3,090 research outputs found
On The Continuous Steering of the Scale of Tight Wavelet Frames
In analogy with steerable wavelets, we present a general construction of
adaptable tight wavelet frames, with an emphasis on scaling operations. In
particular, the derived wavelets can be "dilated" by a procedure comparable to
the operation of steering steerable wavelets. The fundamental aspects of the
construction are the same: an admissible collection of Fourier multipliers is
used to extend a tight wavelet frame, and the "scale" of the wavelets is
adapted by scaling the multipliers. As an application, the proposed wavelets
can be used to improve the frequency localization. Importantly, the localized
frequency bands specified by this construction can be scaled efficiently using
matrix multiplication
Exact reconstruction with directional wavelets on the sphere
A new formalism is derived for the analysis and exact reconstruction of
band-limited signals on the sphere with directional wavelets. It represents an
evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999)
and Wiaux et al. (2005). The translations of the wavelets at any point on the
sphere and their proper rotations are still defined through the continuous
three-dimensional rotations. The dilations of the wavelets are directly defined
in harmonic space through a new kernel dilation, which is a modification of an
existing harmonic dilation. A family of factorized steerable functions with
compact harmonic support which are suitable for this kernel dilation is firstly
identified. A scale discretized wavelet formalism is then derived, relying on
this dilation. The discrete nature of the analysis scales allows the exact
reconstruction of band-limited signals. A corresponding exact multi-resolution
algorithm is finally described and an implementation is tested. The formalism
is of interest notably for the denoising or the deconvolution of signals on the
sphere with a sparse expansion in wavelets. In astrophysics, it finds a
particular application for the identification of localized directional features
in the cosmic microwave background (CMB) data, such as the imprint of
topological defects, in particular cosmic strings, and for their reconstruction
after separation from the other signal components.Comment: 22 pages, 2 figures. Version 2 matches version accepted for
publication in MNRAS. Version 3 (identical to version 2) posted for code
release announcement - "Steerable scale discretised wavelets on the sphere" -
S2DW code available for download at
http://www.mrao.cam.ac.uk/~jdm57/software.htm
Elliptic scaling functions as compactly supported multivariate analogs of the B-splines
In the paper, we present a family of multivariate compactly supported scaling
functions, which we call as elliptic scaling functions. The elliptic scaling
functions are the convolution of elliptic splines, which correspond to
homogeneous elliptic differential operators, with distributions. The elliptic
scaling functions satisfy refinement relations with real isotropic dilation
matrices. The elliptic scaling functions satisfy most of the properties of the
univariate cardinal B-splines: compact support, refinement relation, partition
of unity, total positivity, order of approximation, convolution relation, Riesz
basis formation (under a restriction on the mask), etc. The algebraic
polynomials contained in the span of integer shifts of any elliptic scaling
function belong to the null-space of a homogeneous elliptic differential
operator. Similarly to the properties of the B-splines under differentiation,
it is possible to define elliptic (not necessarily differential) operators such
that the elliptic scaling functions satisfy relations with these operators. In
particular, the elliptic scaling functions can be considered as a composition
of segments, where the function inside a segment, like a polynomial in the case
of the B-splines, vanishes under the action of the introduced operator.Comment: To appear in IJWMI
Complex data processing: fast wavelet analysis on the sphere
In the general context of complex data processing, this paper reviews a
recent practical approach to the continuous wavelet formalism on the sphere.
This formalism notably yields a correspondence principle which relates wavelets
on the plane and on the sphere. Two fast algorithms are also presented for the
analysis of signals on the sphere with steerable wavelets.Comment: 20 pages, 5 figures, JFAA style, paper invited to J. Fourier Anal.
and Appli
Identification of time-varying systems using multiresolution wavelet models
Identification of linear and nonlinear time-varying systems is investigated and a new wavelet model identification algorithm is introduced. By expanding each time-varying coefficient using a multiresolution wavelet expansion, the time-varying problem is reduced to a time invariant problem and the identification reduces to regressor selection and parameter estimation. Several examples are included to illustrate the application of the new algorithm
Scale Invariant Interest Points with Shearlets
Shearlets are a relatively new directional multi-scale framework for signal
analysis, which have been shown effective to enhance signal discontinuities
such as edges and corners at multiple scales. In this work we address the
problem of detecting and describing blob-like features in the shearlets
framework. We derive a measure which is very effective for blob detection and
closely related to the Laplacian of Gaussian. We demonstrate the measure
satisfies the perfect scale invariance property in the continuous case. In the
discrete setting, we derive algorithms for blob detection and keypoint
description. Finally, we provide qualitative justifications of our findings as
well as a quantitative evaluation on benchmark data. We also report an
experimental evidence that our method is very suitable to deal with compressed
and noisy images, thanks to the sparsity property of shearlets
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