165 research outputs found
Hierarchical Riesz bases for Hs(Omega), 1 < s < 5/2
On arbitrary polygonal domains , we construct hierarchical Riesz bases for Sobolev spaces . In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from to . Since the latter range includes , with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned
Algorithms and error bounds for multivariate piecewise constant approximation
We review the surprisingly rich theory of approximation of functions of many vari- ables by piecewise constants. This covers for example the Sobolev-Poincar´e inequalities, parts of the theory of nonlinear approximation, Haar wavelets and tree approximation, as well as recent results about approximation orders achievable on anisotropic partitions
Graph Wedgelets: Adaptive Data Compression on Graphs based on Binary Wedge Partitioning Trees and Geometric Wavelets
We introduce graph wedgelets - a tool for data compression on graphs based on
the representation of signals by piecewise constant functions on adaptively
generated binary graph partitionings. The adaptivity of the partitionings, a
key ingredient to obtain sparse representations of a graph signal, is realized
in terms of recursive wedge splits adapted to the signal. For this, we transfer
adaptive partitioning and compression techniques known for 2D images to general
graph structures and develop discrete variants of continuous wedgelets and
binary space partitionings. We prove that continuous results on best m-term
approximation with geometric wavelets can be transferred to the discrete graph
setting and show that our wedgelet representation of graph signals can be
encoded and implemented in a simple way. Finally, we illustrate that this
graph-based method can be applied for the compression of images as well.Comment: 12 pages, 10 figure
Adaptive multiresolution analysis based on anisotropic triangulations
A simple greedy refinement procedure for the generation of data-adapted
triangulations is proposed and studied. Given a function of two variables, the
algorithm produces a hierarchy of triangulations and piecewise polynomial
approximations on these triangulations. The refinement procedure consists in
bisecting a triangle T in a direction which is chosen so as to minimize the
local approximation error in some prescribed norm between the approximated
function and its piecewise polynomial approximation after T is bisected.
The hierarchical structure allows us to derive various approximation tools
such as multiresolution analysis, wavelet bases, adaptive triangulations based
either on greedy or optimal CART trees, as well as a simple encoding of the
corresponding triangulations. We give a general proof of convergence in the Lp
norm of all these approximations.
Numerical tests performed in the case of piecewise linear approximation of
functions with analytic expressions or of numerical images illustrate the fact
that the refinement procedure generates triangles with an optimal aspect ratio
(which is dictated by the local Hessian of of the approximated function in case
of C2 functions).Comment: 19 pages, 7 figure
A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity
The richness of natural images makes the quest for optimal representations in
image processing and computer vision challenging. The latter observation has
not prevented the design of image representations, which trade off between
efficiency and complexity, while achieving accurate rendering of smooth regions
as well as reproducing faithful contours and textures. The most recent ones,
proposed in the past decade, share an hybrid heritage highlighting the
multiscale and oriented nature of edges and patterns in images. This paper
presents a panorama of the aforementioned literature on decompositions in
multiscale, multi-orientation bases or dictionaries. They typically exhibit
redundancy to improve sparsity in the transformed domain and sometimes its
invariance with respect to simple geometric deformations (translation,
rotation). Oriented multiscale dictionaries extend traditional wavelet
processing and may offer rotation invariance. Highly redundant dictionaries
require specific algorithms to simplify the search for an efficient (sparse)
representation. We also discuss the extension of multiscale geometric
decompositions to non-Euclidean domains such as the sphere or arbitrary meshed
surfaces. The etymology of panorama suggests an overview, based on a choice of
partially overlapping "pictures". We hope that this paper will contribute to
the appreciation and apprehension of a stream of current research directions in
image understanding.Comment: 65 pages, 33 figures, 303 reference
A quadratic finite element wavelet Riesz basis
In this paper, continuous piecewise quadratic finite element wavelets are
constructed on general polygons in . The wavelets are stable in
for and have two vanishing moments. Each wavelet is a
linear combination of 11 or 13 nodal basis functions. Numerically computed
condition numbers for are provided for the unit square.Comment: 13 page
Multilevel Solvers for Unstructured Surface Meshes
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner
A Review of Bandlet Methods for Geometrical Image Representation
International audienceThis article reviews bandlet approaches to geometric image repre- sentations. Orthogonal bandlets using an adaptive segmentation and a local geometric flow well suited to capture the anisotropic regularity of edge struc- tures. They are constructed with a “bandletization” which is a local orthogonal transformation applied to wavelet coeffi cients. The approximation in these bandlet bases exhibits an asymptotically optimal decay for images that are regular outside a set of regular edges. These bandlets can be used to perform image compression and noise removal. More flexible orthogonal bandlets with less vanishing moments are constructed with orthogonal grouplets that group wavelet coeffi cients alon a multiscale association field. Applying a translation invariant grouplet transform over a translation invariant wavelet frame leads to state of the art results for image denoising and super-resolution
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