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Item Learning Fuzzy Set Representations of Partial Shapes on Dual Embedding Spaces(The Eurographics Association and John Wiley & Sons Ltd., 2018) Sung, Minhyuk; Dubrovina, Anastasia; Kim, Vladimir G.; Guibas, Leonidas J.; Ju, Tao and Vaxman, AmirModeling relations between components of 3D objects is essential for many geometry editing tasks. Existing techniques commonly rely on labeled components, which requires substantial annotation effort and limits components to a dictionary of predefined semantic parts. We propose a novel framework based on neural networks that analyzes an uncurated collection of 3D models from the same category and learns two important types of semantic relations among full and partial shapes: complementarity and interchangeability. The former helps to identify which two partial shapes make a complete plausible object, and the latter indicates that interchanging two partial shapes from different objects preserves the object plausibility. Our key idea is to jointly encode both relations by embedding partial shapes as fuzzy sets in dual embedding spaces. We model these two relations as fuzzy set operations performed across the dual embedding spaces, and within each space, respectively. We demonstrate the utility of our method for various retrieval tasks that are commonly needed in geometric modeling interfaces.Item OptCtrlPoints: Finding the Optimal Control Points for Biharmonic 3D Shape Deformation(The Eurographics Association and John Wiley & Sons Ltd., 2023) Kim, Kunho; Uy, Mikaela Angelina; Paschalidou, Despoina; Jacobson, Alec; Guibas, Leonidas J.; Sung, Minhyuk; Chaine, Raphaëlle; Deng, Zhigang; Kim, Min H.We propose OPTCTRLPOINTS, a data-driven framework designed to identify the optimal sparse set of control points for reproducing target shapes using biharmonic 3D shape deformation. Control-point-based 3D deformation methods are widely utilized for interactive shape editing, and their usability is enhanced when the control points are sparse yet strategically distributed across the shape. With this objective in mind, we introduce a data-driven approach that can determine the most suitable set of control points, assuming that we have a given set of possible shape variations. The challenges associated with this task primarily stem from the computationally demanding nature of the problem. Two main factors contribute to this complexity: solving a large linear system for the biharmonic weight computation and addressing the combinatorial problem of finding the optimal subset of mesh vertices. To overcome these challenges, we propose a reformulation of the biharmonic computation that reduces the matrix size, making it dependent on the number of control points rather than the number of vertices. Additionally, we present an efficient search algorithm that significantly reduces the time complexity while still delivering a nearly optimal solution. Experiments on SMPL, SMAL, and DeformingThings4D datasets demonstrate the efficacy of our method. Our control points achieve better template-to-target fit than FPS, random search, and neural-network-based prediction. We also highlight the significant reduction in computation time from days to approximately 3 minutes.Item A Unified Discrete Framework for Intrinsic and Extrinsic Dirac Operators for Geometry Processing(The Eurographics Association and John Wiley & Sons Ltd., 2018) Ye, Zi; Diamanti, Olga; Tang, Chengcheng; Guibas, Leonidas J.; Hoffmann, Tim; Ju, Tao and Vaxman, AmirSpectral mesh analysis and processing methods, namely ones that utilize eigenvalues and eigenfunctions of linear operators on meshes, have been applied to numerous geometric processing applications. The operator used predominantly in these methods is the Laplace-Beltrami operator, which has the often-cited property that it is intrinsic, namely invariant to isometric deformation of the underlying geometry, including rigid transformations. Depending on the application, this can be either an advantage or a drawback. Recent work has proposed the alternative of using the Dirac operator on surfaces for spectral processing. The available versions of the Dirac operator either only focus on the extrinsic version, or introduce a range of mixed operators on a spectrum between fully extrinsic Dirac operator and intrinsic Laplace operator. In this work, we introduce a unified discretization scheme that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds. In this discretization, both operators are very closely related, and preserve their key properties from the smooth case. We showcase various applications of our operators, with improved numerics over prior work.Item Modular Latent Spaces for Shape Correspondences(The Eurographics Association and John Wiley & Sons Ltd., 2018) Ganapathi-Subramanian, Vignesh; Diamanti, Olga; Guibas, Leonidas J.; Ju, Tao and Vaxman, AmirWe consider the problem of transporting shape descriptors across shapes in a collection in a modular fashion, in order to establish correspondences between them. A common goal when mapping between multiple shapes is consistency, namely that compositions of maps along a cycle of shapes should be approximately an identity map. Existing attempts to enforce consistency typically require recomputing correspondences whenever a new shape is added to the collection, which can quickly become intractable. Instead, we propose an approach that is fully modular, where the bulk of the computation is done on each shape independently. To achieve this, we use intermediate nonlinear embedding spaces, computed individually on every shape; the embedding functions use ideas from diffusion geometry and capture how different descriptors on the same shape inter-relate. We then establish linear mappings between the different embedding spaces, via a shared latent space. The introduction of nonlinear embeddings allows for more nuanced correspondences, while the modularity of the construction allows for parallelizable calculation and efficient addition of new shapes. We compare the performance of our framework to standard functional correspondence techniques and showcase the use of this framework to simple interpolation and extrapolation tasks.Item QuadriFlow: A Scalable and Robust Method for Quadrangulation(The Eurographics Association and John Wiley & Sons Ltd., 2018) Huang, Jingwei; Zhou, Yichao; Niessner, Matthias; Shewchuk, Jonathan Richard; Guibas, Leonidas J.; Ju, Tao and Vaxman, AmirQuadriFlow is a scalable algorithm for generating quadrilateral surface meshes based on the Instant Field-Aligned Meshes of Jakob et al. (ACM Trans. Graph. 34(6):189, 2015). We modify the original algorithm such that it efficiently produces meshes with many fewer singularities. Singularities in quadrilateral meshes cause problems for many applications, including parametrization and rendering with Catmull-Clark subdivision surfaces. Singularities can rarely be entirely eliminated, but it is possible to keep their number small. Local optimization algorithms usually produce meshes with many singularities, whereas the best algorithms tend to require non-local optimization, and therefore are slow. We propose an efficient method to minimize singularities by combining the Instant Meshes objective with a system of linear and quadratic constraints. These constraints are enforced by solving a global minimum-cost network flow problem and local boolean satisfiability problems. We have verified the robustness and efficiency of our method on a subset of ShapeNet comprising 17,791 3D objects in the wild. Our evaluation shows that the quality of the quadrangulations generated by our method is as good as, if not better than, those from other methods, achieving about four times fewer singularities than Instant Meshes. Other algorithms that produce similarly few singularities are much slower; we take less than ten seconds to process each model. Our source code is publicly available.Item Topology-Aware Surface Reconstruction for Point Clouds(The Eurographics Association and John Wiley & Sons Ltd., 2020) Brüel-Gabrielsson, Rickard; Ganapathi-Subramanian, Vignesh; Skraba, Primoz; Guibas, Leonidas J.; Jacobson, Alec and Huang, QixingWe present an approach to incorporate topological priors in the reconstruction of a surface from a point scan. We base the reconstruction on basis functions which are optimized to provide a good fit to the point scan while satisfying predefined topological constraints. We optimize the parameters of a model to obtain a likelihood function over the reconstruction domain. The topological constraints are captured by persistence diagrams which are incorporated within the optimization algorithm to promote the correct topology. The result is a novel topology-aware technique which can (i) weed out topological noise from point scans, and (ii) capture certain nuanced properties of the underlying shape which could otherwise be lost while performing surface reconstruction. We show results reconstructing shapes with multiple potential topologies, compare to other classical surface construction techniques, and show the completion of real scan data.