their celerity upon freezing the core electronic degrees of freedom. This review presents the theory of wavelets from a physical perspective, provides a unified and self-contained treatment of non-linear couplings and physical operators and introduces a modern framework for effective single-particle theories of quantum mechanics.Comment: A "how-to from-scratch" book presently in press at Reviews of Modern Physics: 88 pages, 31 figures, 5 tables, 88 references. Significantly IMPROVED version, including (a) new diagrams illustrating algorithms; (b) careful proof-reading of equations and text; (c) expanded bibliography; (d) cosmetic changes including lists of figures and tables and a more reasonable font. Latest changes (Dec. 11, 1998): a more descriptive abstract, and minor lexicographical change