473 research outputs found
A Construction of Biorthogonal Wavelets With a Compact Operator
We present a construction of biorthogonal wavelets using a compact operator
which allows to preserve or increase some properties: regularity/vanishing
moments, parity, compact supported. We build then a simple algorithm which
computes new filters
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
A new class of two-channel biorthogonal filter banks and wavelet bases
We propose a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters
M-channel cosine-modulated wavelet bases
The 13th International Conference on Digital Signal Processing, Santorini, Greece, 2-4 July 1997In this paper, we propose a new M-channel wavelet bases called the cosine-modulated wavelets. We first generalize the theory of two-channel biorthogonal compactly supported wavelet bases to the M-channel case. A sufficient condition for the M-channel perfect reconstruction filter banks to construct M-channel compactly supported wavelet bases is given. By using this condition, a family of orthogonal and biorthogonal M-channel cosine-modulated wavelet bases is constructed by iterations of cosine-modulated filter banks (CMFB). The advantages of the approach are their simple design procedure, efficient implementation and good filter quality. A method for imposing the regularity on the cosine-modulated filter banks is also introduced and design example is given.published_or_final_versio
Variational Approach in Wavelet Framework to Polynomial Approximations of Nonlinear Accelerator Problems
In this paper we present applications of methods from wavelet analysis to
polynomial approximations for a number of accelerator physics problems.
According to variational approach in the general case we have the solution as a
multiresolution (multiscales) expansion in the base of compactly supported
wavelet basis. We give extension of our results to the cases of periodic
orbital particle motion and arbitrary variable coefficients. Then we consider
more flexible variational method which is based on biorthogonal wavelet
approach. Also we consider different variational approach, which is applied to
each scale.Comment: LaTeX2e, aipproc.sty, 21 Page
Wavelets and their use
This review paper is intended to give a useful guide for those who want to
apply discrete wavelets in their practice. The notion of wavelets and their use
in practical computing and various applications are briefly described, but
rigorous proofs of mathematical statements are omitted, and the reader is just
referred to corresponding literature. The multiresolution analysis and fast
wavelet transform became a standard procedure for dealing with discrete
wavelets. The proper choice of a wavelet and use of nonstandard matrix
multiplication are often crucial for achievement of a goal. Analysis of various
functions with the help of wavelets allows to reveal fractal structures,
singularities etc. Wavelet transform of operator expressions helps solve some
equations. In practical applications one deals often with the discretized
functions, and the problem of stability of wavelet transform and corresponding
numerical algorithms becomes important. After discussing all these topics we
turn to practical applications of the wavelet machinery. They are so numerous
that we have to limit ourselves by some examples only. The authors would be
grateful for any comments which improve this review paper and move us closer to
the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh
M-Channel compactly supported biorthogonal cosine-modulated wavelet bases
In this correspondence, we generalize the theory of compactly supported biorthogonal two-channel wavelet bases to M -channel. A sufficient condition for the M-channel perfect reconstruction filter banks to construct M-channel biorthogonal bases of compactly supported wavelets is derived. It is shown that the construction of biorthogonal Af-channel wavelet bases is equivalent to the design of a Af-channel perfect reconstruction filter bank with some added regularity conditions. A family of M-channel biorthogonal wavelet bases based on the cosinemodulated filter bank (CMFB) is proposed. It has the advantages of simple design procedure, efficient implementation, and good filter quality. A new method for imposing the regularity on the CMFB's is also introduced, and several design examples are given. ©1998 IEEE.published_or_final_versio
- âŠ