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Item Computing Surface PolyCube-Maps by Constrained Voxelization(The Eurographics Association and John Wiley & Sons Ltd., 2019) Yang, Yang; Fu, Xiao-Ming; Liu, Ligang; Lee, Jehee and Theobalt, Christian and Wetzstein, GordonWe present a novel method to compute bijective PolyCube-maps with low isometric distortion. Given a surface and its preaxis- aligned shape that is not an exact PolyCube shape, the algorithm contains two steps: (i) construct a PolyCube shape to approximate the pre-axis-aligned shape; and (ii) generate a bijective, low isometric distortion mapping between the constructed PolyCube shape and the input surface. The PolyCube construction is formulated as a constrained optimization problem, where the objective is the number of corners in the constructed PolyCube, and the constraint is to bound the approximation error between the constructed PolyCube and the input pre-axis-aligned shape while ensuring topological validity. A novel erasing-and-filling solver is proposed to solve this challenging problem. Centeral to the algorithm for computing bijective PolyCube-maps is a quad mesh optimization process that projects the constructed PolyCube onto the input surface with high-quality quads. We demonstrate the efficacy of our algorithm on a data set containing 300 closed meshes. Compared to state-of-the-art methods, our method achieves higher practical robustness and lower mapping distortion.Item Accurate Synthesis of Multi-Class Disk Distributions(The Eurographics Association and John Wiley & Sons Ltd., 2019) Ecormier-Nocca, Pierre; Memari, Pooran; Gain, James; Cani, Marie-Paule; Alliez, Pierre and Pellacini, FabioWhile analysing and synthesising 2D distributions of points has been applied both to the generation of textures with discrete elements and for populating virtual worlds with 3D objects, the results are often inaccurate since the spatial extent of objects cannot be expressed.We introduce three improvements enabling the synthesis of more general distributions of elements. First, we extend continuous pair correlation function (PCF) algorithms to multi-class distributions using a dependency graph, thereby capturing interrelationships between distinct categories of objects. Second, we introduce a new normalised metric for disks, which makes the method applicable to both point and possibly overlapping disk distributions. The metric is specifically designed to distinguish perceptually salient features, such as disjoint, tangent, overlapping, or nested disks. Finally, we pay particular attention to convergence of the mean PCF as well as the validity of individual PCFs, by taking into consideration the variance of the input. Our results demonstrate that this framework can capture and reproduce real-life distributions of elements representing a variety of complex semi-structured patterns, from the interaction between trees and the understorey in a forest to droplets of water. More generally, it applies to any category of 2D object whose shape is better represented by bounding circles than points.Item Global Texture Mapping for Dynamic Objects(The Eurographics Association and John Wiley & Sons Ltd., 2019) Kim, Jungeon; Kim, Hyomin; Park, Jaesik; Lee, Seungyong; Lee, Jehee and Theobalt, Christian and Wetzstein, GordonWe propose a novel framework to generate a global texture atlas for a deforming geometry. Our approach distinguishes from prior arts in two aspects. First, instead of generating a texture map for each timestamp to color a dynamic scene, our framework reconstructs a global texture atlas that can be consistently mapped to a deforming object. Second, our approach is based on a single RGB-D camera, without the need of a multiple-camera setup surrounding a scene. In our framework, the input is a 3D template model with an RGB-D image sequence, and geometric warping fields are found using a state-of-the-art non-rigid registration method [GXW*15] to align the template mesh to noisy and incomplete input depth images. With these warping fields, our multi-scale approach for texture coordinate optimization generates a sharp and clear texture atlas that is consistent with multiple color observations over time. Our approach is accelerated by graphical hardware and provides a handy configuration to capture a dynamic geometry along with a clean texture atlas. We demonstrate our approach with practical scenarios, particularly human performance capture. We also show that our approach is resilient on misalignment issues caused by imperfect estimation of warping fields and inaccurate camera parameters.Item Practical Foldover-Free Volumetric Mapping Construction(The Eurographics Association and John Wiley & Sons Ltd., 2019) Su, Jian-Ping; Fu, Xiao-Ming; Liu, Ligang; Lee, Jehee and Theobalt, Christian and Wetzstein, GordonIn this paper, we present a practically robust method for computing foldover-free volumetric mappings with hard linear constraints. Central to this approach is a projection algorithm that monotonically and efficiently decreases the distance from the mapping to the bounded conformal distortion mapping space. After projection, the conformal distortion of the updated mapping tends to be below the given bound, thereby significantly reducing foldovers. Since it is non-trivial to define an optimal bound, we introduce a practical conformal distortion bound generation scheme to facilitate subsequent projections. By iteratively generating conformal distortion bounds and trying to project mappings into bounded conformal distortion spaces monotonically, our algorithm achieves high-quality foldover-free volumetric mappings with strong practical robustness and high efficiency. Compared with existing methods, our method computes mesh-based and meshless volumetric mappings with no prescribed conformal distortion bounds. We demonstrate the efficacy and efficiency of our method through a variety of geometric processing tasks.Item A Family of Barycentric Coordinates for Co-Dimension 1 Manifolds with Simplicial Facets(The Eurographics Association and John Wiley & Sons Ltd., 2019) Yan, Zhipei; Schaefer, Scott; Bommes, David and Huang, HuiWe construct a family of barycentric coordinates for 2D shapes including non-convex shapes, shapes with boundaries, and skeletons. Furthermore, we extend these coordinates to 3D and arbitrary dimension. Our approach modifies the construction of the Floater-Hormann-Kós family of barycentric coordinates for 2D convex shapes.We show why such coordinates are restricted to convex shapes and show how to modify these coordinates to extend to discrete manifolds of co-dimension 1 whose boundaries are composed of simplicial facets. Our coordinates are well-defined everywhere (no poles) and easy to evaluate. While our construction is widely applicable to many domains, we show several examples related to image and mesh deformation.Item Computational Design of Steady 3D Dissection Puzzles(The Eurographics Association and John Wiley & Sons Ltd., 2019) Tang, Keke; Song, Peng; Wang, Xiaofei; Deng, Bailin; Fu, Chi-Wing; Liu, Ligang; Alliez, Pierre and Pellacini, FabioDissection puzzles require assembling a common set of pieces into multiple distinct forms. Existing works focus on creating 2D dissection puzzles that form primitive or naturalistic shapes. Unlike 2D dissection puzzles that could be supported on a tabletop surface, 3D dissection puzzles are preferable to be steady by themselves for each assembly form. In this work, we aim at computationally designing steady 3D dissection puzzles. We address this challenging problem with three key contributions. First, we take two voxelized shapes as inputs and dissect them into a common set of puzzle pieces, during which we allow slightly modifying the input shapes, preferably on their internal volume, to preserve the external appearance. Second, we formulate a formal model of generalized interlocking for connecting pieces into a steady assembly using both their geometric arrangements and friction. Third, we modify the geometry of each dissected puzzle piece based on the formal model such that each assembly form is steady accordingly. We demonstrate the effectiveness of our approach on a wide variety of shapes, compare it with the state-of-the-art on 2D and 3D examples, and fabricate some of our designed puzzles to validate their steadiness.Item Topology Preserving Simplification of Medial Axes in 3D Models(The Eurographics Association and John Wiley & Sons Ltd., 2019) Chu, Yiyao; Hou, Fei; Wang, Wencheng; Li, Lei; Lee, Jehee and Theobalt, Christian and Wetzstein, GordonWe propose an efficient method for topology-preserving simplification of medial axes of 3D models. Existing methods either cannot preserve the topology during medial axes simplification or have the problem of being geometrically inaccurate or computationally expensive. To tackle these issues, we restrict our topology-checking to the areas around the topological holes to avoid unnecessary checks in other areas. Our algorithm can keep high precision even when the medial axis is simplified to be in very few vertices. Furthermore, we parallelize the medial axes simplification procedure to enhance the performance significantly. Experimental results show that our method can preserve the topology with highly efficient performance, much superior to the existing methods in terms of topology preservation, accuracy and performance.Item Procedural Tectonic Planets(The Eurographics Association and John Wiley & Sons Ltd., 2019) Cortial, Yann; Peytavie, Adrien; Galin, Eric; Guérin, Eric; Alliez, Pierre and Pellacini, FabioWe present a procedural method for authoring synthetic tectonic planets. Instead of relying on computationally demanding physically-based simulations, we capture the fundamental phenomena into a procedural method that faithfully reproduces largescale planetary features generated by the movement and collision of the tectonic plates. We approximate complex phenomena such as plate subduction or collisions to deform the lithosphere, including the continental and oceanic crusts. The user can control the movement of the plates, which dynamically evolve and generate a variety of landforms such as continents, oceanic ridges, large scale mountain ranges or island arcs. Finally, we amplify the large-scale planet model with either procedurallydefined or real-world elevation data to synthesize coherent detailed reliefs. Our method allows the user to control the evolution of an entire planet interactively, and to trigger specific events such as catastrophic plate rifting.Item Exact Constraint Satisfaction for Truly Seamless Parametrization(The Eurographics Association and John Wiley & Sons Ltd., 2019) Mandad, Manish; Campen, Marcel; Alliez, Pierre and Pellacini, FabioIn the field of global surface parametrization a recent focus has been on so-called seamless parametrization. This term refers to parametrization approaches which, while using an atlas of charts to enable the handling of surfaces of arbitrary topology, relate the parametrization across the cuts between charts via transition functions from special classes of transformations. This effectively makes the cuts invisible to applications which are invariant to these specific transformations in some sense. In actual implementations of these parametrization approaches, however, these restrictions are obeyed only approximately; errors stem from the tolerances of numerical solvers employed and, ultimately, from the limited accuracy of floating point arithmetic. In practice, robustness issues arise from these flaws in the seamlessness of a parametrization, no matter how small. We present a robust global algorithm that turns a given approximately seamless parametrization into an exactly seamless one - that still is representable by standard floating point numbers. It supports common practically relevant additional constraints regarding boundary and feature curve alignment or isocurve connectivity, and ensures that these are likewise fulfilled exactly. This allows subsequent algorithms to operate robustly on the resulting truly seamless parametrization. We believe that the core of our method will furthermore be of benefit in a broader range of applications involving linearly constrained numerical optimization.