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ORTHOGONAL FUNCTIONS AND ZERNIKE POLYNOMIALS—A RANDOM VARIABLE INTERPRETATION

Part of: Harmonic analysis in one variable

Published online by Cambridge University Press:  03 November 2009

C. S. WITHERS
C. S. WITHERS*
Affiliation:
Applied Mathematics Group, Industrial Research Ltd, Box 31-310, Lower Hutt, New Zealand (email: c.withers@irl.cri.nz)
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Abstract

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There are advantages in viewing orthogonal functions as functions generated by a random variable from a basis set of functions. Let Y be a random variable distributed uniformly on [0,1]. We give two ways of generating the Zernike radial polynomials with parameter l, {Zll+2n(x), n≥0}. The first is using the standard basis {xn,n≥0} and the random variable Y1/(l+1). The second is using the nonstandard basis {xl+2n,n≥0} and the random variable Y1/2. Zernike polynomials are important in the removal of lens aberrations, in characterizing video images with a small number of numbers, and in automatic aircraft identification.

Keywords

Zernike polynomialsorthogonal functions

MSC classification

Secondary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 3: This Special Issue is dedicated to Dr Stephen White , January 2009 , pp. 435 - 444
Copyright
Copyright © Australian Mathematical Society 2009

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