49,267 research outputs found
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
Tensor product approximations of high dimensional potentials
The paper is devoted to the efficient computation of high-order cubature
formulas for volume potentials obtained within the framework of approximate
approximations. We combine this approach with modern methods of structured
tensor product approximations. Instead of performing high-dimensional discrete
convolutions the cubature of the potentials can be reduced to a certain number
of one-dimensional convolutions leading to a considerable reduction of
computing resources. We propose one-dimensional integral representions of
high-order cubature formulas for n-dimensional harmonic and Yukawa potentials,
which allow low rank tensor product approximations.Comment: 20 page
Efficient computation of highly oscillatory integrals by using QTT tensor approximation
We propose a new method for the efficient approximation of a class of highly
oscillatory weighted integrals where the oscillatory function depends on the
frequency parameter , typically varying in a large interval. Our
approach is based, for fixed but arbitrary oscillator, on the pre-computation
and low-parametric approximation of certain -dependent prototype
functions whose evaluation leads in a straightforward way to recover the target
integral. The difficulty that arises is that these prototype functions consist
of oscillatory integrals and are itself oscillatory which makes them both
difficult to evaluate and to approximate. Here we use the quantized-tensor
train (QTT) approximation method for functional -vectors of logarithmic
complexity in in combination with a cross-approximation scheme for TT
tensors. This allows the accurate approximation and efficient storage of these
functions in the wide range of grid and frequency parameters. Numerical
examples illustrate the efficiency of the QTT-based numerical integration
scheme on various examples in one and several spatial dimensions.Comment: 20 page
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