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Added by
Chai Shuangming
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Low isometric distortion is often required for mesh parameterizations. A configuration of some vertices, where the distortion is concentrated, provides a way to mitigate isometric distortion, but determining the number and placement of these vertices is non-trivial. We call these vertices distortion points. We present a novel and automatic method to detect distortion points using a voting strategy. Our method integrates two components: candidate generation and candidate voting. Given a closed triangular mesh, we generate candidate distortion points by executing a three-step procedure repeatedly: (1) randomly cut an input to a disk topology; (2) compute a low conformal distortion parameterization; and (3) detect the distortion points. Finally, we count the candidate points and generate the final distortion points by voting. We demonstrate that our algorithm succeeds when employed on various closed meshes with a genus of zero or higher. The distortion points generated by our method are utilized in three applications, including planar parameterization, semi-automatic landmark correspondence, and isotropic remeshing. Compared to other state-of-the-art methods, our method demonstrates stronger practical robustness in distortion point detection.
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We introduce a robust optimization method for flip-free distortion energies used, for example, in parametrization, deformation, and volume correspondence. This method can minimize a variety of distortion energies, such as the symmetric Dirichlet energy and our new symmetric gradient energy. We identify and exploit the special structure of distortion energies to employ an operator splitting technique, leading us to propose a novel Alternating Direction Method of Multipliers (ADMM) algorithm to deal with the non-convex, non-smooth nature of distortion energies. The scheme results in an efficient method where the global step involves a single matrix multiplication and the local steps are closed-form per-triangle/per-tetrahedron expressions that are highly parallelizable. The resulting general-purpose optimization algorithm exhibits robustness to flipped triangles and tetrahedra in initial data as well as during the optimization. We establish the convergence of our proposed algorithm under certain conditions and demonstrate applications to parametrization, deformation, and volume correspondence.
Mesh distortion optimization is a popular research topic and has wide range of applications in computer graphics, including geometry modeling, variational shape interpolation, UV parameterization, elastoplastic simulation, etc. In recent years, many solvers have been proposed to solve this nonlinear optimization efficiently, among which projected Newton has been shown to have best convergence rate and work well in both 2D and 3D applications. Traditional Newton approach suffers from ill conditioning and indefiniteness of local energy approximation. A crucial step in projected Newton is to fix this issue by projecting energy Hessian onto symmetric positive definite (SPD) cone so as to guarantee the search direction always pointing to decrease the energy locally. Such step relies on time consuming eigen decomposition of element Hessian, which has been addressed by several work before on how to obtain a conjugacy that is as diagonal as possible. In this report, we demonstrate an analytic form of Hessian eigen system for distortion energy defined using principal stretches, which is the most general representation. Compared with existing projected Newton diagonalization approaches, our formulation is more general as it doesnt require the energy to be representable by tensor invariants. In this report, we will only show the derivation for 3D and the extension to 2D case is straightforward.
Geometric model fitting is a fundamental task in computer graphics and computer vision. However, most geometric model fitting methods are unable to fit an arbitrary geometric model (e.g. a surface with holes) to incomplete data, due to that the similarity metrics used in these methods are unable to measure the rigid partial similarity between arbitrary models. This paper hence proposes a novel rigid geometric similarity metric, which is able to measure both the full similarity and the partial similarity between arbitrary geometric models. The proposed metric enables us to perform partial procedural geometric model fitting (PPGMF). The task of PPGMF is to search a procedural geometric model space for the model rigidly similar to a query of non-complete point set. Models in the procedural model space are generated according to a set of parametric modeling rules. A typical query is a point cloud. PPGMF is very useful as it can be used to fit arbitrary geometric models to non-complete (incomplete, over-complete or hybrid-complete) point cloud data. For example, most laser scanning data is non-complete due to occlusion. Our PPGMF method uses Markov chain Monte Carlo technique to optimize the proposed similarity metric over the model space. To accelerate the optimization process, the method also employs a novel coarse-to-fine model dividing strategy to reject dissimilar models in advance. Our method has been demonstrated on a variety of geometric models and non-complete data. Experimental results show that the PPGMF method based on the proposed metric is able to fit non-complete data, while the method based on other metrics is unable. It is also shown that our method can be accelerated by several times via early rejection.
In this paper, we propose a stochastic geometric iterative method to approximate the high-resolution 3D models by finite Loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite Loop subdivision surface approximates the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of our method and verify it by demonstrating error curves in the experiment. In addition, compared with an existing geometric iterative method, our method has a faster fitting speed and higher fitting precision.
We present a method for the efficient processing of contact and collision in volumetric elastic models simulated using the Projective Dynamics paradigm. Our approach enables interactive simulation of tetrahedral meshes with more than half a million elements, provided that the model satisfies two fundamental properties: the region of the models surface that is susceptible to collision events needs to be known in advance, and the simulation degrees of freedom associated with that surface region should be limited to a small fraction (e.g. 5%) of the total simulation nodes. Despite this conscious delineation of scope, our hypotheses hold true for common animation subjects, such as simulated models of the human face and parts of the body. In such scenarios, a partial Cholesky factorization can abstract away the behavior of the collision-safe subset of the face into the Schur Complement matrix with respect to the collision-prone region. We demonstrate how fast and accurate updates of penalty-based collision terms can be incorporated into this representation, and solved with high efficiency on the GPU. We also demonstrate the opportunity to iterate a partial update of the element rotations, akin to a selective application of the local step, specifically on the smaller collision-prone region without explicitly paying the cost associated with the rest of the simulation mesh. We demonstrate efficient and robust interactive simulation in detailed models from animation and medical applications.
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