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Register a new userStochastic Geometric Iterative Method for Loop Subdivision Surface Fitting
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Added by
Hongwei Lin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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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.
We investigate the isogeometric analysis for surface PDEs based on the extended Loop subdivision approach. The basis functions consisting of quartic box-splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly, and construct the solution space for dependent variables as well, which is consistent with the concept of isogeometric analysis. The subdivision process is equivalent to the $h$-refinement of NURBS-based isogeometric analysis. The performance of the proposed method is evaluated by solving various surface PDEs, such as surface Laplace-Beltrami harmonic/biharmonic/triharmonic equations, which are defined on different limit surfaces of the extended Loop subdivision for different initial control meshes. Numerical experiments demonstrate that the proposed method has desirable performance in terms of the accuracy, convergence and computational cost for solving the above surface PDEs defined on both open and closed surfaces. The proposed approach is proved to be second-order accuracy in the sense of $L^2$-norm by theoretical and/or numerical results, which is also outperformed over the standard linear finite element by several numerical comparisons.
This paper introduces Neural Subdivision, a novel framework for data-driven coarse-to-fine geometry modeling. During inference, our method takes a coarse triangle mesh as input and recursively subdivides it to a finer geometry by applying the fixed topological updates of Loop Subdivision, but predicting vertex positions using a neural network conditioned on the local geometry of a patch. This approach enables us to learn complex non-linear subdivision schemes, beyond simple linear averaging used in classical techniques. One of our key contributions is a novel self-supervised training setup that only requires a set of high-resolution meshes for learning network weights. For any training shape, we stochastically generate diverse low-resolution discretizations of coarse counterparts, while maintaining a bijective mapping that prescribes the exact target position of every new vertex during the subdivision process. This leads to a very efficient and accurate loss function for conditional mesh generation, and enables us to train a method that generalizes across discretizations and favors preserving the manifold structure of the output. During training we optimize for the same set of network weights across all local mesh patches, thus providing an architecture that is not constrained to a specific input mesh, fixed genus, or category. Our network encodes patch geometry in a local frame in a rotation- and translation-invariant manner. Jointly, these design choices enable our method to generalize well, and we demonstrate that even when trained on a single high-resolution mesh our method generates reasonable subdivisions for novel shapes.
Customized grippers have broad applications in industrial assembly lines. Compared with general parallel grippers, the customized grippers have specifically designed fingers to increase the contact area with the workpieces and improve the grasp robustness. However, grasp planning for customized grippers is challenging due to the object variations, surface contacts and structural constraints of the grippers. In this paper, an iterative surface fitting (ISF) algorithm is proposed to plan grasps for customized grippers. ISF simultaneously searches for optimal gripper transformation and finger displacement by minimizing the surface fitting error. A guided sampling is introduced to avoid ISF getting stuck in local optima and improve the collision avoidance performance. The proposed algorithm is able to consider the structural constraints of the gripper and plan optimal grasps in real-time. The effectiveness of the algorithm is verified by both simulations and experiments. The experimental videos are available at: http://me.berkeley.edu/%7Eyongxiangfan/CASE2018/caseisf.html
In recent years, mesh subdivision---the process of forging smooth free-form surfaces from coarse polygonal meshes---has become an indispensable production instrument. Although subdivision performance is crucial during simulation, animation and rendering, state-of-the-art approaches still rely on serial implementations for complex parts of the subdivision process. Therefore, they often fail to harness the power of modern parallel devices, like the graphics processing unit (GPU), for large parts of the algorithm and must resort to time-consuming serial preprocessing. In this paper, we show that a complete parallelization of the subdivision process for modern architectures is possible. Building on sparse matrix linear algebra, we show how to structure the complete subdivision process into a sequence of algebra operations. By restructuring and grouping these operations, we adapt the process for different use cases, such as regular subdivision of dynamic meshes, uniform subdivision for immutable topology, and feature-adaptive subdivision for efficient rendering of animated models. As the same machinery is used for all use cases, identical subdivision results are achieved in all parts of the production pipeline. As a second contribution, we show how these linear algebra formulations can effectively be translated into efficient GPU kernels. Applying our strategies to $sqrt{3}$, Loop and Catmull-Clark subdivision shows significant speedups of our approach compared to state-of-the-art solutions, while we completely avoid serial preprocessing.
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