Abstract: We introduce a new method for computing conformal transformations of triangle meshes in R³. Conformal maps are desirable in digital geometry processing because they do not exhibit shear, and therefore preserve texture fidelity as well as the quality of the mesh itself. Traditional discretizations consider maps into the complex plane, which are useful only for problems such as surface parameterization and planar shape deformation where the target surface is flat. We instead consider maps into the quaternions, which allows us to work directly with surfaces sitting in R³. In particular, we introduce a quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. Our discretization of this condition results in a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications.

Abstract: This paper presents a straightforward algorithm for constructing connections on discrete surfaces that are as smooth as possible everywhere but on a set of isolated singularities with given index. We compute these connections by solving a single linear system built from standard operators. The solution can be used to design rotationally symmetric direction fields with user-specified singularities and directional constraints.

Abstract: Connections provide a way to compare local quantities defined at different points of a geometric space. This thesis develops a discrete theory of connections that naturally leads to practical, efficient numerical algorithms for geometry processing. Our formulation is motivated by real-world applications where meshes may be noisy or coarsely discretized. Further, because our discrete framework closely parallels the smooth theory, we can draw upon a huge wealth of existing knowledge to develop and interpret mesh processing algorithms.
The main contribution of this thesis is a new algorithm for computing trivial connections on discrete surfaces that are as smooth as possible everywhere but on a set of isolated singularities of given index. A connection is represented via an angle associated with each dual edge, i.e., a discrete angle-valued 1-form. These angles are determined by the solution to a linear system, and are globally optimal in the sense that they describe the trivial connection closest to Levi-Civita among all solutions with the prescribed set of singularities. Relative to previous methods our algorithm is surprisingly simple, and can be implemented using standard operations from mesh processing and linear algebra. The solution can be used to construct rotationally symmetric direction fields with a prescribed set of singularities and directional constraints, which are essential in applications such as quadrilateral remeshing and texture synthesis.

Abstract: Numerical viscosity has long been a problem in fluid animation. Existing methods suffer from intrinsic artificial dissipation and often apply complicated computational mechanisms to combat such effects. Consequently, dissipative behavior cannot be controlled or modeled explicitly in a manner independent of time step size, complicating the use of coarse previews and adaptive time-stepping methods. This paper proposes simple, unconditionally stable, fully Eulerian integration schemes with no numerical viscosity that are capable of maintaining the liveliness of fluid motion without recourse to corrective devices. Pressure and fluxes are solved efficiently and simultaneously in a time-reversible manner on simplicial grids, and the energy is preserved exactly over long time scales in the case of inviscid fluids. These integrators can be viewed as an extension of the classical energy-preserving Harlow-Welch / Crank- Nicolson scheme to simplicial grids.

Abstract: This paper is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars. Motivated by recent developments in discrete geometric mechanics we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preserving their state-space geometry and motion invariants. We demonstrate that the resulting integration schemes are superior to standard methods in numerical robustness and efficiency, and can be applied to many types of vehicles. In addition, we show how to use this framework in an optimal control setting to automatically compute accurate and realistic motions for arbitrary user-specified constraints.

Abstract: Reference grids are commonly used in design software to help users to judge distances and to understand the orientation of the virtual workspace. However, despite their ubiquity in 3D graphics authoring applications, little research has gone into important design considerations of the 3D reference grids themselves, which directly impact their usefulness. In an effort to resolve some of these outstanding issues, we have developed two new techniques; the multiscale reference grid and position pegs that form a consistent foundation for presenting relative scale and position information to the user.
Our design of a multiscale reference grid consistently subdivides and coalesces gridlines, based on the computation of a closeness metric, while ensuring that there are neither too many nor too few subdivisions. Position pegs extend the grid so that objects that are lying above or below the ground plane can be brought into a common environmental frame of reference without interfering with the grid or object data. We introduce an analytical system, similar to MIP mapping, to provide a stable viewpoint-determined result. Our design solves several depth cue problems in a way that is independent of viewing projection.

Abstract: We capture the shape of moving cloth using a custom set of color markers printed on the surface of the cloth. The output is a sequence of triangle meshes with static connectivity and with detail at the scale of individual markers in both smooth and folded regions. We compute markers' coordinates in space using marker correspondence across multiple synchronized video cameras. Correspondence is determined from color information in small neighborhoods and refined using a novel strain pruning process. Final correspondence does not require neighborhood information. We use a novel data driven hole-filling technique to fill occluded regions. Our results include several challenging examples: a wrinkled shirt sleeve, a dancing pair of pants, and a rag tossed onto a cup. Finally, we demonstrate that cloth capture is reusable by animating a pair of pants using human motion capture data.

Abstract: We present a new method for cloth animation based on data driven synthesis. In contrast to approaches that focus on physical simulation, we animate cloth by manipulating short sequences of existing cloth animation. While our source of data is cloth animation captured using video cameras, the method is equally applicable to simulation data. The approach has benefits in both cases: current cloth capture is limited because small tweaks to the data require filming an entirely new sequence. Likewise, simulation suffers from long computation times and complications such as tangling. In this sketch we create new animations by fitting cloth animation to human motion capture data, i.e., we drive the cloth with a skeleton.

Abstract: Physically based animation of fluids such as smoke, water, and fire provides some of the most stunning visuals in computer graphics, but has historically been the domain of high-quality offline rendering due to great computational cost. In this chapter we not only show how these effects can be simulated and rendered in real time, but also how they can be seamlessly integrated into real time applications.

Abstract: Many mesh parameterization algorithms have focused on minimizing distortion and utilizing texture area, but few have addressed issues related to processing a signal on the mesh surface. We present an algorithm which partitions a mesh into rectangular charts while preserving a one-to-one texel correspondence across chart boundaries. This mapping permits any computation on the mesh surface which is typically carried out on a regular grid, and prevents seams by ensuring resolution continuity along the boundary. These features are also useful for traditional texture applications such as surface painting where continuity is important. Distortion is comparable to other parameterization schemes, and the rectangular charts yield efficient packing into a texture atlas. We apply this parameterization to texture synthesis, fluid simulation, mesh processing and storage, and locating geodesics.

Abstract: Using the GPU to accelerate ray tracing may seem like a natural choice due to the highly parallel nature of the problem. However, determining the most versatile GPU data structure for scene storage and traversal is a challenge. In this paper, we introduce a new method for quick intersection of triangular meshes on the GPU. The method uses a threaded bounding volume hierarchy built from a geometry image, which can be efficiently traversed and constructed entirely on the GPU. This acceleration scheme is highly competitive with other GPU ray tracing methods, while allowing for both dynamic geometry and an efficient level of detail scheme at no extra cost.

Abstract: Pluto and its satellite Charon regularly occulted or transited each other's disks from 1985 through 1990. The light curves resulting from these events (collectively called "mutual events") have been used to determine albedo maps of Pluto's sub-Charon hemisphere. We now use a data set of four light curves that were obtained in both B and V Johnson filters to construct a two-color map of Pluto's surface. We are able to resolve the central part of Pluto's sub-Charon hemisphere. We find that the dark albedo feature that forms a band below Pluto's equator is comprised of several distinct color units. We detect ratios of V-filter/B-filter normal reflectances ranging from 1.15 to 1.39 on Pluto's sub-Charon hemisphere.