92 research outputs found
On a new conformal functional for simplicial surfaces
We introduce a smooth quadratic conformal functional and its weighted version
where
is the extrinsic intersection angle of the circumcircles of the
triangles of the mesh sharing the edge and is the valence of
vertex . Besides minimizing the squared local conformal discrete Willmore
energy this functional also minimizes local differences of the angles
. We investigate the minimizers of this functionals for simplicial
spheres and simplicial surfaces of nontrivial topology. Several remarkable
facts are observed. In particular for most of randomly generated simplicial
polyhedra the minimizers of and are inscribed polyhedra. We
demonstrate also some applications in geometry processing, for example, a
conformal deformation of surfaces to the round sphere. A partial theoretical
explanation through quadratic optimization theory of some observed phenomena is
presented.Comment: 14 pages, 8 figures, to appear in the proceedings of "Curves and
Surfaces, 8th International Conference", June 201
On a linear programming approach to the discrete Willmore boundary value problem and generalizations
We consider the problem of finding (possibly non connected) discrete surfaces
spanning a finite set of discrete boundary curves in the three-dimensional
space and minimizing (globally) a discrete energy involving mean curvature.
Although we consider a fairly general class of energies, our main focus is on
the Willmore energy, i.e. the total squared mean curvature Our purpose is to
address the delicate task of approximating global minimizers of the energy
under boundary constraints.
The main contribution of this work is to translate the nonlinear boundary
value problem into an integer linear program, using a natural formulation
involving pairs of elementary triangles chosen in a pre-specified dictionary
and allowing self-intersection.
Our work focuses essentially on the connection between the integer linear
program and its relaxation. We prove that: - One cannot guarantee the total
unimodularity of the constraint matrix, which is a sufficient condition for the
global solution of the relaxed linear program to be always integral, and
therefore to be a solution of the integer program as well; - Furthermore, there
are actually experimental evidences that, in some cases, solving the relaxed
problem yields a fractional solution. Due to the very specific structure of the
constraint matrix here, we strongly believe that it should be possible in the
future to design ad-hoc integer solvers that yield high-definition
approximations to solutions of several boundary value problems involving mean
curvature, in particular the Willmore boundary value problem
Non-Iterative, Feature-Preserving Mesh Smoothing
With the increasing use of geometry scanners to create 3D models, there is a rising need for fast and robust mesh smoothing to remove inevitable noise in the measurements. While most previous work has favored diffusion-based iterative techniques for feature-preserving smoothing, we propose a radically different approach, based on robust statistics and local first-order predictors of the surface. The robustness of our local estimates allows us to derive a non-iterative feature-preserving filtering technique applicable to arbitrary "triangle soups". We demonstrate its simplicity of implementation and its efficiency, which make it an excellent solution for smoothing large, noisy, and non-manifold meshes.Singapore-MIT Alliance (SMA
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