480 research outputs found
Almost Isometric Mesh Parameterization through Abstract Domains
In this paper, we propose a robust, automatic technique to build a global hi-quality parameterization of a two-manifold triangular mesh. An adaptively chosen 2D domain of the parameterization is built as part of the process. The produced parameterization exhibits very low isometric distortion, because it is globally optimized to preserve both areas and angles. The domain is a collection of equilateral triangular 2D regions enriched with explicit adjacency relationships (it is abstract in the sense that no 3D embedding is necessary). It is tailored to minimize isometric distortion, resulting in excellent parameterization qualities, even when meshes with complex shape and topology are mapped into domains composed of a small number of large continuous regions. Moreover, this domain is, in turn, remapped into a collection of 2D square regions, unlocking many advantages found in quad-based domains (e. g., ease of packing). The technique is tested on a variety of cases, including challenging ones, and compares very favorably with known approaches. An open-source implementation is made available
Surface parameterization over regular domains
Surface parameterization has been widely studied and it has been playing a critical role in many geometric processing tasks in graphics, computer-aided design, visualization, vision, physical simulation and etc. Regular domains, such as polycubes, are favored due to their structural regularity and geometric simplicity. This thesis focuses on studying the surface parameterization over regular domains, i.e. polycubes, and develops effective computation algorithms. Firstly, the motivation for surface parameterization and polycube mapping is introduced. Secondly, we briefly review existing surface parameterization techniques, especially for extensively studied parameterization algorithms for topological disk surfaces and parameterizations over regular domains for closed surfaces. Then we propose a polycube parameterization algorithm for closed surfaces with general topology. We develop an efficient optimization framework to minimize the angle and area distortion of the mapping. Its applications on surface meshing, inter-shape morphing and volumetric polycube mapping are also discussed
On the isometric embedding problem for length metric spaces
We prove that every proper -dimensional length metric space admits an
"approximate isometric embedding" into Lorentzian space .
By an "approximate isometric embedding" we mean an embedding which preserves
the energy functional on a prescribed set of geodesics connecting a dense set
of points.Comment: 40 pages, 10 figure
A Low-Dimensional Representation for Robust Partial Isometric Correspondences Computation
Intrinsic isometric shape matching has become the standard approach for pose
invariant correspondence estimation among deformable shapes. Most existing
approaches assume global consistency, i.e., the metric structure of the whole
manifold must not change significantly. While global isometric matching is well
understood, only a few heuristic solutions are known for partial matching.
Partial matching is particularly important for robustness to topological noise
(incomplete data and contacts), which is a common problem in real-world 3D
scanner data. In this paper, we introduce a new approach to partial, intrinsic
isometric matching. Our method is based on the observation that isometries are
fully determined by purely local information: a map of a single point and its
tangent space fixes an isometry for both global and the partial maps. From this
idea, we develop a new representation for partial isometric maps based on
equivalence classes of correspondences between pairs of points and their
tangent spaces. From this, we derive a local propagation algorithm that find
such mappings efficiently. In contrast to previous heuristics based on RANSAC
or expectation maximization, our method is based on a simple and sound
theoretical model and fully deterministic. We apply our approach to register
partial point clouds and compare it to the state-of-the-art methods, where we
obtain significant improvements over global methods for real-world data and
stronger guarantees than previous heuristic partial matching algorithms.Comment: 17 pages, 12 figure
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