Smoothness of Stationary Subdivision on Irregular Meshes

We derive necessary and sufficient conditions for tangent plane and C ^k -continuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the universal surface . Any subdivision surface can be locally represented as a projection of the universal surface, which is uniquely defined by the subdivision scheme. This approach provides us with a more intuitive geometric understanding of subdivision near extraordinary vertices. February 16, 1998. Date revised: January 27, 1999. Date accepted: April 2, 1999.     

Orthogonal and Nonorthogonal Multiresolution Analysis, Scale Discrete and Exact Fully Discrete Wavelet Transform on the Sphere

Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in the case of band-limited wavelets. June 18, 1996. Date revised: January 14, 1997.     

Convexity Preserving Interpolatory Subdivision Schemes

We construct local subdivision schemes that interpolate functional univariate data and that preserve convexity. The resulting limit function of these schemes is continuous and convex for arbitrary convex data. Moreover this class of schemes is restricted to a subdivision scheme that generates a limit function that is convex and continuously differentiable for strictly convex data. The approximation order of this scheme is four. Some generalizations, such as tension control and piecewise convexity preservation, are briefly discussed. November 29, 1996. Date revised: May 28, 1997.     

Biorthogonal Wavelet Expansions

This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular, we address the close connection of this issue with stationary subdivision schemes. Date received: May 20, 1995. Date revised: March 2, 1996.     

Biorthogonal M -Channel Compactly Supported Wavelets

We study a class of M -channel subband coding schemes with perfect reconstruction. Along the lines of [8] and [10], we construct compactly supported biorthogonal wavelet bases of L ² (R) , with dilation factor M , associated to these schemes. In particular, we study the case of splines, and obtain explicitly simple expressions for all the relevant filters. The resulting wavelets have arbitrarily large regularity and we also obtain asymptotic estimates for the regularity exponent. September 17, 1998. Date revised: June 14, 1999. Date accepted: June 25, 1999.     

Hyperbolic Wavelet Approximation

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT]. October 16, 1995. Date revised: August 28, 1996.     

Dependence Relations Among the Shifts of a Multivariate Refinable Distribution

Refinable functions are an intrinsic part of subdivision schemes and wavelet constructions. The relevant properties of such functions must usually be determined from their refinement masks. In this paper, we provide a characterization of linear independence for the shifts of a multivariate refinable vector of distributions in terms of its (finitely supported) refinement mask. March 14, 1998. Dates revised: February 3, 1999 and August 6, 1999. Date accepted: November 16, 1999.     

Regularity of Irregular Subdivision

We study the smoothness of the limit function for one-dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural generalization of the four-point scheme introduced by Dubuc and Dyn, Levin, and Gregory, we show that, under some geometric restrictions, the limit function is always C ¹ ; under slightly stronger restrictions we show that the limit function is almost C ² , the same regularity as in the regularly spaced case. May 27, 1997. Date revised: March 10, 1998. Date accepted: March 28, 1998.     

Daubechies–Matzinger wavelets and Lorentz polynomials

We construct new compactly supported wavelets and investigate their asymptotic regularity; they appear to be more regular than the Daubechies ones. These new wavelets are associated to Bernstein–Lorentz polynomials (Daubechies–Volkmer’s wavelets) and Hermite–Féjer polynomials (Lemarié–Matzinger’s wavelets) and this property enables us to derive some improved regularity ratio bounds. This revised version was published online in June 2006 with corrections to the Cover Date.     

Smoothness of subdivision surfaces at extraordinary points

A stationary subdivision scheme such as Catmull and Clark's is described by a matrix iteration around an extraordinary point. We show how higher order smoothness of a limiting surface obtained by a stationary subdivision algorithm for tri- or quadrilateral nets depends on the spectral properties of the matrix and give necessary and sufficient conditions. The results are also useful to construct subdivision algorithms for surfaces of any smoothness order at extraordinary points. This revised version was published online in August 2006 with corrections to the Cover Date.