We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topology. The singular vertices are chosen automatically. As opposed to most previous methods, our flattening process involves only the solution of linear systems of Poisson equations, thus is very efficient. Our method is shown to be faster than existing methods, yet generates parameterizations having comparable quasi-conformal distortion.

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

3D surface matching is fundamental for shape analysis. As a powerful method in geometric analysis, Ricci flow can flexibly design metrics by prescribed target curvature. In this paper we describe a novel approach for matching surfaces with complicated topologies based on hyperbolic Ricci flow. For surfaces with negative Euler characteristics, such as a human face with holes (eye contours), the canonical hyperbolic metric is conformal to the original and can be efficiently computed. Then the surface can be canonically decomposed to hyperbolic hexagons. By matching the corresponding hyperbolic hexagons, the matching between surfaces can be easily established. Compared to existing methods, hyperbolic Ricci flow induces diffeomorphisms between surfaces with complicated topologies with negative Euler characteristics, while avoiding singularities. Furthermore, all the boundaries are intrinsically mapped to hyperbolic lines as alignment constraints. Finally, we demonstrate the applicability of this intrinsic shape representation for 3D face matching and registration. local isometric mapping [28], summation invariants [21], landmark-sliding [7], physics-based deformable models [30], Free-Form Deformation (FFD) [14], and Level-Set based methods [23]. However, many surface representations that use local geometric invariants can not guarantee a global convergence and might suffer from local minima in the presence of non-rigid deformations. To address this issue, many global parameterization methods have been developed recently based on conformal geometric maps

Abstract—Designing rotational symmetry fields on surfaces is an important task for a wide range of graphics applications. This work introduces a rigorous and practical approach for automatic N-RoSy field design on arbitrary surfaces with user defined field topologies. The user has full control of the number, positions and indices of the singularities (as long as they are compatible with necessary global constraints), the turning numbers of the loops, and is able to edit the field interactively. We formulate N-RoSy field construction as designing a Riemannian metric, such that the holonomy along any loop is compatible with the local symmetry of N-RoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of N-RoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility, such that, it can design N-RoSy fields with single singularity, and mixed-RoSy fields. This approach can also be generalized to construct regular remeshing on surfaces. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing. Furthermore, based on our remeshing results with high global symmetry, we generate Celtic knots on surfaces directly.

We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topology. The singular vertices are chosen automatically. As opposed to most previous methods, our flattening process involves only the solution of linear systems of Poisson equations, thus is very efficient. Our method is shown to be faster than existing methods, yet generates parameterizations having comparable quasi-conformal distortion.

In this paper, we present a novel approach for efficiently evolving meshes using mean-curvature flow. We use a finite-elements hierarchy that supports an efficient multigrid solver for performing the semi-implicit time-stepping. Though expensive to compute, we show that it is possible to track this hierarchy through the process of surface evolution. As a result, we provide a way to efficiently flow the surface through the evolution, without requiring a costly initialization at the beginning of each time-step. Using our approach, we demonstrate a factor of nearly seven-fold improvement over the non-tracking implementation, supporting the evolution of surfaces consisting of 1M triangles at a rate of just a few seconds per update.

Scientific visualization provides useful insights and analysis for scientists through graphical representations of data. As numerical simulations become more sophisticated, interpretation of the underlying physical phenomena requires new tools to be developed that perform complex feature analysis and cross correlation of data parameters. Towards this end, this thesis presents two new frameworks that contribute to more effective data understanding. The first is a segmentation framework that can be used to effectively identify regions of interest in data sets. A format of higher-level data is defined from which meaningful derived features can be efficiently extracted and visualized. Furthermore, a higher-order interpolation scheme is presented that allows for extraction of smoother feature surfaces from scalar data. The second framework robustly computes cross parameterizations between two triangulated meshes of arbitrary and possibly unequal genus. Cross parameterizations can be used to describe the evolution of surfaces over time as well as to establish a shape deviation metric. Results are presented that demonstrate the effectiveness of both of the frameworks in a

In this paper we present a robust approach to construct a map between two triangulated meshes, M and M ’ of arbitrary and possibly unequal genus. We introduce a novel initial alignment scheme that allows the user to identify “landmark tunnels” and/or a “constrained silhouette ” in addition to the standard landmark vertices. To describe the evolution of non-landmark tunnels we automatically derive a continuous deformation from M to M ’ using a variational implicit approach. Overall, we achieve a cross parameterization scheme that is provably robust in the sense that it can map M to M ’ without constraints on their relative genus. We provide a number of examples to demonstrate the practical effectiveness of our scheme between meshes of different genus and shape.