Discrete Exponential Map Research Articles

Surface Simplification using Intrinsic Error Metrics

This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM) , we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature "drifts" during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation---a feature unique to the intrinsic setting. The overall payoff is a "black box" approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map.

Lightweight preprocessing and fast query of geodesic distance via proximity graph

Computing geodesic distance on a mesh surface S efficiently and accurately is a central task in numerous computer graphics applications. In order to deal with high-resolution mesh surfaces, a lightweight preprocessing is a proper choice to make a balance between query accuracy and speed. In the preprocessing stage, we build a proximity graph G with regard to a set of sample points and keep the exact geodesic distance between any pair of nearby sample points. In the query stage, given two query points s and t, we augment the proximity graph G by adding s and t on-the-fly, and then use the shortest path between s and t on the augmented proximity graph to approximate the exact geodesic path between s and t. We establish an empirical relationship between the number of samples and expected accuracy (measured in relative error), which facilitates fast and accurate query of geodesic distance with a lightweight processing cost. We exhibit the uses of the new approach in two applications—real-time computation of discrete exponential map for texture mapping and interactive design of spline curves on surfaces.

Smooth interpolation of key frames in a Riemannian shell space

Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation.

Intrinsic computation of centroidal Voronoi tessellation (CVT) on meshes

Centroidal Voronoi tessellation (CVT) is a special type of Voronoi diagram such that the generating point of each Voronoi cell is also its center of mass. The CVT has broad applications in computer graphics, such as meshing, stippling, sampling, etc. The existing methods for computing CVTs on meshes either require a global parameterization or compute it in the restricted sense (that is, intersecting a 3D CVT with the surface). Therefore, these approaches often fail on models with complicated geometry and/or topology. This paper presents two intrinsic algorithms for computing CVT on triangle meshes. The first algorithm adopts the Lloyd framework, which iteratively moves the generator of each geodesic Voronoi diagram to its mass center. Based on the discrete exponential map, our method can efficiently compute the Riemannian center and the center of mass for any geodesic Voronoi diagram. The second algorithm uses the L-BFGS method to accelerate the intrinsic CVT computation. Thanks to the intrinsic feature, our methods are independent of the embedding space, and work well for models with arbitrary topology and complicated geometry, where the existing extrinsic approaches often fail. The promising experimental results show the advantages of our method.

Variational time discretization of geodesic calculus

We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps, and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct methods in the calculus of variation, on $\Gamma$-convergence, and on weighted finite element error estimation. The convergence results of the discrete geodesic calculus are experimentally confirmed for a basic model on a two-dimensional Riemannian manifold. This provides a theoretical basis for applications in computer vision: In particular, discrete geodesics offer an effective tool for shape morphing and the computation of a distance between shapes, the discrete logarithm allows a linear representation of strongly nonlinear shape variability, the discrete exponential map provides a robust tool for shape extrapolation, and the discrete parallel transport can be used to transfer geometric details from one shape to another. The basic operations are exemplarily illustrated for two different shape spaces, the space of viscous volumetric objects and the space of discrete viscous shells.

A Magnus Series-Based Modified Sweep Method for Optimal Control

This paper presents variations of the Successive Backward Sweep (SBS) method for solving nonlinear optimal control problems with terminal constraints. The state-costate differential equations are linearized and discretized explicitly in time by a truncated Magnus series representation of the state transition. These discrete exponential maps are propagated, backwards in time, by the sweep method. Two variants of the method are implemented to test for the Jacobi sufficient condition and achieve convergence to local optimal solutions. Where required, the problems are regularized to ensure satisfaction of the convexity conditions by modifying the associated Hessian. The performance of the Magnus series-based SBS method is compared with that of a non-symplectic 5 th order Runge-Kutta method on three examples: a highly nonlinear, two-dimensional problem, a hypersensitive problem, and an atmospheric reentry guidance problem.

Combustion Waves on the Point Set Surface

This paper introduces a combustion model of heat transfer and fuel consumption for the propagation of a fire front on a point cloud surface. The heat transfer includes the heat advection by the airflow as well as diffusion, chemical reaction, and heat loss to generate complex, but controllable heat flows with a designed airflow velocity. For the stable heat advection, we solve a semi-Lagrangian method on point samples using discrete exponential maps to trace the position from which the wind blows while preserving the geodesic distance. We also propose angular Voronoi weights for a discrete Laplace-Beltrami operator that shows better isotropic diffusion on the inhomogeneous distribution of point clouds than the cotangent or moving least-squares schemes. We demonstrate a diversity of burning scenarios by incorporating factors affecting the fire spreading such as buoyancy and object geometries in the airflow velocity fields, or by synthesizing patterns.

Stroke Parameterization

AbstractWe present a novel algorithm for generating a planar parameterization of the region surrounding a curve embedded in a 3D surface, which we call a stroke parameterization. The technique, which extends the well‐known Discrete Exponential Map [SGW06], uses the same basic geometric transformations and hence is both efficient and easy‐to‐implement. We also handle self‐intersecting curves, for which a 1–1 map between the original surface and the plane is not possible. Stroke parameterizations provide an ideal coordinate space for solving a variety of computer graphics problems. We present applications including tiling texture and displacement along 3D brush strokes, procedural texturing along 3D paths, and user‐guided crease extraction.

Discrete Geodesic Calculus in Shape Space and Applications in the Space of Viscous Fluidic Objects

Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessian reproduces the underlying Riemannian metric, and it is used to define length and energy of discrete paths in shape space. The notion of discrete geodesics defined as energy minimizing paths gives rise to a discrete logarithmic map, a variational definition of a discrete exponential map, and a time discrete parallel transport. This new concept is developed in the context of shape spaces with shapes that are described via deformations of a given reference shape, and it is applied to a particular shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving shape morphing, the representation of paths in shape space via local shape variations as path generators, shape extrapolation via discrete geodesic flow, and the transfer of geometric features.

COUNTING FIXED POINTS, TWO-CYCLES, AND COLLISIONS OF THE DISCRETE EXPONENTIAL FUNCTION USING p-ADIC METHODS

AbstractBrizolis asked for which primes p greater than 3 there exists a pair (g,h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Various authors have contributed to the understanding of this problem. In this paper, we use p-adic methods, primarily Hensel’s lemma and p-adic interpolation, to count fixed points, two-cycles, collisions, and solutions to related equations modulo powers of a prime p.

GeoBrush: Interactive Mesh Geometry Cloning

AbstractWe propose a method for interactive cloning of 3D surface geometry using a paintbrush interface, similar to the continuous cloning brush popular in image editing. Existing interactive mesh composition tools focus on atomic copy‐and‐paste of preselected feature areas, and are either limited to copying surface displacements, or require the solution of variational optimization problems, which is too expensive for an interactive brush interface. In contrast, our GeoBrush method supports real‐time continuous copying of arbitrary high‐resolution surface features between irregular meshes, including topological handles. We achieve this by first establishing a correspondence between the source and target geometries using a novel generalized discrete exponential map parameterization. Next we roughly align the source geometry with the target shape using Green Coordinates with automatically‐constructed cages. Finally, we compute an offset membrane to smoothly blend the pasted patch with C continuity before stitching it into the target. The offset membrane is a solution of a bi‐harmonic PDE, which is computed on the GPU in real time by exploiting the regular parametric domain. We demonstrate the effectiveness of GeoBrush with various editing scenarios, including detail enrichment and completion of scanned surfaces.

Interactive decal compositing with discrete exponential maps

A method is described for texturing surfaces using decals , images placed on the surface using local parameterizations. Decal parameterizations are generated with a novel O ( N log N ) discrete approximation to the exponential map which requires only a single additional step in Dijkstra's graph-distance algorithm. Decals are dynamically composited in an interface that addresses many limitations of previous work. Tools for image processing, deformation/feature-matching, and vector graphics are implemented using direct surface interaction. Exponential map decals can contain holes and can also be combined with conformal parameterization to reduce distortion. The exponential map approximation can be computed on any point set, including meshes and sampled implicit surfaces, and is relatively stable under resampling. The decals stick to the surface as it is interactively deformed, allowing the texture to be preserved even if the surface changes topology. These properties make exponential map decals a suitable approach for texturing animated implicit surfaces.

Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought

Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation.