Nira Dyn

A butterfly subdivision scheme for surface interpolation with tension control

A new interpolatory subdivision scheme for surface design is presented. The new scheme is designed for a general triangulation of control points and has a tension parameter that provides design flexibility. The resulting limit surface is C1 for a specified range of the tension parameter, with a few exceptions. Application of the butterfly scheme and the role of the tension parameter are demonstrated by several examples.

A 4-point interpolatory subdivision scheme for curve design

A 4-point interpolatory subdivision scheme with a tension parameter is analysed. It is shown that for a certain range of the tension parameter the resulting curve is C^1. The role of the tension parameter is demonstrated by a few examples. The application to surfaces and some further potential generalizations are discussed.

Numerical Procedures for Surface Fitting of Scattered Data by Radial Functions

In many applications one encounters the problem of approximating surfaces from data given on a set of scattered points in a two-dimensional domain. The global interpolation methods with Duchons “thin plate splines” and Hardys multiquadrics are considered to be of high quality; however, their application is limited, due to computational difficulties, to

Subdivision schemes in geometric modelling

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Image warping by radial basis functions: applications to facial expressions

data points. In this work we develop some efficient iterative schemes for computing global approximation surfaces interpolating a given smooth data. The suggested iterative procedures can, in principle, handle any number of data points, according to computer capacity. These procedures are extensions of a previous work by Dyn and Levin on iterative methods for computing thin-plate spline interpolants for data given on a square grid. Here the procedures are improved significantly and generalized to the case of data given in a general configuration.The major theme of this work is the development of an iterative scheme for the construction of a smooth surface, presented by global basis functions, which approximates only the smooth components of a set of scattered noisy data. The novelty in the suggested method is in the construction of an iterative procedure for low-pass filtering based on detailed spectral properties of a preconditioned matrix. The general concepts of this approach can also be used in designing iterative computation procedures for many other problems.The interpolation and smoothing procedures are tested, and the theoretical results are verified, by many numerical experiments.

Analysis of uniform binary subdivision schemes for curve design

Subdivision schemes are efficient computational methods for the design and representation of 3D surfaces of arbitrary topology. They are also a tool for the generation of refinable functions, which are instrumental in the construction of wavelets. This paper presents various flavours of subdivision, seasoned by the personal viewpoint of the authors, which is mainly concerned with geometric modelling. Our starting point is the general setting of scalar multivariate nonstationary schemes on regular grids. We also briefly review other classes of schemes, such as schemes on general nets, matrix schemes, non-uniform schemes and nonlinear schemes. Different representations of subdivision schemes, and several tools for the analysis of convergence, smoothness and approximation order are discussed, followed by explanatory examples.

Convergence and C1 analysis of subdivision schemes on manifolds by proximity

Abstract The human face is an elastic object. A natural paradigm for representing facial expressions is to form a complete 3D model of facial muscles and tissues. However, determining the actual parameter values for synthesizing and animating facial expressions is tedious; evaluating these parameters for facial expression analysis out of gray-level images is ahead of the state of the art in computer vision. Using only 2D face images and a small number of anchor points, we show that the method of radial basis functions provides a powerful mechanism for processing facial expressions. Although constructed specifically for facial expressions, our method is applicable to other elastic objects as well.

INTERPOLATION OF SCATTERED DATA BY RADIAL FUNCTIONS

AbstractThe paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form

Quasilinear subdivision schemes with applications to ENO interpolation

\begin{array}{*{20}c} {f_{2i}^{k + 1} = \sum\limits_{j = 0}^m {a_j f_{i + j}^k } ,} & {f_{2i + 1}^{k + 1} = \sum\limits_{j = 0}^m {b_j f_{i + j}^k ,} } & {i \in Z,k = 0,1,2,....} \\ \end{array}