Browse Prior Art Database

Least Squares Synthesis of Wavelet Basis Functions

IP.com Disclosure Number: IPCOM000119342D
Original Publication Date: 1991-Jan-01
Included in the Prior Art Database: 2005-Apr-01
Document File: 2 page(s) / 39K

Publishing Venue

IBM

Related People

Feig, E: AUTHOR [+2]

Abstract

Wavelets have recently become very popular tools in signal analysis and processing. Presented here is a method for designing wavelets which best match some predetermined wavelet transform in the least squares sense.

This text was extracted from an ASCII text file.
This is the abbreviated version, containing approximately 100% of the total text.

Least Squares Synthesis of Wavelet Basis Functions

      Wavelets have recently become very popular tools in
signal analysis and processing.  Presented here is a method for
designing wavelets which best match some predetermined wavelet
transform in the least squares sense.

      For a given function g, the wavelet transform is an operator
acting on functions f by the formula

                            (Image Omitted)

 where             .  Define
similarly,             . Setting, Wg(f)(u,v), this article presents a
solution to the problem of approximating a given kernel K on
by W(f,g) in          relative to Lebeque measure; that is, minimize

      The solution is constructive, yielding an algorithm for
designing wavelets which are optimal in the sense described above.
Proof of the assertion below can be found in [*]

      For K(x,y) a function defined on        such that the integral
operators T[+] given by are compact operators mapping
into L2 .  The minimum of equation (2) over

      Reference
(*)  E. Feig and C. A. Micchelli, "L2-Synthesis by Ambiguity
Functions, Multivariate approximation theory IV," Proc . Conf .
Oberwolfach/FRG ISNM 90, 143-156 (1989).