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Item Cascading Upper Bounds for Triangle Soup Pompeiu-Hausdorff Distance(The Eurographics Association and John Wiley & Sons Ltd., 2024) Sacht, Leonardo; Jacobson, Alec; Hu, Ruizhen; Lefebvre, SylvainWe propose a new method to accurately approximate the Pompeiu-Hausdorff distance from a triangle soup A to another triangle soup B up to a given tolerance. Based on lower and upper bound computations, we discard triangles from A that do not contain the maximizer of the distance to B and subdivide the others for further processing. In contrast to previous methods, we use four upper bounds instead of only one, three of which newly proposed by us. Many triangles are discarded using the simpler bounds, while the most difficult cases are dealt with by the other bounds. Exhaustive testing determines the best ordering of the four upper bounds. A collection of experiments shows that our method is faster than all previous accurate methods in the literature.Item Search Me Knot, Render Me Knot: Embedding Search and Differentiable Rendering of Knots in 3D(The Eurographics Association and John Wiley & Sons Ltd., 2024) Gangopadhyay, Aalok; Gupta, Paras; Sharma, Tarun; Singh, Prajwal; Raman, Shanmuganathan; Hu, Ruizhen; Lefebvre, SylvainWe introduce the problem of knot-based inverse perceptual art. Given multiple target images and their corresponding viewing configurations, the objective is to find a 3D knot-based tubular structure whose appearance resembles the target images when viewed from the specified viewing configurations. To solve this problem, we first design a differentiable rendering algorithm for rendering tubular knots embedded in 3D for arbitrary perspective camera configurations. Utilizing this differentiable rendering algorithm, we search over the space of knot configurations to find the ideal knot embedding. We represent the knot embeddings via homeomorphisms of the desired template knot, where the weights of an invertible neural network parametrize the homeomorphisms. Our approach is fully differentiable, making it possible to find the ideal 3D tubular structure for the desired perceptual art using gradient-based optimization. We propose several loss functions that impose additional physical constraints, enforcing that the tube is free of self-intersection, lies within a predefined region in space, satisfies the physical bending limits of the tube material, and the material cost is within a specified budget. We demonstrate through results that our knot representation is highly expressive and gives impressive results even for challenging target images in both single-view and multiple-view constraints. Through extensive ablation study, we show that each proposed loss function effectively ensures physical realizability. We construct a real-world 3D-printed object to demonstrate the practical utility of our approach.Item Mesh Parameterization Meets Intrinsic Triangulations(The Eurographics Association and John Wiley & Sons Ltd., 2024) Akalin, Koray; Finnendahl, Ugo; Sorkine-Hornung, Olga; Alexa, Marc; Hu, Ruizhen; Lefebvre, SylvainA parameterization of a triangle mesh is a realization in the plane so that all triangles have positive signed area. Triangle mesh parameterizations are commonly computed by minimizing a distortion energy, measuring the distortions of the triangles as they are mapped into the parameter domain. It is assumed that the triangulation is fixed and the triangles are mapped affinely. We consider a more general setup and additionally optimize among the intrinsic triangulations of the piecewise linear input geometry. This means the distortion energy is computed for the same geometry, yet the space of possible parameterizations is enlarged. For minimizing the distortion energy, we suggest alternating between varying the parameter locations of the vertices and intrinsic flipping. We show that this process improves the mapping for different distortion energies at moderate additional cost. We also find intrinsic triangulations that are better starting points for the optimization of positions, offering a compromise between the full optimization approach and exploiting the additional freedom of intrinsic triangulations.