1,788 research outputs found
Discrete spherical means of directional derivatives and Veronese maps
We describe and study geometric properties of discrete circular and spherical
means of directional derivatives of functions, as well as discrete
approximations of higher order differential operators. For an arbitrary
dimension we present a general construction for obtaining discrete spherical
means of directional derivatives. The construction is based on using the
Minkowski's existence theorem and Veronese maps. Approximating the directional
derivatives by appropriate finite differences allows one to obtain finite
difference operators with good rotation invariance properties. In particular,
we use discrete circular and spherical means to derive discrete approximations
of various linear and nonlinear first- and second-order differential operators,
including discrete Laplacians. A practical potential of our approach is
demonstrated by considering applications to nonlinear filtering of digital
images and surface curvature estimation
Self-similar prior and wavelet bases for hidden incompressible turbulent motion
This work is concerned with the ill-posed inverse problem of estimating
turbulent flows from the observation of an image sequence. From a Bayesian
perspective, a divergence-free isotropic fractional Brownian motion (fBm) is
chosen as a prior model for instantaneous turbulent velocity fields. This
self-similar prior characterizes accurately second-order statistics of velocity
fields in incompressible isotropic turbulence. Nevertheless, the associated
maximum a posteriori involves a fractional Laplacian operator which is delicate
to implement in practice. To deal with this issue, we propose to decompose the
divergent-free fBm on well-chosen wavelet bases. As a first alternative, we
propose to design wavelets as whitening filters. We show that these filters are
fractional Laplacian wavelets composed with the Leray projector. As a second
alternative, we use a divergence-free wavelet basis, which takes implicitly
into account the incompressibility constraint arising from physics. Although
the latter decomposition involves correlated wavelet coefficients, we are able
to handle this dependence in practice. Based on these two wavelet
decompositions, we finally provide effective and efficient algorithms to
approach the maximum a posteriori. An intensive numerical evaluation proves the
relevance of the proposed wavelet-based self-similar priors.Comment: SIAM Journal on Imaging Sciences, 201
Local Kernels and the Geometric Structure of Data
We introduce a theory of local kernels, which generalize the kernels used in
the standard diffusion maps construction of nonparametric modeling. We prove
that evaluating a local kernel on a data set gives a discrete representation of
the generator of a continuous Markov process, which converges in the limit of
large data. We explicitly connect the drift and diffusion coefficients of the
process to the moments of the kernel. Moreover, when the kernel is symmetric,
the generator is the Laplace-Beltrami operator with respect to a geometry which
is influenced by the embedding geometry and the properties of the kernel. In
particular, this allows us to generate any Riemannian geometry by an
appropriate choice of local kernel. In this way, we continue a program of
Belkin, Niyogi, Coifman and others to reinterpret the current diverse
collection of kernel-based data analysis methods and place them in a geometric
framework. We show how to use this framework to design local kernels invariant
to various features of data. These data-driven local kernels can be used to
construct conformally invariant embeddings and reconstruct global
diffeomorphisms
The Semi Implicit Gradient Augmented Level Set Method
Here a semi-implicit formulation of the gradient augmented level set method
is presented. By tracking both the level set and it's gradient accurate subgrid
information is provided,leading to highly accurate descriptions of a moving
interface. The result is a hybrid Lagrangian-Eulerian method that may be easily
applied in two or three dimensions. The new approach allows for the
investigation of interfaces evolving by mean curvature and by the intrinsic
Laplacian of the curvature. In this work the algorithm, convergence and
accuracy results are presented. Several numerical experiments in both two and
three dimensions demonstrate the stability of the scheme.Comment: 19 Pages, 14 Figure
The Geometrical Structure of 2d Bond-Orientational Order
We study the formulation of bond-orientational order in an arbitrary two
dimensional geometry. We find that bond-orientational order is properly
formulated within the framework of differential geometry with torsion. The
torsion reflects the intrinsic frustration for two-dimensional crystals with
arbitrary geometry. Within a Debye-Huckel approximation, torsion may be
identified as the density of dislocations. Changes in the geometry of the
system cause a reorganization of the torsion density that preserves
bond-orientational order. As a byproduct, we are able to derive several
identities involving the topology, defect density and geometric invariants such
as Gaussian curvature. The formalism is used to derive the general free energy
for a 2D sample of arbitrary geometry, both in the crystalline and hexatic
phases. Applications to conical and spherical geometries are briefly addressed.Comment: 22 pages, LaTeX, 4 eps figures Published versio
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