ORTHOGONAL FUNCTIONS AND ZERNIKE POLYNOMIALS-A RANDOM VARIABLE INTERPRETATION

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 1, {Z(l+2n)(l)(x), n >= 0}. The first is using the standard basis {x(n), n >= 0} and the random variable Y(1/(l+1)). The second is using the nonstandard basis {x(l+2n), n >= 0} 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.