This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit d-cube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo-differential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions. Key Words: Biorthogonal wavelets, norm equivalences, boundary element methods, composite multiresolution, multiscale methods fo...

The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. However, for realistic domain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes in this sense. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach considered here is conceptually different though from o...

Subdivision surfaces are considered which consist of tri- or quadrilateral patches in a mostly regular arrangement with finitely many irregularities. A sharp estimate on the lowest possible degree of the patches is given. It depends on the smoothness and flexibility of the underlying subdivision scheme. Keywords Subdivision, piecewise polynomial surface, arbitrary topology, extraordinary point. 1 Introduction This paper deals with subdivision algorithms acting on arbitrary 2D control nets as the schemes described in [Doo78, DS78, CC78, Loo87, Qu90, DGL90, Kob94, PR96]. Here we will generalize the degree estimate in [Rei94]. A surface S generated by one of the above mentioned algorithms consists of infinitely many either tri- or quadrilateral patches which are locally arranged as illustrated in Figure 1 schematically. Note that these patches may consist of smaller patches themselves as indicated by the broken lines. For our analysis it suffices to consider such a local patch configur...

By explicitly deriving the curvature of subdivision surfaces in the extraordinary points, we give an alternative, more direct account of the criteria necessary and sufficient for achieving curvature continuity than earlier approaches that locally parametrize the surface by eigenfunctions. The approach allows us to rederive and thus survey the important lower bound results on piecewise polynomial subdivision surfaces by Prautzsch, Reif, Sabin and Zorin, as well as explain the beauty of curvature continuous constructions like Prautzsch’s. The parametrization neutral perspective gives also additional insights into the inherent constraints and stiffness of subdivision surfaces. 1

This report summarizes the findings and recommendations of the authors concerning the usefulness of subdivision surfaces for geometric modeling, and in particular for engineering applications. The work described is a result of a three-month collaboration of the authors during the visit of the second author to Boeing in the Summer of 2001.

Given a closed triangular mesh, we construct a smooth free–form surface which is described as a collection of rational tensor–product and triangular surface patches. The surface is obtained by a special manifold surface construction, which proceeds by blending together geometry functions for each vertex. The transition functions between the charts, which are associated with the vertices of the mesh, are obtained via subchart parameterization.

Abstract. We present a construction of a piecewise rational free-form surface of arbitrary topological genus which may contain sharp features: creases, corners or cusps. The surface is automatically generated from a given closed triangular mesh. Some of the edges are tagged as sharp ones, defining the features on the surface. The surface is C s smooth, for an arbitrary value of s, except for the sharp features defined by the user. Our method is based on the manifold construction and follows the blending approach.

We introduce curvature continuous regular free-form surfaces with triangular control nets. These surfaces are composed of quartic box spline surfaces and are piecewise polynomial multisided patches of total degree 8 which minimize some energy integral. The Bezier nets can be computed efficiently form the spline control net by some fixed masks, i.e. matrix multiplications.