Point-Normal Subdivision Curves and Surfaces

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.Comment: 30 pages, 17 figures, 22.5M

PARAMETRIZATION AND SHAPE RECONSTRUCTION TECHNIQUES FOR DOO-SABIN SUBDIVISION SURFACES

This thesis presents a new technique for the reconstruction of a smooth surface from a set of 3D data points. The reconstructed surface is represented by an everywhere -continuous subdivision surface which interpolates all the given data points. And the topological structure of the reconstructed surface is exactly the same as that of the data points. The new technique consists of two major steps. First, use an efficient surface reconstruction method to produce a polyhedral approximation to the given data points. Second, construct a Doo-Sabin subdivision surface that smoothly passes through all the data points in the given data set. A new technique is presented for the second step in this thesis. The new technique iteratively modifies the vertices of the polyhedral approximation 1CM until a new control meshM, whose Doo-Sabin subdivision surface interpolatesM, is reached. It is proved that, for any mesh M with any size and any topology, the iterative process is always convergent with Doo-Sabin subdivision scheme. The new technique has the advantages of both a local method and a global method, and the surface reconstruction process can reproduce special features such as edges and corners faithfully

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

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

Subdivision surface fitting to a dense mesh using ridges and umbilics

Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of the control mesh for subdivision, so that the edges follow salient features on the surface. Furthermore, the chosen features and connectivity characterise the overall shape of the original mesh, since ridges capture extreme principal curvatures and ridges start and end at umbilics. A metric of Hausdorff distance including curvature vectors is proposed and implemented in a distance transform algorithm to construct the connectivity. Ridge-colour matching is introduced as a criterion for edge flipping to improve feature alignment. Several examples are provided to demonstrate the feature-preserving capability of the proposed approach

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

Lifting-based subdivision wavelets with geometric constraints.

Qin, Guiming."August 2010."Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (p. 72-74).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 1.1 --- B splines and B-splines surfaces --- p.5Chapter 1. 2 --- Box spline --- p.6Chapter 1. 3 --- Biorthogonal subdivision wavelets based on the lifting scheme --- p.7Chapter 1.4 --- Geometrically-constrained subdivision wavelets --- p.9Chapter 1.5 --- Contributions --- p.9Chapter 2 --- Explicit symbol formulae for B-splines --- p.11Chapter 2. 1 --- Explicit formula for a general recursion scheme --- p.11Chapter 2. 2 --- Explicit formulae for de Boor algorithms of B-spline curves and their derivatives --- p.14Chapter 2.2.1 --- Explicit computation of de Boor Algorithm for Computing B-Spline Curves --- p.14Chapter 2.2.2 --- Explicit computation of Derivatives of B-Spline Curves --- p.15Chapter 2. 3 --- Explicit power-basis matrix fomula for non-uniform B-spline curves --- p.17Chapter 3 --- Biorthogonal subdivision wavelets with geometric constraints --- p.23Chapter 3. 1 --- Primal subdivision and dual subdivision --- p.23Chapter 3. 2 --- Biorthogonal Loop-subdivision-based wavelets with geometric constraints for triangular meshes --- p.24Chapter 3.2.1 --- Loop subdivision surfaces and exact evaluation --- p.24Chapter 3.2.2 --- Lifting-based Loop subdivision wavelets --- p.24Chapter 3.2.3 --- Biorthogonal Loop-subdivision wavelets with geometric constraints --- p.26Chapter 3. 3 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.35Chapter 3.3.1 --- Catmull-Clark subdivision and Doo-Sabin subdivision surfaces --- p.35Chapter 3.3.1.1 --- Catmull-Clark subdivision --- p.36Chapter 3.3.1.2 --- Doo-Sabin subdivision --- p.37Chapter 3.3.2 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.38Chapter 3.3.2.1 --- Biorthogonal Doo-Sabin subdivision wavelets with geometric constraints --- p.38Chapter 3.3.2.2 --- Biorthogonal Catmull-Clark subdivision wavelets with geometric constraints --- p.44Chapter 4 --- Experiments and results --- p.49Chapter 5 --- Conclusions and future work --- p.60Appendix A --- p.62Appendix B --- p.67Appendix C --- p.69Appendix D --- p.71References --- p.7