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GENERAL INFORMATION
chapter 5, Ocular Wavefront Conversion
Table of Contents
- 1. Introduction
Chapter Contents
- 5.1 General Discussion of Wavefront Conversion
- 5.1.1 Completeness of Basis Functions
- 5.1.2 Conversions from the Coefficients of a Complete Set of Basis Functions to the Coefficients of Orthonormal Basis Functions
- 5.1.3 Conversions between the Coefficients of a Complete Set of Basis Functions and Fourier Coefficients
- 5.2 Conversions of Zernike Polynomials and Seidel Series
- 5.2.1 Seidel Coefficients to Zernike Coefficients
- 5.2.2 Zernike Coefficients to Seidel Coefficients
- 5.3 Conversions of Zernike Polynomials and Fourier Series
- 5.3.1 Zernike Coefficients to Fourier Coefficients
- 5.3.2 Fourier Coefficients to Zernike Coefficients
- 5.4 Conversions of Taylor Monomials and Zernike Polynomials
- 5.4.1 Taylor Coefficients to Zernike Coefficients
- 5.4.2 Zernike Coefficients to Taylor Coefficients
- 5.5 Conversions of Fourier Series and Taylor Monomials
- 5.5.1 Taylor Coefficients to Fourier Coefficients
- 5.5.2 Fourier Coefficients to Taylor Coefficients
- Appendix 5.A Derivation of Eq. (5.3)
- Appendix 5.B Derivation of Eqs. (5.6) and (5.7)
- Appendix 5.C Derivation of Conversion Matrices Cs2z and Cz2s
- Appendix 5.D Proof of Eq. (5.15)
- Appendix 5.E Derivation of Conversion Matrices Ct2z and Cz2t
- Appendix 5.F Matlab Code for Conversions of Zernike and Taylor
- Appendix 5.G Derivation of Qpq(k,ϕ)
- Bibliography
Excerpt
After an ocular wavefront is obtained from an aberrometer with wavefront sensing and reconstruction, the wavefront often needs to be manipulated. In the previous chapter, we discussed the modal wavefront reconstruction with different basis functions, such as Zernike polynomials, Fourier series, and Taylor monomials. When different sets of basis functions are used in the modal wavefront reconstruction, we must have a way to compare these wavefronts. Therefore, there is a need to convert the coefficients between two different sets of basis functions. In this chapter, we discuss the conversions of coefficients of the following basis functions: Seidel series, Zernike polynomials, Fourier series, and Taylor monomials.
Among these four sets of basis functions, the set of Zernike polynomials and the set of Fourier series are orthonormal. Zernike polynomials are orthonormal over circular pupils, and Fourier series are orthonormal over rectangular pupils. Because they are orthonormal, they are also complete. The set of Taylor monomials is not orthonormal over any pupils, however, they are a complete set of basis functions. Any complete set of basis functions can be used to accurately represent any well-behaved functions, such as an ocular wavefront. The set of Seidel series, which is an extension of the classical aberrations, is not orthonormal over any pupils. Nor is it a complete set of basis functions. The inclusion of the set of Seidel series is mainly for the relation to classical aberrations. Because Zernike polynomials are also related to classical aberrations, we discuss the conversion between Zernike polynomials and Seidel series. The conversion of Seidel series with the other sets of basis functions is not discussed. For the other three complete sets of basis functions, namely Zernike polynomials, Fourier series, and Taylor monomials, the conversions between any two of them are dicussed.
5.1 General Discussion of Wavefront Conversion
For ocular wavefront representation, we expect an accurate account when an analytical approach is used. Therefore, for ocular wavefront conversions, there should be no error when an ocular wavefront is converted from one representation to another. From a mathematical viewpoint, only a complete set of basis functions can meet that requirement.
©2008 Society of Photo-Optical Instrumentation Engineers
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