@inproceedings{9771149b4ffa4e23b9aa47bd87836fb9,
title = "Fitting high-order Zernike polynomials to finite data",
abstract = "While the use of Zernike polynomials to represent simulated or measured data on a grid of points is common, the accuracy of the coefficients can be limited by the non-orthogonality of the functions over the pixelated domains. The Zernike polynomials are defined to be analytically orthogonal over a circular domain, but this breaks down for discrete data. A simple correction is presented that uses a weighted scalar product to determine coefficients. This method preserves the meaning of the Zernike polynomials and allows efficient calculations using an inner product. The algorithm for defining the weighting function is provided, and simulations are included that demonstrate nearly an order of magnitude improvement in accuracy when the new weighted scalar product is used.",
keywords = "Edge effects, Edge weighting, Finite data, Fitting, Gram-schmidt, Orthogonal, Weight mapping, Zernike polynomials",
author = "Benjamin Lewis and Burge, {James H.}",
year = "2012",
doi = "10.1117/12.930774",
language = "English (US)",
isbn = "9780819492104",
series = "Proceedings of SPIE - The International Society for Optical Engineering",
booktitle = "Interferometry XVI",
note = "Interferometry XVI: Techniques and Analysis ; Conference date: 13-08-2012 Through 15-08-2012",
}