Open AccessOrthonormal curvature polynomials over a unit circle: basis set derived from curvatures of Zernike polynomials
Author(s) -
Chunyu Zhao,
James H. Burge
Publication year - 2013
Publication title -
optics express
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.394
H-Index - 271
ISSN - 1094-4087
DOI - 10.1364/oe.21.031430
Subject(s) - zernike polynomials , orthonormal basis , curvature , jacobi polynomials , classical orthogonal polynomials , mathematics , wavefront , basis function , mathematical analysis , surface (topology) , gegenbauer polynomials , orthogonal polynomials , discrete orthogonal polynomials , geometry , optics , physics , quantum mechanics
Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.