Ternary shape-preserving subdivision schemes

We analyze the shape-preserving properties of ternary subdivision schemes generated by bell-shaped masks. We prove that any bell-shaped mask, satisfying the basic sum rules, gives rise to a convergent monotonicity preserving subdivision scheme, but convexity preservation is not guaranteed. We show that to reach convexity preservation the first order divided difference scheme needs to be bell-shaped, too. Finally, we show that ternary subdivision schemes associated with certain refinable functions with dilation 3 have shape-preserving properties of higher order

The Mask of Odd Points n

We present an explicit formula for the mask of odd points n-ary, for any odd n⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-point a-ary schemes introduced by Lian, 2008, and (2m+1)-point a-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd point n-ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented

Convexity preserving interpolatory subdivision with conic precision

The paper is concerned with the problem of shape preserving interpolatory subdivision. For arbitrarily spaced, planar input data an efficient non-linear subdivision algorithm is presented that results in $G^1$ limit curves, reproduces conic sections and respects the convexity properties of the initial data. Significant numerical examples illustrate the effectiveness of the proposed method

A UNIQUE COMBINATION OF MASK IN BINARY FOUR-POINT SUBDIVISION SCHEME

A unique binary four-point approximating subdivision scheme has been developed in which one part of binary formula have stationary mask and other part have the non-stationary mask. The resulting curves have the smoothness of C3 continuous for the wider range of shape control parameter. The role of the parameter has been depicted using the square form of discrete control points

Convexity-preserving Bernstein–Be´ zier quartic scheme

A C1 convex surface data interpolation scheme is presented to preserve the shape of scattered data arranged over a triangular grid. Bernstein–Be´ zier quartic function is used for interpolation. Lower bound of the boundary and inner Be´zier ordinates is determined to guarantee convexity of surface. The developed scheme is flexible and involves more relaxed constraints

Preserving monotone or convex data using quintic trigonometric Bézier curves

Bézier curves are essential for data interpolation. However, traditional Bézier curves often fail to detect special features that may exist in a data set, such as monotonicity or convexity, leading to invalid interpolations. This study aims to improve the deficiency of Bézier curves by imposing monotonicity or convexity-preserving conditions on the shape parameter and control points. For this purpose, the quintic trigonometric Bézier curves with two shape parameters are used. These techniques constrain only one of the shape parameters, leaving the other free to provide users with more freedom and flexibility in modifying the final curve. To guarantee smooth interpolation, the curvature profiles of the curves are analyzed, which aids in selecting the optimal shape parameter values. The effectiveness of the developed schemes was evaluated by implementing real-life data and data obtained from the existing schemes. Compared with the existing schemes, the developed schemes produce low-curvature interpolation curves with unnoticeable wiggles and turns. The proposed methods also work effectively for both nonuniformly spaced data and negative-valued convex data in real-life applications. When the shape parameter is correctly chosen, the developed interpolants exhibit continuous curvature plots, assuring $C^2$ continuity

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

The translation operator. Applications to nonlinear reconstruction operators on nonuniform grids

In this paper, we define a translation operator in a general form to allow for the application of the weighted harmonic mean in different applications. We outline the main steps to follow to define adapted methods using this tool. We give a practical example by improving the behavior of a nonlinear reconstruction operator defined in nonuniform grids, which was initially meant to work well with strictly convex data. With this improvement, the reconstruction can be now applied to data coming from smooth functions, retaining the expected maximum approximation order even around the inflection point areas, and maintaining convexity properties of the initial data. This adaptation can be carried out for general nonuniform grids, although to get the theoretical results about the approximation order, we require to work with quasi-uniform grids. We check the theoretical results through some numerical experiments

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum