944 research outputs found
H\"older Regularity of Geometric Subdivision Schemes
We present a framework for analyzing non-linear -valued
subdivision schemes which are geometric in the sense that they commute with
similarities in . It admits to establish
-regularity for arbitrary schemes of this type, and
-regularity for an important subset thereof, which includes all
real-valued schemes. Our results are constructive in the sense that they can be
verified explicitly for any scheme and any given set of initial data by a
universal procedure. This procedure can be executed automatically and
rigorously by a computer when using interval arithmetics.Comment: 31 pages, 1 figur
Second Order Differences of Cyclic Data and Applications in Variational Denoising
In many image and signal processing applications, as interferometric
synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis or
color image restoration in HSV or LCh spaces the data has its range on the
one-dimensional sphere . Although the minimization of total
variation (TV) regularized functionals is among the most popular methods for
edge-preserving image restoration such methods were only very recently applied
to cyclic structures. However, as for Euclidean data, TV regularized
variational methods suffer from the so called staircasing effect. This effect
can be avoided by involving higher order derivatives into the functional.
This is the first paper which uses higher order differences of cyclic data in
regularization terms of energy functionals for image restoration. We introduce
absolute higher order differences for -valued data in a sound way
which is independent of the chosen representation system on the circle. Our
absolute cyclic first order difference is just the geodesic distance between
points. Similar to the geodesic distances the absolute cyclic second order
differences have only values in [0,{\pi}]. We update the cyclic variational TV
approach by our new cyclic second order differences. To minimize the
corresponding functional we apply a cyclic proximal point method which was
recently successfully proposed for Hadamard manifolds. Choosing appropriate
cycles this algorithm can be implemented in an efficient way. The main steps
require the evaluation of proximal mappings of our cyclic differences for which
we provide analytical expressions. Under certain conditions we prove the
convergence of our algorithm. Various numerical examples with artificial as
well as real-world data demonstrate the advantageous performance of our
algorithm.Comment: 32 pages, 16 figures, shortened version of submitted manuscrip
Subdivision schemes with general dilation in the geometric and nonlinear setting
AbstractWe establish results on convergence and smoothness of subdivision rules operating on manifold-valued data which are based on a general dilation matrix. In particular we cover irregular combinatorics. For the regular grid case results are not restricted to isotropic dilation matrices. The nature of the results is that intrinsic subdivision rules which operate on geometric data inherit smoothness properties of their linear counterparts
Single basepoint subdivision schemes for manifold-valued data: Timesymmetry without space-symmetry.
Abstract. This paper establishes smoothness results for a class of nonlinear subdivision schemes, known as the single basepoint manifold-valued subdivision schemes, which shows up in the construction of waveletlike transform for manifold-valued data. This class includes the (single basepoint) Log-Exp subdivision scheme as a special case. In these schemes, the exponential map is replaced by a so-called retraction map f from the tangent bundle of a manifold to the manifold. It is known that any choice of retraction map yields a C 2 scheme, provided the underlying linear scheme is C 2 (this is called "C 2 equivalence"). But when the underlying linear scheme is C 3 , Navayazdani and Yu have shown that to guarantee C 3 equivalence, a certain tensor P f associated to f must vanish. They also show that P f vanishes when the underlying manifold is a symmetric space and f is the exponential map. In the present paper, a geometric interpretation of the tensor P f is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponential map. The condition P f = 0 is shown to be equivalent to the condition that f agrees with the exponential map of the connection up to the 3rd order. In particular, when f is the exponential map of a connection, one recovers the original connection and P f vanishes. It then follows that the condition P f = 0 is satisfied by a wider class of manifolds than was previously known. Under the additional assumption that the subdivision rule satisfies a time-symmetry, it is shown that the vanishing of P f also guarantees C 4 equivalence. Finally, the analysis in the paper strongly indicates that vanishing curvature of the connection associated to f is a necessary condition for C k equivalence for k ≥ 5
Point-Normal Subdivision Curves and Surfaces
This paper proposes to generalize linear subdivision schemes to nonlinear
subdivision schemes for curve and surface modeling by refining vertex positions
together with refinement of unit control normals at the vertices. For each
round of subdivision, new control normals are obtained by projections of
linearly subdivided normals onto unit circle or sphere while new vertex
positions are obtained by updating linearly subdivided vertices along the
directions of the newly subdivided normals. Particularly, the new position of
each linearly subdivided vertex is computed by weighted averages of end points
of circular or helical arcs that interpolate the positions and normals at the
old vertices at one ends and the newly subdivided normal at the other ends.
The main features of the proposed subdivision schemes are three folds:
(1) The point-normal (PN) subdivision schemes can reproduce circles, circular
cylinders and spheres using control points and control normals;
(2) PN subdivision schemes generalized from convergent linear subdivision
schemes converge and can have the same smoothness orders as the linear schemes;
(3) PN subdivision schemes generalizing linear subdivision schemes that
generate subdivision surfaces with flat extraordinary points can generate
visually subdivision surfaces with non-flat extraordinary points.
Experimental examples have been given to show the effectiveness of the
proposed techniques for curve and surface modeling.Comment: 30 pages, 17 figures, 22.5M
Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection
By utilizing diffusion maps embedding and transition matrix analysis we
investigate sparse temperature measurement time-series data from
Rayleigh--B\'enard convection experiments in a cylindrical container of aspect
ratio between its diameter () and height (). We consider
the two cases of a cylinder at rest and rotating around its cylinder axis. We
find that the relative amplitude of the large-scale circulation (LSC) and its
orientation inside the container at different points in time are associated to
prominent geometric features in the embedding space spanned by the two dominant
diffusion-maps eigenvectors. From this two-dimensional embedding we can measure
azimuthal drift and diffusion rates, as well as coherence times of the LSC. In
addition, we can distinguish from the data clearly the single roll state (SRS),
when a single roll extends through the whole cell, from the double roll state
(DRS), when two counter-rotating rolls are on top of each other. Based on this
embedding we also build a transition matrix (a discrete transfer operator),
whose eigenvectors and eigenvalues reveal typical time scales for the stability
of the SRS and DRS as well as for the azimuthal drift velocity of the flow
structures inside the cylinder. Thus, the combination of nonlinear dimension
reduction and dynamical systems tools enables to gain insight into turbulent
flows without relying on model assumptions
Numerical Methods in Shape Spaces and Optimal Branching Patterns
The contribution of this thesis is twofold. The main part deals with numerical methods in the context of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a local approximation of the squared Riemannian distance on the manifold. On physical shape spaces this approximation can be derived e.g. from a dissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell model the time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered as Riemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the optimization of the spline functional—subject to appropriate constraints—can be used to solve the multiple interpolation problem in shape space, e.g. to realize keyframe animation. Based on the spline functional, we further develop a simple regression model which generalizes linear regression to nonlinear shape spaces. Numerical examples based on real data from anatomy and botany show the capability of the model. Finally, we apply the statistical analysis of elastic shape spaces presented by Rumpf and Wirth [RW09, RW11] to the space of discrete shells. To this end, we compute a Fréchet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the metric induced by the Hessian of an elastic shell energy. The last part of this thesis deals with the optimization of microstructures arising e.g. at austenite-martensite interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and Müller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two different martensite phases. We perform a finite element simulation based on subdivision surfaces that suggests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and show—numerically and analytically—that our new branching pattern results in a significantly better upper energy bound
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