1,721,080 research outputs found
Jacobson, Alec, WX10171
No full textThis record was harvested from a previous catalogue system and will be withdrawn in 2025. Information in this record may be superseded or incomplete. Visit this record in UMA's new catalogue at: https://archives.library.unimelb.edu.au/nodes/view/394780Surname: JACOBSON. Given Name(s) or Initials: ALEC. Military Service Number or Last Known Location: WX10171. Missing, Wounded and Prisoner of War Enquiry Card Index Number: 17977.227921
Item: [2016.0049.27073] "Jacobson, Alec, WX10171
A Simple Discretization of the Vector Dirichlet Energy
No full textWe present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix-Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy.Computer Graphics ForumDiscrete Differential Geometry39
A mixed finite element method with piecewise linear elements for the biharmonic equation on surfaces
No full textThe biharmonic equation with Dirichlet and Neumann boundary conditions
discretized using the mixed finite element method and piecewise linear (with
the possible exception of boundary triangles) finite elements on triangular
elements has been well-studied for domains in R2. Here we study the analogous
problem on polyhedral surfaces. In particular, we provide a convergence proof
of discrete solutions to the corresponding smooth solution of the biharmonic
equation. We obtain convergence rates that are identical to the ones known for
the planar setting. Our proof focuses on three different problems: solving the
biharmonic equation on the surface, solving the biharmonic equation in a
discrete space in the metric of the surface, and solving the biharmonic
equation in a discrete space in the metric of the polyhedral approximation of
the surface. We employ inverse discrete Laplacians to bound the error between
the solutions of the two discrete problems, and generalize a flat strategy to
bound the remaining error between the discrete solutions and the exact solution
on the curved surface
Natural Boundary Conditions for Smoothing in Geometry Processing
No full text
In geometry processing, smoothness energies are commonly used to model scattered data interpolation, dense data denoising, and regularization during shape optimization. The squared Laplacian energy is a popular choice of energy and has a corresponding standard implementation: squaring the discrete Laplacian matrix. For compact domains, when values along the boundary are not known in advance, this construction
bakes in
low-order boundary conditions. This causes the geometric shape of the boundary to strongly bias the solution. For many applications, this is undesirable. Instead, we propose using the squared Frobenius norm of the Hessian as a smoothness energy. Unlike the squared Laplacian energy, this energy’s
natural boundary conditions
(those that best minimize the energy) correspond to meaningful high-order boundary conditions. These boundary conditions model free boundaries where the shape of the boundary should not bias the solution locally. Our analysis begins in the smooth setting and concludes with discretizations using finite-differences on 2D grids or mixed finite elements for triangle meshes. We demonstrate the core behavior of the squared Hessian as a smoothness energy for various tasks.
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OptCtrlPoints: Finding the Optimal Control Points for Biharmonic 3D Shape Deformation
Full text linkWe propose OptCtrlPoints, a data-driven framework designed to identify the
optimal sparse set of control points for reproducing target shapes using
biharmonic 3D shape deformation. Control-point-based 3D deformation methods are
widely utilized for interactive shape editing, and their usability is enhanced
when the control points are sparse yet strategically distributed across the
shape. With this objective in mind, we introduce a data-driven approach that
can determine the most suitable set of control points, assuming that we have a
given set of possible shape variations. The challenges associated with this
task primarily stem from the computationally demanding nature of the problem.
Two main factors contribute to this complexity: solving a large linear system
for the biharmonic weight computation and addressing the combinatorial problem
of finding the optimal subset of mesh vertices. To overcome these challenges,
we propose a reformulation of the biharmonic computation that reduces the
matrix size, making it dependent on the number of control points rather than
the number of vertices. Additionally, we present an efficient search algorithm
that significantly reduces the time complexity while still delivering a nearly
optimal solution. Experiments on SMPL, SMAL, and DeformingThings4D datasets
demonstrate the efficacy of our method. Our control points achieve better
template-to-target fit than FPS, random search, and neural-network-based
prediction. We also highlight the significant reduction in computation time
from days to approximately 3 minutes.Comment: Pacific Graphics 2023 (Full Paper). Project page:
https://soulmates2.github.io/publications/OptCtrlPoints
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