We give a formula for the duals of the masks associated with trivariate box spline functions. We show how to construct trivariate nonseparable compactly supported biorthogonal wavelets associated with box spline functions. The biorthogonal wavelets may have arbitrarily high regularities.

We study the reconstruction of cardinal splines f(t), from their average samples yn = f ∗h(n), n ∈ Z, when the average function h(t) has support in [−1/2, 1/2]. We investigate the existence and uniqueness of the solution of the following problem: for given dates yn, find a cardinal spline f(t), of a given degree, satisfying yn = f ∗ h(n), n ∈ Z. 1 Spline interpolation and average sampling First...

In both applications and wavelet theory, the spline wavelets are especially interesting, in part because of their simple structure. In a previous paper we proved that the function m;l is an m th order spline wavelet having an l th order spline dual wavelet. This enabled us to derive biorthogonal spline wavelet bases. In this paper we rst study the general structure of cardinal spline wavelets, ...

An algebraic approach to the design of different binds of discrete wavelet transforma (orthogonal and biorthogonal single-/multiwavelet transforms, multiwavelet-like transforma) is taken. The different transfomrs are anaL ysed with respect to computational efforts, approzimation properties and symmetry. The design of the orthogonal and biorthogonal single-/multiwavelets requires the solution of...

The derivatives of the Gaussian function, G(x) = 1 √ 2π e−x 2/2, produce the Hermite polynomials by the relation, (−1)mG(m)(x) = Hm(x)G(x), m = 0, 1, . . . , where Hm(x) are Hermite polynomials of degree m. The orthonormal property of the Hermite polynomials, 1 m! ∫∞ −∞Hm(x)Hn(x)G(x)dx = δmn, can be considered as a biorthogonal relation between the derivatives of the Gaussian, {(−1)nG(n) : n = ...

We show how to combine, into one unified spline complex, C2 tensorproduct bi-cubic NURBS and G2 bi-cubic rational splines. The G2 splines are capable of exactly representing basic shapes such as (pieces of) quadrics and surfaces of revolution, including tori and cyclides. The main challenge is to bridge the differing continuity. We transform the G2 splines to splines that are C2 in homogeneous ...

This paper presents a new approach to computation of geometric continuity for parametric bi-cubic patches, based on a simple mathematical reformulation which leads to simple additional conditions to be applied in the patching computation. The paper presents an Hermite formulation of a bicubic parametric patch, but reformulations can be made also for Bézier and B-Spline patches as well. The pres...

Inverse solutions to interpolation techniques in computer graphics may increase the accuracy of the rendered frames. An example is the use of an arbitrary time stamp in time interpolation to evaluate the t-parameter, which in turn can be used for time-variant spatial interpolation. In this paper, an analytic inverse is presented for a shape-preserving piecewise cubic Hermite interpolant, used i...

The Euclidean Algorithm is the often forgotten key to rational approximation techniques, including Taylor, Lagrange, Hermite, osculating, cubic spline, Chebyshev, Padé and other interpolation schemes. A unified view of these various interpolation techniques is eloquently expressed in terms of the concept of the spectral basis of a factor ring of polynomials. When these methods are applied to th...