A new method for noise removal of arbitrary surfaces
meshes is presented which focuses on the preservation
and sharpening of non-linear geometric features such
as curved surface regions and feature lines. Our method
uses a prescribed mean curvature flow (PMC) for simplicial
surfaces which is based on three new contributions:
1. the definition and efficient calculation of a
discrete shape operator and principal curvature properties
on simplicial surfaces that is fully consistent with
the well-known discrete mean curvature formula, 2. an
anisotropic discrete mean curvature vector that combines
the advantages of the mean curvature normal with
the special anisotropic behaviour along feature lines of
a surface, and 3. an anisotropic prescribed mean curvature
flow which converges to surfaces with an estimated
mean curvature distribution and with preserved nonlinear
features. Additionally, the PMC flow prevents
boundary shrinkage at constrained and free boundary
segments.
In this work, we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis. In particular, we consider a physical model that describes the vibration modes and frequencies of a surface through the eigenfunctions and eigenvalues of the Hessian of a deformation energy, and we
derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Furthermore, we design a quadratic energy, such that the eigenmodes of the Hessian of
this energy are sensitive to the extrinsic curvature of the surface. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that
can be used to identify features of the surface. In addition, we discuss a
spectral quadrangulation scheme for surfaces.
We propose a framework for deformation-based surface modeling that is interactive, robust and intuitive to use. The deformations are described by a non-linear optimization problem that models static states of elastic shapes under external forces which implement the user input. Interactive response is achieved by a combination of model reduction, a robust energy approximation, and an efficient quasi-Newton solver. Motivated by the observation that a typical modeling session requires only a fraction of the full shape
space of the underlying model, we use second and third derivatives of a deformation energy to construct a low-dimensional shape space that forms the feasible set for the optimization. Based on mesh coarsening, we propose an
energy approximation scheme with adjustable approximation quality. The quasi-Newton solver guarantees superlinear convergence without the need of costly Hessian evaluations during modeling. We demonstrate the effectiveness of the approach on different examples including the test suite introduced in [Botsch and Sorkine 2008].
This work concerns the approximation of the shape operator of smooth surfaces in R^3 from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator. We show experimental results that confirm our estimates.
Discrete Laplace--Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace--Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace--Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace--Beltrami operators.
This work concerns the approximation of the shape operator of smooth surfaces in $\mathbb{R}^{3}$ from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator, including mean and Gaussian curvature, principal curvatures, and principal curvature directions.
The estimates are confirmed by numerical experiments.
We provide conditions for convergence of polyhedral surfaces and their
discrete geometric properties to smooth surfaces embedded in R^3. The
notion of totally normal convergence is shown to be equivalent to the convergence
of either one of the following: surface area, intrinsic metric, and
Laplace-Beltrami operators. We further show that totally normal convergence
implies convergence results for shortest geodesics, mean curvature,
and solutions to the Dirichlet problem. This work provides the justifi-
cation for a discrete theory of differential geometric operators defined on
polyhedral surfaces based on a variational formulation.
In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies or deformation energies. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved
surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of surfaces.