4,881 research outputs found
Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization
A versatile method is described for the practical computation of the discrete
Fourier transforms (DFT) of a continuous function given by its values
at the points of a uniform grid generated by conjugacy classes
of elements of finite adjoint order in the fundamental region of
compact semisimple Lie groups. The present implementation of the method is for
the groups SU(2), when is reduced to a one-dimensional segment, and for
in multidimensional cases. This simplest case
turns out to result in a transform known as discrete cosine transform (DCT),
which is often considered to be simply a specific type of the standard DFT.
Here we show that the DCT is very different from the standard DFT when the
properties of the continuous extensions of these two discrete transforms from
the discrete grid points to all points are
considered. (A) Unlike the continuous extension of the DFT, the continuous
extension of (the inverse) DCT, called CEDCT, closely approximates
between the grid points . (B) For increasing , the derivative of CEDCT
converges to the derivative of . And (C), for CEDCT the principle of
locality is valid. Finally, we use the continuous extension of 2-dimensional
DCT to illustrate its potential for interpolation, as well as for the data
compression of 2D images.Comment: submitted to JMP on April 3, 2003; still waiting for the referee's
Repor
(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms
Four families of special functions, depending on n variables, are studied. We
call them symmetric and antisymmetric multivariate sine and cosine functions.
They are given as determinants or antideterminants of matrices, whose matrix
elements are sine or cosine functions of one variable each. These functions are
eigenfunctions of the Laplace operator, satisfying specific conditions at the
boundary of a certain domain F of the n-dimensional Euclidean space. Discrete
and continuous orthogonality on F of the functions within each family, allows
one to introduce symmetrized and antisymmetrized multivariate Fourier-like
transforms, involving the symmetric and antisymmetric multivariate sine and
cosine functions.Comment: 25 pages, no figures; LaTaX; corrected typo
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
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