Denis Zorin

Interpolating Subdivision for meshes with arbitrary topology

Subdivision is a powerful paradigm for the generation of surfaces of arbitrary topology. Given an initial triangular mesh the goal is to produce a smooth and visually pleasing surface whose shape is controlled by the initial mesh. Of particular interest are interpolating schemes since they match the original data exactly, and play an important role in fast multiresolution and wavelet techniques. Dyn, Gregory, and Levin introduced the Butterfly scheme, which yields C^1 surfaces in the topologically regular setting. Unfortunately it exhibits undesirable artifacts in the case of an irregular topology. We examine these failures and derive an improved scheme, which retains the simplicity of the Butterfly scheme, is interpolating, and results in smoother surfaces.

Illustrating smooth surfaces

We present a new set of algorithms for line-art rendering of smooth surfaces. We introduce an efficient, deterministic algorithm for finding silhouettes based on geometric duality, and an algorithm for segmenting the silhouette curves into smooth parts with constant visibility. These methods can be used to find all silhouettes in real time in software. We present an automatic method for generating hatch marks in order to convey surface shape. We demonstrate these algorithms with a drawing style inspired by A Topological Picturebook by G. Francis.

Interactive multiresolution mesh editing

We describe a multiresolution representation for meshes based on subdivision, which is a natural extension of the existing patch-based surface representations. Combining subdivision and the smoothing algorithms of Taubin [26] allows us to construct a set of algorithms for interactive multiresolution editing of complex hierarchical meshes of arbitrary topology. The simplicity of the underlying algorithms for refinement and coarsification enables us to make them local and adaptive, thereby considerably improving their efficiency. We have built a scalable interactive multiresolution editing system based on such algorithms.

4-8 Subdivision

In this paper we introduce 4-8 subdivision, a new scheme that generalizes the four-directional box spline of class C^4 to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more commonly used face or vertex splits. In the uniform case, bisection refinement results in doubling, rather than quadrupling of the number of faces in a mesh. Adaptive bisection refinement automatically generates conforming variable-resolution meshes in contrast to face and vertex split methods which require a postprocessing step to make an adaptively refined mesh conforming. The fact that the size of faces decreases more gradually with refinement allows one to have greater control over the resolution of a refined mesh. It also makes it possible to achieve higher smoothness while using small stencils (the size of the stencils used by our scheme is similar to Loop subdivision). We show that the subdivision surfaces produced by the 4-8 scheme are C^4 continuous almost everywhere, except at extraordinary vertices where they are is C^1-continuous.Most commonly used subdivision schemes are of primal type, i.e., they split faces. Examples include the schemes of Catmull-Clark (quadrilaterals) and Loop (triangles). In contrast, dual subdivision schemes such as Doo-Sabin, are based on vertex splits. Triangle based subdivision does not admit primal and dual schemes as the latter are based on hexagons. Quadrilateral schemes on the other hand come in both primal and dual varieties allowing for the possibility of a unified treatment and common implementation. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning with a vertex split step we successively construct variants of Doo-Sabin and Catmull-Clark schemes followed by novel schemes generalizing B-splines of bi-degree up to nine. We prove the schemes to be C at extraordinary points, and analyze the behavior of the schemes as the number of averaging steps increases. We discuss a number of implementation issues common to all quadrilateral schemes; in particular we describe a simple algorithm for adaptive subdivision of dual schemes. Because of the unified construction framework and common algorithmic treatment of primal and dual quadrilateral subdivision schemes it is straightforward to support multiple schemes in the same application. This is useful for more flexible geometric modeling as well as in p-versions of the Subdivision Element Method.

Cut-and-paste editing of multiresolution surfaces

Cutting and pasting to combine different elements into a common structure are widely used operations that have been successfully adapted to many media types. Surface design could also benefit from the availability of a general, robust, and efficient cut-and-paste tool, especially during the initial stages of design when a large space of alternatives needs to be explored. Techniques to support cut-and-paste operations for surfaces have been proposed in the past, but have been of limited usefulness due to constraints on the type of shapes supported and the lack of real-time interaction. In this paper, we describe a set of algorithms based on multiresolution subdivision surfaces that perform at interactive rates and enable intuitive cut-and-paste operations.

Quad-Mesh Generation and Processing: A Survey

Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semi‐regular ones, have advantages for many applications, and significant progress was made in quadrilateral mesh generation and processing during the last several years. In this survey we discuss the advantages and problems of techniques operating on quadrilateral meshes, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrisation and remeshing.

Texture and shape synthesis on surfaces

We present a novel method for texture synthesis on surfaces from examples. We consider a very general type of textures, including color, transparency and displacements. Our method synthesizes the texture directly on the surface, rather than synthesizing a texture image and then mapping it to the surface. The synthesized textures have the same qualitative visual appearance as the example texture, and cover the surfaces without the distortion or seams of conventional texture-mapping. We describe two synthesis methods, based on the work of Wei and Levoy and Ashikhmin; our techniques produce similar results, but directly on surfaces.

A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D

We present a new method for the evolution of inextensible vesicles immersed in a Stokesian fluid. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the non-local hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present two semi-implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep third-order accurate scheme in time. We use the fast multipole method (FMM) to efficiently compute vesicle-vesicle interaction forces in a suspension with a large number of vesicles. We report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme.

Petascale Direct Numerical Simulation of Blood Flow on 200K Cores and Heterogeneous Architectures

We present a fast, petaflop-scalable algorithm for Stokesian particulate flows. Our goal is the direct simulation of blood, which we model as a mixture of a Stokesian fluid (plasma) and red blood cells (RBCs). Directly simulating blood is a challenging multiscale, multiphysics problem. We report simulations with up to 200 million deformable RBCs. The largest simulation amounts to 90 billion unknowns in space. In terms of the number of cells, we improve the state-of-the art by several orders of magnitude: the previous largest simulation, at the same physical fidelity as ours, resolved the flow of O(1,000-10,000) RBCs. Our approach has three distinct characteristics: (1) we faithfully represent the physics of RBCs by using nonlinear solid mechanics to capture the deformations of each cell; (2) we accurately resolve the long-range, N-body, hydrodynamic interactions between RBCs (which are caused by the surrounding plasma); and (3) we allow for the highly non-uniform distribution of RBCs in space. The new method has been implemented in the software library MOBO (for “Moving Boundaries”). We designed MOBO to support parallelism at all levels, including inter-node distributed memory parallelism, intra-node shared memory parallelism, data parallelism (vectorization), and fine-grained multithreading for GPUs. We have implemented and optimized the majority of the computation kernels on both Intel/AMD x86 and NVidias Tesla/Fermi platforms for single and double floating point precision. Overall, the code has scaled on 256 CPU-GPUs on the Teragrids Lincoln cluster and on 200,000 AMD cores of the Oak Ridge national Laboratorys Jaguar PF system. In our largest simulation, we have achieved 0.7 Petaflops/s of sustained performance on Jaguar.

A simple manifold-based construction of surfaces of arbitrary smoothness

We present a smooth surface construction based on the manifold approach of Grimm and Hughes. We demonstrate how this approach can relatively easily produce a number of desirable properties which are hard to achieve simultaneously with polynomial patches, subdivision or variational surfaces. Our surfaces are C∞-continuous with explicit nonsingular C∞ parameterizations, high-order flexible at control vertices, depend linearly on control points, have fixed-size local support for basis functions, and have good visual quality.