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

Analysis of uniform binary subdivision schemes for curve design

The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form .0,1,2,...kz,ikj,ifjbm0j1k12ifjam0j1k2if=∈+Σ==++Σ==+ The convergence of the control polygons to a Cu curve is analysed in terms of the convergence to zero of a derived scheme for the differences - . The analysis of the smoothness of the limit curve is reduced to kif the convergence analysis of "differentiated" schemes which correspond to divided differences of {/i ∈Z} with respect to the diadic parameteriz- kif ation = i/2kitk . The inverse process of "integration" provides schemes with limit curves having additional orders of smoothness

Polynomial-based non-uniform interpolatory subdivision with features control

Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that the most convenient parameter values may be chosen as well as the intervals for insertion. Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control

Joint segmentation of color and depth data based on splitting and merging driven by surface fitting

This paper proposes a segmentation scheme based on the joint usage of color and depth data together with a 3D surface estimation scheme. Firstly a set of multi-dimensional vectors is built from color, geometry and surface orientation information. Normalized cuts spectral clustering is then applied in order to recursively segment the scene in two parts thus obtaining an over-segmentation. This procedure is followed by a recursive merging stage where close segments belonging to the same object are joined together. At each step of both procedures a NURBS model is fitted on the computed segments and the accuracy of the fitting is used as a measure of the plausibility that a segment represents a single surface or object. By comparing the accuracy to the one at the previous step, it is possible to determine if each splitting or merging operation leads to a better scene representation and consequently whether to perform it or not. Experimental results show how the proposed method provides an accurate and reliable segmentation

AlSub: Fully Parallel and Modular Subdivision

In recent years, mesh subdivision---the process of forging smooth free-form surfaces from coarse polygonal meshes---has become an indispensable production instrument. Although subdivision performance is crucial during simulation, animation and rendering, state-of-the-art approaches still rely on serial implementations for complex parts of the subdivision process. Therefore, they often fail to harness the power of modern parallel devices, like the graphics processing unit (GPU), for large parts of the algorithm and must resort to time-consuming serial preprocessing. In this paper, we show that a complete parallelization of the subdivision process for modern architectures is possible. Building on sparse matrix linear algebra, we show how to structure the complete subdivision process into a sequence of algebra operations. By restructuring and grouping these operations, we adapt the process for different use cases, such as regular subdivision of dynamic meshes, uniform subdivision for immutable topology, and feature-adaptive subdivision for efficient rendering of animated models. As the same machinery is used for all use cases, identical subdivision results are achieved in all parts of the production pipeline. As a second contribution, we show how these linear algebra formulations can effectively be translated into efficient GPU kernels. Applying our strategies to $\sqrt{3}$, Loop and Catmull-Clark subdivision shows significant speedups of our approach compared to state-of-the-art solutions, while we completely avoid serial preprocessing.Comment: Changed structure Added content Improved description