Volume 42 (2023)
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Browsing Volume 42 (2023) by Author "Alexa, Marc"
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Item ARAP Revisited Discretizing the Elastic Energy using Intrinsic Voronoi Cells(© 2023 Eurographics ‐ The European Association for Computer Graphics and John Wiley & Sons Ltd., 2023) Finnendahl, Ugo; Schwartz, Matthias; Alexa, Marc; Hauser, Helwig and Alliez, PierreAs‐rigid‐as‐possible (ARAP) surface modelling is widely used for interactive deformation of triangle meshes. We show that ARAP can be interpreted as minimizing a discretization of an elastic energy based on non‐conforming elements defined over dual orthogonal cells of the mesh. Using the Voronoi cells rather than an orthogonal dual of the extrinsic mesh guarantees that the energy is non‐negative over each cell. We represent the intrinsic Delaunay edges extrinsically as polylines over the mesh, encoded in barycentric coordinates relative to the mesh vertices. This modification of the original ARAP energy, which we term , remedies problems stemming from non‐Delaunay edges in the original approach. Unlike the spokes‐and‐rims version of the ARAP approach it is less susceptible to the triangulation of the surface. We provide examples of deformations generated with iARAP and contrast them with other versions of ARAP. We also discuss the properties of the Laplace‐Beltrami operator implicitly introduced with the new discretization.Item Poisson Manifold Reconstruction - Beyond Co-dimension One(The Eurographics Association and John Wiley & Sons Ltd., 2023) Kohlbrenner, Maximilian; Lee, Singchun; Alexa, Marc; Kazhdan, Misha; Memari, Pooran; Solomon, JustinScreened Poisson Surface Reconstruction creates 2D surfaces from sets of oriented points in 3D (and can be extended to codimension one surfaces in arbitrary dimensions). In this work we generalize the technique to manifolds of co-dimension larger than one. The reconstruction problem consists of finding a vector-valued function whose zero set approximates the input points. We argue that the right extension of screened Poisson Surface Reconstruction is based on exterior products: the orientation of the point samples is encoded as the exterior product of the local normal frame. The goal is to find a set of scalar functions such that the exterior product of their gradients matches the exterior products prescribed by the input points. We show that this setup reduces to the standard formulation for co-dimension 1, and leads to more challenging multi-quadratic optimization problems in higher co-dimension. We explicitly treat the case of co-dimension 2, i.e., curves in 3D and 2D surfaces in 4D. We show that the resulting bi-quadratic problem can be relaxed to a set of quadratic problems in two variables and that the solution can be made effective and efficient by leveraging a hierarchical approach.