Locally supported, piecewise polynomial biorthogona wavelets on non-uniform meshes

In this paper, biorthogonal wavelets are constructed on non-uniform meshes. Both primal and dual wavelets are explicitly given locally supported, continuous piecewise polynomials. The wavelets generate Riesz bases for the Sobolev spaces H s for j s j < 3 2 . The wavelets at the primal side span standard Lagrange nite element spaces

Wavelet representation of functions defined on tetrahedrical grids

In this paper, a method for representing scalar functions on volumes is presented. The method is based on wavelets and it can be used for representing volumetric data (geometric or scalar) defifined on non structured grids. The basic contribution is the extension of wavelets to represent scalar functions on volumetric domains of arbitrary topological type. This extension is made by constructing a wavelet basis defifined on any tetrahedrized volume. This basis construction is achieved using multiresolution analysis and the lifting schemeFacultad de Informátic

Multiresolution editing for B-spline curves and surfaces

Since 1980 surface modeling has been used in industrial design, CAD and entertainment to create and represent complex forms. Even with this comparatively long history of development, challenges remain in free-form surface modeling. One such challenge is building surface creation and editing techniques that effectively balance the need for local control with the need to control the overall global shape, or sweep of the surface. This dissertation presents a multiresolution approach to the creation of surfaces that allows a designer to more easily manage this balance between local and global control. The techniques presented in this dissertation utilize a wavelet decomposition of B-spline curves and surfaces to allow a designer to easily develop the basic shape using lower level representations, and then seamlessly switch to higher level representations to achieve fine control over local features. The algorithms described in the dissertation are implemented in an interactive software system that is used to demonstrate their effectiveness in comparison to existing methods

Wavelet-based multiresolution data representations for scalable distributed GIS services

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2002.Includes bibliographical references (p. 155-160).Demand for providing scalable distributed GIS services has been growing greatly as the Internet continues to boom. However, currently available data representations for these services are limited by a deficiency of scalability in data formats. In this research, four types of multiresolution data representations based on wavelet theories have been put forward. The designed Wavelet Image (WImg) data format helps us to achieve dynamic zooming and panning of compressed image maps in a prototype GIS viewer. The Wavelet Digital Elevation Model (WDEM) format is developed to deal with cell-based surface data. A WDEM is better than a raster pyramid in that a WDEM provides a non-redundant multiresolution representation. The Wavelet Arc (WArc) format is developed for decomposing curves into a multiresolution format through the lifting scheme. The Wavelet Triangulated Irregular Network (WTIN) format is developed to process general terrain surfaces based on the second generation wavelet theory. By designing a strategy to resample a terrain surface at subdivision points through the modified Butterfly scheme, we achieve the result: only one wavelet coefficient needs to be stored for each point in the final representation. In contrast to this result, three wavelet coefficients need to be stored for each point in a general 3D object wavelet-based representation. Our scheme is an interpolation scheme and has much better performance than the Hat wavelet filter on a surface. Boundary filters are designed to make the representation consistent with the rectangular boundary constraint.(cont.) We use a multi-linked list and a quadtree array as the data structures for computing. A method to convert a high resolution DEM to a WTIN is also provided. These four wavelet-based representations provide consistent and efficient multiresolution formats for online GIS. This makes scalable distributed GIS services more efficient and implementable.by Jingsong Wu.Ph.D

Lifting-based subdivision wavelets with geometric constraints.

Qin, Guiming."August 2010."Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (p. 72-74).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 1.1 --- B splines and B-splines surfaces --- p.5Chapter 1. 2 --- Box spline --- p.6Chapter 1. 3 --- Biorthogonal subdivision wavelets based on the lifting scheme --- p.7Chapter 1.4 --- Geometrically-constrained subdivision wavelets --- p.9Chapter 1.5 --- Contributions --- p.9Chapter 2 --- Explicit symbol formulae for B-splines --- p.11Chapter 2. 1 --- Explicit formula for a general recursion scheme --- p.11Chapter 2. 2 --- Explicit formulae for de Boor algorithms of B-spline curves and their derivatives --- p.14Chapter 2.2.1 --- Explicit computation of de Boor Algorithm for Computing B-Spline Curves --- p.14Chapter 2.2.2 --- Explicit computation of Derivatives of B-Spline Curves --- p.15Chapter 2. 3 --- Explicit power-basis matrix fomula for non-uniform B-spline curves --- p.17Chapter 3 --- Biorthogonal subdivision wavelets with geometric constraints --- p.23Chapter 3. 1 --- Primal subdivision and dual subdivision --- p.23Chapter 3. 2 --- Biorthogonal Loop-subdivision-based wavelets with geometric constraints for triangular meshes --- p.24Chapter 3.2.1 --- Loop subdivision surfaces and exact evaluation --- p.24Chapter 3.2.2 --- Lifting-based Loop subdivision wavelets --- p.24Chapter 3.2.3 --- Biorthogonal Loop-subdivision wavelets with geometric constraints --- p.26Chapter 3. 3 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.35Chapter 3.3.1 --- Catmull-Clark subdivision and Doo-Sabin subdivision surfaces --- p.35Chapter 3.3.1.1 --- Catmull-Clark subdivision --- p.36Chapter 3.3.1.2 --- Doo-Sabin subdivision --- p.37Chapter 3.3.2 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.38Chapter 3.3.2.1 --- Biorthogonal Doo-Sabin subdivision wavelets with geometric constraints --- p.38Chapter 3.3.2.2 --- Biorthogonal Catmull-Clark subdivision wavelets with geometric constraints --- p.44Chapter 4 --- Experiments and results --- p.49Chapter 5 --- Conclusions and future work --- p.60Appendix A --- p.62Appendix B --- p.67Appendix C --- p.69Appendix D --- p.71References --- p.7