Uniform subdivision algorithms for curves and surfaces

A convergence analysis for studying the continuity and differentiability of limit curves generated by uniform subdivision algorithms is presented. The analysis is based on the study of corresponding difference and divided difference algorithms. The alternative process of "integrating" the algorithms is considered. A specific example of a 4-point interpolatory curve algorithm is described and its generalization to a surface algorithm defined over a subdivision of a regular triangular partition is illustrated

Analysis of uniform binary subdivision schemes for curve design

The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form .0,1,2,...kz,ikj,ifjbm0j1k12ifjam0j1k2if=∈+Σ==++Σ==+ The convergence of the control polygons to a Cu curve is analysed in terms of the convergence to zero of a derived scheme for the differences - . The analysis of the smoothness of the limit curve is reduced to kif the convergence analysis of "differentiated" schemes which correspond to divided differences of {/i ∈Z} with respect to the diadic parameteriz- kif ation = i/2kitk . The inverse process of "integration" provides schemes with limit curves having additional orders of smoothness

A necessary condition for best approximation in monotone and sign-monotone norms

AbstractBest approximation to ƒ ϵ C[a, b] by elements of an n-dimensional Tchebycheff space in monotone norms (norms defined on C[a, b] for which ¦ƒ(x)¦ ⩽ ¦ g(x)¦, a ⩽ x ⩽ b, implies ∥ƒ∥ ⩽ ∥g∥) is studied. It is proved that the error function has at least n zeroes in [a, b], counting twice interior zeroes with no change of sign. This result is best possible for monotone norms in general, and improves the one in [5]. The proof follows from the observation that, for any monotone norm, sgn ƒ(x) = sgn g(x), a ⩽ x ⩽ b, implies ∥ƒ− λg ∥ < ∥ƒ∥ for λ > 0 small enough. This property is shown to characterize a class of norms wider than the class of monotone norms, namely “sign-monotone” norms defined by: ¦ƒ(x)¦ ⩽ ¦g(x)¦, ƒ(x) g(x) ⩾ 0, a ⩽ x ⩽ b, implies ∥ƒ∥ ⩽ ∥g∥. It is noted that various results concerning approximation in monotone norms, are actually valid for approximation in sign-monotone norms

Non-uniform interpolatory subdivision schemes with improved smoothness

Subdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For non-stationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Non-uniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of non-stationary and non-uniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a non-stationary or non-uniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that non-uniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2-point and 4-point schemes that generate C-1 and C-2 limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one. (C) 2022 The Author(s). Published by Elsevier B.V

Generalized Refinement Equations and Subdivision Processes

AbstractThe concept of subdivision schemes is generalized to schemes with a continuous mask, generating compactly supported solutions of corresponding functional equations in integral form. A necessary and a sufficient condition for uniform convergence of these schemes are derived. The equivalence of weak convergence of subdivision schemes with the existence of weak compactly supported solutions to the corresponding functional equations is shown for both the discrete and integral cases. For certain non-negative masks stronger results are derived by probabilistic methods. The solution of integral functional equations whose continuous masks solve discrete functional equations, are shown to be limits of discrete nonstationary schemes with masks of increasing support. Interesting functions created by these schemes are C∞ functions of compact support including the up-function of Rvachev

An Interpolatory Subdivision Scheme for Triangular Meshes and Progressive Transmission

4 authors, including: chen ren Guangzhou cool-smart electronical inform

Fourier analysis of 2-point Hermite interpolatory subdivision schemes

Two subdivision schemes with Hermite data on Z are studied. These schemes use 2 or 7 parameters respectively depending on whether Hermite data involve only first derivatives or include second derivatives. For a large region in the parameters space, the schemes are C1 or C2 convergent or at least are convergent on the space of Schwartz distributions. The Fourier transform of any interpolating function can be computed through products of matrices of order 2 or 3. The Fourier transform is related to a specific system of functional equations whose analytic solution is unique except for a multiplicative constant. The main arguments for these results come from Paley-Wiener-Schwartz theorem on the characterization of the Fourier transforms of distributions with compact support and a theorem of Artzrouni about convergent products of matrices