A New Four Point Circular-Invariant Corner-Cutting Subdivision for Curve Design

A 4-point nonlinear corner-cutting subdivision scheme is established. It is induced from a special C-shaped biarc circular spline structure. The scheme is circular-invariant and can be effectively applied to 2-dimensional (2D) data sets that are locally convex. The scheme is also extended adaptively to non-convex data. Explicit examples are demonstrated

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

Kurven – Verstehen – Entwerfen – Berechnen – Darstellen

Das Buch ist für einen breiten Leserkreis gedacht und enthält eine Einführung in verschiedene praktisch relevante Aspekte von Kurven mit vielen Abbildungen: So werden differentialgeometrische Grundlagen behandelt, Bézier- und B-Spline-Darstellungen, rationale und algebraische Kurven, sowie einige speziele Kurven, die aus historischen Gründen oder ihrer Anwendungen wegen von Bedeutung sind. Auf Beweise, die tiefer gehende mathematische Vorkenntnisse voraussetzen, wird verzichtet

Non-linear subdivision of univariate signals and discrete surfaces

During the last 20 years, the joint expansion of computing power, computer graphics, networking capabilities and multiresolution analysis have stimulated several research domains, and developed the need for new types of data such as 3D models, i.e. discrete surfaces. In the intersection between multiresolution analysis and computer graphics, subdivision methods, i.e. iterative refinement procedures of curves or surfaces, have a non-negligible place, since they are a basic component needed to adapt existing multiresolution techniques dedicated to signals and images to more complicated data such as discrete surfaces represented by polygonal meshes. Such representations are of great interest since they make polygonal meshes nearly as exible as higher level 3D model representations, such as piecewise polynomial based surfaces (e.g. NURBS, B-splines...). The generalization of subdivision methods from univariate data to polygonal meshes is relatively simple in case of a regular mesh but becomes less straightforward when handling irregularities. Moreover, in the linear univariate case, obtaining a smoother limit curve is achieved by increasing the size of the support of the subdivision scheme, which is not a trivial operation in the case of a surface subdivision scheme without a priori assumptions on the mesh. While many linear subdivision methods are available, the studies concerning more general non-linear methods are relatively sparse, whereas such techniques could be used to achieve better results without increasing the size support. The goal of this study is to propose and to analyze a binary non-linear interpolatory subdivision method. The proposed technique uses local polar coordinates to compute the positions of the newly inserted points. It is shown that the method converges toward continuous limit functions. The proposed univariate scheme is extended to triangular meshes, possibly with boundaries. In order to evaluate characteristics of the proposed scheme which are not proved analytically, numerical estimates to study convergence, regularity of the limit function and approximation order are studied and validated using known linear schemes of identical support. The convergence criterion is adapted to surface subdivision via a Hausdorff distance-based metric. The evolution of Gaussian and mean curvature of limit surfaces is also studied and compared against theoretical values when available. An application of surface subdivision to build a multiresolution representation of 3D models is also studied. In particular, the efficiency of such a representation for compression and in terms of rate-distortion of such a representation is shown. An alternate to the initial SPIHT-based encoding, based on the JPEG 2000 image compression standard method. This method makes possible partial decoding of the compressed model in both SNR-progressive and level-progressive ways, while adding only a minimal overhead when compared to SPIHT

Nonuniform corner cutting

10.1016/0167-8396(96)00008-8Computer Aided Geometric Design138763-772CAGD

Conditions

Abstract. Sufficient conditions are given for C 1 and C 2 (subdivision) curves generated by a particular non-uniform, interpolatory, variational refinement scheme. The ‘energy ’ functional being minimized is a discretization of the standard linearized spline functional over piecewise linear curves – a generalization of the minimizing functional used for the uniform scheme in [10]. The conditions used are uniform bounds on either the energy functional or certain divided differences, along with a condition on the knots, forcing them to be dense and uniform in the limit. To establish C 2, a certain ‘bootstrap ’ argument is applied. The argument is based on a generalization of a result in [6] used to show smoothness of curves generated by nonuniform corner cutting. In this paper we investigate the smoothness of the limiting curves generated by a particular non-uniform, interpolatory, variational refinemen