2,784 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
Signal Flow Graph Approach to Efficient DST I-IV Algorithms
In this paper, fast and efficient discrete sine transformation (DST)
algorithms are presented based on the factorization of sparse, scaled
orthogonal, rotation, rotation-reflection, and butterfly matrices. These
algorithms are completely recursive and solely based on DST I-IV. The presented
algorithms have low arithmetic cost compared to the known fast DST algorithms.
Furthermore, the language of signal flow graph representation of digital
structures is used to describe these efficient and recursive DST algorithms
having points signal flow graph for DST-I and points signal flow
graphs for DST II-IV
Fourier Policy Gradients
We propose a new way of deriving policy gradient updates for reinforcement
learning. Our technique, based on Fourier analysis, recasts integrals that
arise with expected policy gradients as convolutions and turns them into
multiplications. The obtained analytical solutions allow us to capture the low
variance benefits of EPG in a broad range of settings. For the critic, we treat
trigonometric and radial basis functions, two function families with the
universal approximation property. The choice of policy can be almost arbitrary,
including mixtures or hybrid continuous-discrete probability distributions.
Moreover, we derive a general family of sample-based estimators for stochastic
policy gradients, which unifies existing results on sample-based approximation.
We believe that this technique has the potential to shape the next generation
of policy gradient approaches, powered by analytical results
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs
This paper presents a systematic methodology based on the algebraic theory of
signal processing to classify and derive fast algorithms for linear transforms.
Instead of manipulating the entries of transform matrices, our approach derives
the algorithms by stepwise decomposition of the associated signal models, or
polynomial algebras. This decomposition is based on two generic methods or
algebraic principles that generalize the well-known Cooley-Tukey FFT and make
the algorithms' derivations concise and transparent. Application to the 16
discrete cosine and sine transforms yields a large class of fast algorithms,
many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar
Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions
Properties of the four families of recently introduced special functions of
two real variables, denoted here by , and , are studied. The
superscripts and refer to the symmetric and antisymmetric functions
respectively. The functions are considered in all details required for their
exploitation in Fourier expansions of digital data, sampled on square grids of
any density and for general position of the grid in the real plane relative to
the lattice defined by the underlying group theory. Quality of continuous
interpolation, resulting from the discrete expansions, is studied, exemplified
and compared for some model functions.Comment: 22 pages, 10 figure
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