67 research outputs found
Steklov Spectral Geometry for Extrinsic Shape Analysis
We propose using the Dirichlet-to-Neumann operator as an extrinsic
alternative to the Laplacian for spectral geometry processing and shape
analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator,
cannot capture the spatial embedding of a shape up to rigid motion, and many
previous extrinsic methods lack theoretical justification. Instead, we consider
the Steklov eigenvalue problem, computing the spectrum of the
Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable
property of this operator is that it completely encodes volumetric geometry. We
use the boundary element method (BEM) to discretize the operator, accelerated
by hierarchical numerical schemes and preconditioning; this pipeline allows us
to solve eigenvalue and linear problems on large-scale meshes despite the
density of the Dirichlet-to-Neumann discretization. We further demonstrate that
our operators naturally fit into existing frameworks for geometry processing,
making a shift from intrinsic to extrinsic geometry as simple as substituting
the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde
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Routing with Guaranteed Delivery on Virtual Coordinates
We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for non-triangular graphs, and their virtual coordinates are easy to construct.Engineering and Applied Science
BPM: Blended Piecewise Moebius Maps
We propose a novel Moebius interpolator that takes as an input a discrete map
between the vertices of two planar triangle meshes, and outputs a smooth map on
the input domain. The output map interpolates the discrete map, is continuous
between triangles, and has low quasi-conformal distortion when the input map is
discrete conformal. Our map leads to considerably smoother texture transfer
compared to the alternatives, even on very coarse triangulations. Furthermore,
our approach has a closed-form expression, is local, applicable to any discrete
map, and leads to smooth results even for extreme deformations. Finally, by
working with local intrinsic coordinates, our approach is easily generalizable
to discrete maps between a surface triangle mesh and a planar mesh, i.e., a
planar parameterization. We compare our method with existing approaches, and
demonstrate better texture transfer results, and lower quasi-conformal errors
AmiGo: Computational Design of Amigurumi Crochet Patterns
We propose an approach for generating crochet instructions (patterns) from an
input 3D model. We focus on Amigurumi, which are knitted stuffed toys. Given a
closed triangle mesh, and a single point specified by the user, we generate
crochet instructions, which when knitted and stuffed result in a toy similar to
the input geometry. Our approach relies on constructing the geometry and
connectivity of a Crochet Graph, which is then translated into a crochet
pattern. We segment the shape automatically into chrochetable components, which
are connected using the join-as-you-go method, requiring no additional sewing.
We demonstrate that our method is applicable to a large variety of shapes and
geometries, and yields easily crochetable patterns.Comment: 11 pages, 10 figures, SCF 202
Distributed Computation of Virtual Coordinates
International audienceSensor networks are emerging as a paradigm for future computing, but pose a number of challenges in the fields of networking and distributed computation. One challenge is to devise a greedy routing protocol – one that routes messages through the network using only information available at a node or its neighbors. Modeling the connectivity graph of a sensor network as a 3-connected planar graph, we describe how to compute on the network in a distributed and local manner a special geometric embedding of the graph. This embedding supports a geometric routing protocol based on the ”virtual” coordinates of the nodes derived from the embedding
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