460,069 research outputs found
Multi-dimensional real Fourier transform
Four subroutines compute one-dimensional and multi-dimensional Fourier transforms for real data, multi-dimensional complex Fourier transforms, and multi-dimensional sine, cosine and sine-cosine transforms. Subroutines use Cooley-Tukey fast Fourier transform. In all but one-dimensional case, transforms are calculated in up to six dimensions
Watermarking for multimedia security using complex wavelets
This paper investigates the application of complex wavelet transforms to the field of digital data hiding. Complex wavelets offer improved directional selectivity and shift invariance over their discretely sampled counterparts allowing for better adaptation of watermark distortions to the host media. Two methods of deriving visual models for the watermarking system are adapted to the complex wavelet transforms and their performances are compared. To produce improved capacity a spread transform embedding algorithm is devised, this combines the robustness of spread spectrum methods with the high capacity of quantization based methods. Using established information theoretic methods, limits of watermark capacity are derived that demonstrate the superiority of complex wavelets over discretely sampled wavelets. Finally results for the algorithm against commonly used attacks demonstrate its robustness and the improved performance offered by complex wavelet transforms
A Poisson transform adapted to the Rumin complex
Let be a semisimple Lie group with finite center, a maximal
compact subgroup, and a parabolic subgroup. Following ideas of
P.Y.\ Gaillard, one may use -invariant differential forms on
to construct -equivariant Poisson transforms mapping differential forms on
to differential forms on . Such invariant forms can be constructed
using finite dimensional representation theory. In this general setting, we
first prove that the transforms that always produce harmonic forms are exactly
those that descend from the de Rham complex on to the associated
Bernstein-Gelfand-Gelfand (or BGG) complex in a well defined sense.
The main part of the article is devoted to an explicit construction of such
transforms with additional favorable properties in the case that .
Thus is with its natural CR structure and the relevant BGG
complex is the Rumin complex, while is complex hyperbolic space of
complex dimension . The construction is carried out both for complex and
for real differential forms and the compatibility of the transforms with the
natural operators that are available on their sources and targets are analyzed
in detail.Comment: 36 pages, comments are welcome, v2: final version, to appear in J.
Topol. Ana
Complex and Hypercomplex Discrete Fourier Transforms Based on Matrix Exponential Form of Euler's Formula
We show that the discrete complex, and numerous hypercomplex, Fourier
transforms defined and used so far by a number of researchers can be unified
into a single framework based on a matrix exponential version of Euler's
formula , and a matrix root of -1
isomorphic to the imaginary root . The transforms thus defined can be
computed using standard matrix multiplications and additions with no
hypercomplex code, the complex or hypercomplex algebra being represented by the
form of the matrix root of -1, so that the matrix multiplications are
equivalent to multiplications in the appropriate algebra. We present examples
from the complex, quaternion and biquaternion algebras, and from Clifford
algebras Cl1,1 and Cl2,0. The significance of this result is both in the
theoretical unification, and also in the scope it affords for insight into the
structure of the various transforms, since the formulation is such a simple
generalization of the classic complex case. It also shows that hypercomplex
discrete Fourier transforms may be computed using standard matrix arithmetic
packages without the need for a hypercomplex library, which is of importance in
providing a reference implementation for verifying implementations based on
hypercomplex code.Comment: The paper has been revised since the second version to make some of
the reasons for the paper clearer, to include reviews of prior hypercomplex
transforms, and to clarify some points in the conclusion
Programs for high-speed Fourier, Mellin and Fourier-Bessel transforms
Several FORTRAN program modules for performing one-dimensional and two-dimensional discrete Fourier transforms, Mellin, and Fourier-Bessel transforms are described along with programs that realize the algebra of high speed Fourier transforms on a computer. The programs can perform numerical harmonic analysis of functions, synthesize complex optical filters on a computer, and model holographic image processing methods
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