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

Multiresolution editing for B-spline curves and surfaces

Since 1980 surface modeling has been used in industrial design, CAD and entertainment to create and represent complex forms. Even with this comparatively long history of development, challenges remain in free-form surface modeling. One such challenge is building surface creation and editing techniques that effectively balance the need for local control with the need to control the overall global shape, or sweep of the surface. This dissertation presents a multiresolution approach to the creation of surfaces that allows a designer to more easily manage this balance between local and global control. The techniques presented in this dissertation utilize a wavelet decomposition of B-spline curves and surfaces to allow a designer to easily develop the basic shape using lower level representations, and then seamlessly switch to higher level representations to achieve fine control over local features. The algorithms described in the dissertation are implemented in an interactive software system that is used to demonstrate their effectiveness in comparison to existing methods

Operator-adapted wavelets for finite-element differential forms

We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations

Grid-based Finite Elements System for Solving Laplace-Beltrami Equations on 2-Manifolds

Solving the Poisson equation has numerous important applications. On a Riemannian 2-manifold, the task is most often formulated in terms of finite elements and two challenges commonly arise: discretizing the space of functions and solving the resulting system of equations. In this work, we describe a finite elements system that simultaneously addresses both aspects. The idea is to define a space of functions in 3D and then restrict the 3D functions to the mesh. Unlike traditional approaches, our method is tessellation-independent and has a direct control over system complexity. More importantly, the resulting function space comes with a multi-resolution structure supporting an efficient multigrid solver, and the regularity of the function space can be leveraged in parallelizing/streaming the computation. We evaluate our framework by conducting several experiments. These include a spectral analysis that reveals the embedding-invariant robustness of our discretization, and a benchmark for solver convergence/performance that reveals the competitiveness of our approach against other state-of-the-art methods. We apply our work to several geometry-processing applications. Using curvature flows, we show that we can support efficient surface evolution where the embedding changes with time. Formulating surface filtering as a solution to the screened-Poisson equation, we demonstrate that we can support an anisotropic surface editing system that processes high resolution meshes in real time

Operator-adapted wavelets for finite-element differential forms

We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations

Subdivision and manifold techniques for isogeometric design and analysis of surfaces

Design of surfaces and analysis of partial differential equations defined on them are of great importance in engineering applications, e.g., structural engineering, automotive and aerospace. This thesis focuses on isogeometric design and analysis of surfaces, which aims to integrate engineering design and analysis by using the same representation for both. The unresolved challenge is to develop a desirable surface representation that simultaneously satisfies certain favourable properties on meshes of arbitrary topology around the extraordinary vertices (EVs), i.e., vertices not shared by four quadrilaterals or three triangles. These properties include high continuity (geometric or parametric), optimal convergence in finite element analysis as well as simplicity in terms of implementation. To overcome the challenge, we further develop subdivision and manifold surface modelling techniques, and explore a possible scheme to combine the distinct appealing properties of the two. The unique advantages of the developed techniques have been confirmed with numerical experiments. Subdivision surfaces generate smooth surfaces from coarse control meshes of arbitrary topology by recursive refinement. Around EVs the optimal refinement weights are application-dependent. We first review subdivision-based finite elements. We then proceed to derive the optimal subdivision weights that minimise finite element errors and can be easily incorporated into existing implementations of subdivision schemes to achieve the same accuracy with much coarser meshes in engineering computations. To this end, the eigenstructure of the subdivision matrix is extensively used and a novel local shape decomposition approach is proposed to choose the optimal weights for each EV independently. Manifold-based basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. This thesis derives novel manifold-based basis functions with arbitrary prescribed smoothness using quasi-conformal maps, enabling us to model and analyse surfaces with sharp features, such as creases and corners. Their practical utility in finite element simulation of hinged or rigidly joined structures is demonstrated with Kirchhoff-Love thin shell examples. We also propose a particular manifold basis reproducing subdivision surfaces away from EVs, i.e., B-splines, providing a way to combine the appealing properties of subdivision (available in industrial software) for design and manifold basis (relatively new) for analysis.Cambridge International Scholarship Scheme (CISS) by Cambridge Trus

Real-time rendering and simulation of trees and snow

Tree models created by an industry used package are exported and the structure extracted in order to procedurally regenerate the geometric mesh, addressing the limitations of the application's standard output. The structure, once extracted, is used to fully generate a high quality skeleton for the tree, individually representing each section in every branch to give the greatest achievable level of freedom of deformation and animation. Around the generated skeleton, a new geometric mesh is wrapped using a single, continuous surface resulting in the removal of intersection based render artefacts. Surface smoothing and enhanced detail is added to the model dynamically using the GPU enhanced tessellation engine. A real-time snow accumulation system is developed to generate snow cover on a dynamic, animated scene. Occlusion techniques are used to project snow accumulating faces and map exposed areas to applied accumulation maps in the form of dynamic textures. Accumulation maps are xed to applied surfaces, allowing moving objects to maintain accumulated snow cover. Mesh generation is performed dynamically during the rendering pass using surface o�setting and tessellation to enhance required detail

Surface modelling for 2D imagery

Vector graphics provides powerful tools for drawing scalable 2D imagery. With the rise of mobile computers, of different types of displays and image resolutions, vector graphics is receiving an increasing amount of attention. However, vector graphics is not the leading framework for creating and manipulating 2D imagery. The reason for this reluctance of employing vector graphical frameworks is that it is difficult to handle complex behaviour of colour across the 2D domain. A challenging problem within vector graphics is to define smooth colour functions across the image. In previous work, two approaches exist. The first approach, known as diffusion curves, diffuses colours from a set of input curves and points. The second approach, known as gradient meshes, defines smooth colour functions from control meshes. These two approaches are incompatible: diffusion curves do not support the local behaviour provided by gradient meshes and gradient meshes do not support freeform curves as input. My research aims to narrow the gap between diffusion curves and gradient meshes. With this aim in mind, I propose solutions to create control meshes from freeform curves. I demonstrate that these control meshes can be used to render a vector primitive similar to diffusion curves using subdivision surfaces. With the use of subdivision surfaces, instead of a diffusion process, colour gradients can be locally controlled using colour-gradient curves associated with the input curves. The advantage of local control is further explored in the setting of vector-centric image processing. I demonstrate that a certain contrast enhancement profile, known as the Cornsweet profile, can be modelled via surfaces in images. This approach does not produce saturation artefacts related with previous filter-based methods. Additionally, I demonstrate various approaches to artistic filtering, where the artist locally models given artistic effects. Gradient meshes are restricted to rectangular topology of the control meshes. I argue that this restriction hinders the applicability of the approach and its potential to be used with control meshes extracted from freeform curves. To this end, I propose a mesh-based vector primitive that supports arbitrary manifold topology of the mesh

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest