9,343 research outputs found
ShearLab 3D: Faithful Digital Shearlet Transforms based on Compactly Supported Shearlets
Wavelets and their associated transforms are highly efficient when
approximating and analyzing one-dimensional signals. However, multivariate
signals such as images or videos typically exhibit curvilinear singularities,
which wavelets are provably deficient of sparsely approximating and also of
analyzing in the sense of, for instance, detecting their direction. Shearlets
are a directional representation system extending the wavelet framework, which
overcomes those deficiencies. Similar to wavelets, shearlets allow a faithful
implementation and fast associated transforms. In this paper, we will introduce
a comprehensive carefully documented software package coined ShearLab 3D
(www.ShearLab.org) and discuss its algorithmic details. This package provides
MATLAB code for a novel faithful algorithmic realization of the 2D and 3D
shearlet transform (and their inverses) associated with compactly supported
universal shearlet systems incorporating the option of using CUDA. We will
present extensive numerical experiments in 2D and 3D concerning denoising,
inpainting, and feature extraction, comparing the performance of ShearLab 3D
with similar transform-based algorithms such as curvelets, contourlets, or
surfacelets. In the spirit of reproducible reseaerch, all scripts are
accessible on www.ShearLab.org.Comment: There is another shearlet software package
(http://www.mathematik.uni-kl.de/imagepro/members/haeuser/ffst/) by S.
H\"auser and G. Steidl. We will include this in a revisio
Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts
We compute minimal bases of solutions for a general interpolation problem,
which encompasses Hermite-Pad\'e approximation and constrained multivariate
interpolation, and has applications in coding theory and security.
This problem asks to find univariate polynomial relations between vectors
of size ; these relations should have small degree with respect to an
input degree shift. For an arbitrary shift, we propose an algorithm for the
computation of an interpolation basis in shifted Popov normal form with a cost
of field operations, where
is the exponent of matrix multiplication and the notation
indicates that logarithmic terms are omitted.
Earlier works, in the case of Hermite-Pad\'e approximation and in the general
interpolation case, compute non-normalized bases. Since for arbitrary shifts
such bases may have size , the cost bound
was feasible only with restrictive
assumptions on the shift that ensure small output sizes. The question of
handling arbitrary shifts with the same complexity bound was left open.
To obtain the target cost for any shift, we strengthen the properties of the
output bases, and of those obtained during the course of the algorithm: all the
bases are computed in shifted Popov form, whose size is always . Then, we design a divide-and-conquer scheme. We recursively reduce
the initial interpolation problem to sub-problems with more convenient shifts
by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms
Stable normal forms for polynomial system solving
This paper describes and analyzes a method for computing border bases of a
zero-dimensional ideal . The criterion used in the computation involves
specific commutation polynomials and leads to an algorithm and an
implementation extending the one provided in [MT'05]. This general border basis
algorithm weakens the monomial ordering requirement for \grob bases
computations. It is up to date the most general setting for representing
quotient algebras, embedding into a single formalism Gr\"obner bases, Macaulay
bases and new representation that do not fit into the previous categories. With
this formalism we show how the syzygies of the border basis are generated by
commutation relations. We also show that our construction of normal form is
stable under small perturbations of the ideal, if the number of solutions
remains constant. This new feature for a symbolic algorithm has a huge impact
on the practical efficiency as it is illustrated by the experiments on
classical benchmark polynomial systems, at the end of the paper
A trivariate interpolation algorithm using a cube-partition searching procedure
In this paper we propose a fast algorithm for trivariate interpolation, which
is based on the partition of unity method for constructing a global interpolant
by blending local radial basis function interpolants and using locally
supported weight functions. The partition of unity algorithm is efficiently
implemented and optimized by connecting the method with an effective
cube-partition searching procedure. More precisely, we construct a cube
structure, which partitions the domain and strictly depends on the size of its
subdomains, so that the new searching procedure and, accordingly, the resulting
algorithm enable us to efficiently deal with a large number of nodes.
Complexity analysis and numerical experiments show high efficiency and accuracy
of the proposed interpolation algorithm
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