Bivariate Hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in $\mathbb{R}^3$ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points $\{(x_i, y_i)\}_{i=1}^N$ from which none of the pairs $(x_i,y_i)$ and $(x_j,y_j)$ with $i\neq j$ coincide, it is proved that the resulting surface (function) is $C^1$. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is $C^2$ if the data are not 'too irregular'. Finally the approximation properties of the methods are investigated

Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

Recursive subdivision algorithms for curve and surface design

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.The Chinese Educational Commission and The British Council (SBFSS/1987

Smooth Subdivision Surfaces: Mesh Blending and Local Interpolation

Subdivision surfaces are widely used in computer graphics and animation. Catmull-Clark subdivision (CCS) is one of the most popular subdivision schemes. It is capable of modeling and representing complex shape of arbitrary topology. Polar surface, working on a triangle-quad mixed mesh structure, is proposed to solve the inherent ripple problem of Catmull-Clark subdivision surface (CCSS). CCSS is known to be C1 continuous at extraordinary points. In this work, we present a G2 scheme at CCS extraordinary points. The work is done by revising CCS subdivision step with Extraordinary-Points-Avoidance model together with mesh blending technique which selects guiding control points from a set of regular sub-meshes (named dominative control meshes) iteratively at each subdivision level. A similar mesh blending technique is applied to Polar extraordinary faces of Polar surface as well. Both CCS and Polar subdivision schemes are approximating. Traditionally, one can obtain a CCS limit surface to interpolate given data mesh by iteratively solving a global linear system. In this work, we present a universal interpolating scheme for all quad subdivision surfaces, called Bezier Crust. Bezier Crust is a specially selected bi-quintic Bezier surface patch. With Bezier Crust, one can obtain a high quality interpolating surface on CCSS by parametrically adding CCSS and Bezier Crust. We also show that with a triangle/quad conversion process one can apply Bezier Crust on Polar surfaces as well. We further show that Bezier Crust can be used to generate hollowed 3D objects for applications in rapid prototyping. An alternative interpolating approach specifically designed for CCSS is developed. This new scheme, called One-Step Bi-cubic Interpolation, uses bicubic patches only. With lower degree polynomial, this scheme is appropriate for interpolating large-scale data sets. In sum, this work presents our research on improving surface smoothness at extraordinary points of both CCS and Polar surfaces and present two local interpolating approaches on approximating subdivision schemes. All examples included in this work show that the results of our research works on subdivision surfaces are of high quality and appropriate for high precision engineering and graphics usage

Pseudoedge--a hierarchical skeletal modeler for the computer aided design of structural components

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1991.Includes bibliographical references (p. 121-122).by David Leo Bonner.M.S

Subdivision Surface based One-Piece Representation

Subdivision surfaces are capable of modeling and representing complex shapes of arbi-trary topology. However, methods on how to build the control mesh of a complex surfaceare not studied much. Currently, most meshes of complicated objects come from trian-gulation and simplification of raster scanned data points, like the Stanford 3D ScanningRepository. This approach is costly and leads to very dense meshes.Subdivision surface based one-piece representation means to represent the final objectin a design process with only one subdivision surface, no matter how complicated theobject\u27s topology or shape. Hence the number of parts in the final representation isalways one.In this dissertation we present necessary mathematical theories and geometric algo-rithms to support subdivision surface based one-piece representation. First, an explicitparametrization method is presented for exact evaluation of Catmull-Clark subdivisionsurfaces. Based on it, two approaches are proposed for constructing the one-piece rep-resentation of a given object with arbitrary topology. One approach is to construct theone-piece representation by using the interpolation technique. Interpolation is a naturalway to build models, but the fairness of the interpolating surface is a big concern inprevious methods. With similarity based interpolation technique, we can obtain bet-ter modeling results with less undesired artifacts and undulations. Another approachis through performing Boolean operations. Up to this point, accurate Boolean oper-ations over subdivision surfaces are not approached yet in the literature. We presenta robust and error controllable Boolean operation method which results in a one-piecerepresentation. Because one-piece representations resulting from the above two methodsare usually dense, error controllable simplification of one-piece representations is needed.Two methods are presented for this purpose: adaptive tessellation and multiresolutionanalysis. Both methods can significantly reduce the complexity of a one-piece represen-tation and while having accurate error estimation.A system that performs subdivision surface based one-piece representation was im-plemented and a lot of examples have been tested. All the examples show that our ap-proaches can obtain very good subdivision based one-piece representation results. Eventhough our methods are based on Catmull-Clark subdivision scheme, we believe they canbe adapted to other subdivision schemes as well with small modifications

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

Modelado jerárquico de objetos 3D con superficies de subdivisión

Las SSs (Superficies de Subdivisión) son un potente paradigma de modelado de objetos 3D (tridimensionales) que establece un puente entre los dos enfoques tradicionales a la aproximación de superficies, basados en mallas poligonales y de parches alabeados, que conllevan problemas uno y otro. Los esquemas de subdivisión permiten definir una superficie suave (a tramos), como las más frecuentes en la práctica, como el límite de un proceso recursivo de refinamiento de una malla de control burda, que puede ser descrita muy compactamente. Además, la recursividad inherente a las SSs establece naturalmente una relación de anidamiento piramidal entre las mallas / NDs (Niveles de Detalle) generadas/os sucesivamente, por lo que las SSs se prestan extraordinariamente al AMRO (Análisis Multiresolución mediante Ondículas) de superficies, que tiene aplicaciones prácticas inmediatas e interesantísimas, como la codificación y la edición jerárquicas de modelos 3D. Empezamos describiendo los vínculos entre las tres áreas que han servido de base a nuestro trabajo (SSs, extracción automática de NDs y AMRO) para explicar como encajan estas tres piezas del puzzle del modelado jerárquico de objetos de 3D con SSs. El AMRO consiste en descomponer una función en una versión burda suya y un conjunto de refinamientos aditivos anidados jerárquicamente llamados "coeficientes ondiculares". La teoría clásica de ondículas estudia las señales clásicas nD: las definidas sobre dominios paramétricos homeomorfos a R" o (0,1)n como el audio (n=1), las imágenes (n=2) o el vídeo (n=3). En topologías menos triviales, como las variedades 2D) (superficies en el espacio 3D), el AMRO no es tan obvio, pero sigue siendo posible si se enfoca desde la perspectiva de las SSs. Basta con partir de una malla burda que aproxime a un bajo ND la superficie considerada, subdividirla recursivamente y, al hacerlo, ir añadiendo los coeficientes ondiculares, que son los detalles 3D necesarios para obtener aproximaciones más y más finas a la superficie original. Pasamos después a las aplicaciones prácticas que constituyen nuestros principal desarrollo original y, en particular, presentamos una técnica de codificación jerárquica de modelos 3D basada en SSs, que actúa sobre los detalles 3D mencionados: los expresa en un referencial normal loscal; los organiza según una estructura jerárquica basada en facetas; los cuantifica dedicando menos bits a sus componentes tangenciales, menos energéticas, y los "escalariza"; y los codifica dinalmente gracias a una técnica similar al SPIHT (Set Partitioning In Hierarchical Tress) de Said y Pearlman. El resultado es un código completamente embebido y al menos dos veces más compacto, para superficies mayormente suaves, que los obtenidos con técnicas de codificación progresiva de mallas 3D publicadas previamente, en las que además los NDs no están anidados piramidalmente. Finalmente, describimos varios métodos auxiliares que hemos desarrollado, mejorando técnicas previas y creando otras propias, ya que una solución completa al modelado de objetos 3D con SSs requiere resolver otros dos problemas. El primero es la extracción de una malla base (triangular, en nuestro caso) de la superficie original, habitualmente dada por una malla triangular fina con conectividad arbitraria. El segundo es la generación de un remallado recursivo con conectividad de subdivisión de la malla original/objetivo mediante un refinamiento recursivo de la malla base, calculando así los detalles 3D necesarios para corregir las posiciones predichas por la subdivisión para nuevos vértices