orthogonal functions and Zernike polynomials---a random variable interpretation

Abstract

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 $l$, $\{Z^l_{l+2n}(x), x\geq 0\}$. The first is using the standard basis $\{x^n, n\geq 0\}$ and the random variable $Y^{1/(l+1)}$. The second is using the non-standard basis $\{x^{l+2n}, n\geq0\}$ and the random variable $Y^{1/2}$. 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. doi:10.1017/S1446181109000169