Volume 40 (2021)
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Browsing Volume 40 (2021) by Author "Alexa, Marc"
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Item The Diamond Laplace for Polygonal and Polyhedral Meshes(The Eurographics Association and John Wiley & Sons Ltd., 2021) Bunge, Astrid; Botsch, Mario; Alexa, Marc; Digne, Julie and Crane, KeenanWe introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge together with its corresponding primal element - an edge for surface meshes and a face for volumetric meshes. We call the operator resulting from taking the divergence of the gradient Diamond Laplacian. Additional vertices used for the construction are represented as affine combinations of the original vertices, so that the Laplacian operator maps from values at vertices to values at vertices, as is common in geometry processing applications. The construction is local, exactly the same for all types of meshes, and results in a symmetric negative definite operator with linear precision. We show that the accuracy of the Diamond Laplacian is similar or better compared to other discretizations. The greater versatility and generally good behavior come at the expense of an increase in the number of non-zero coefficients that depends on the degree of the mesh elements.Item Fast Updates for Least-Squares Rotational Alignment(The Eurographics Association and John Wiley & Sons Ltd., 2021) Zhang, Jiayi Eris; Jacobson, Alec; Alexa, Marc; Mitra, Niloy and Viola, IvanAcross computer graphics, vision, robotics and simulation, many applications rely on determining the 3D rotation that aligns two objects or sets of points. The standard solution is to use singular value decomposition (SVD), where the optimal rotation is recovered as the product of the singular vectors. Faster computation of only the rotation is possible using suitable parameterizations of the rotations and iterative optimization. We propose such a method based on the Cayley transformations. The resulting optimization problem allows better local quadratic approximation compared to the Taylor approximation of the exponential map. This results in both faster convergence as well as more stable approximation compared to other iterative approaches. It also maps well to AVX vectorization. We compare our implementation with a wide range of alternatives on real and synthetic data. The results demonstrate up to two orders of magnitude of speedup compared to a straightforward SVD implementation and a 1.5-6 times speedup over popular optimized code.Item Gauss Stylization: Interactive Artistic Mesh Modeling based on Preferred Surface Normals(The Eurographics Association and John Wiley & Sons Ltd., 2021) Kohlbrenner, Maximilian; Finnendahl, Ugo; Djuren, Tobias; Alexa, Marc; Digne, Julie and Crane, KeenanExtending the ARAP energy with a term that depends on the face normal, energy minimization becomes an effective stylization tool for shapes represented as meshes. Our approach generalizes the possibilities of Cubic Stylization: the set of preferred normals can be chosen arbitrarily from the Gauss sphere, including semi-discrete sets to model preference for cylinder- or cone-like shapes. The optimization is designed to retain, similar to ARAP, the constant linear system in the global optimization. This leads to convergence behavior that enables interactive control over the parameters of the optimization. We provide various examples demonstrating the simplicity and versatility of the approach.