Regression Analysis Of Zernike Polynomials

Louis D. Grey

doi:10.1117/12.978912

1 January 1987 Regression Analysis Of Zernike Polynomials

Louis D. Grey

Proceedings Volume 0818, Current Developments in Optical Engineering II; (1987) https://doi.org/10.1117/12.978912
Event: 31st Annual Technical Symposium on Optical and Optoelectronic Applied Sciences and Engineering, 1987, San Diego, CA, United States

Abstract

In evaluating the performance of a mirror by interferometric methods, a standard procedure used by the optical engineer is to obtain the optical path difference (OPD) map from the fringes of the interferogram and to express the differences in terms of a least squares fit of the classical Zernike polynomials. Since the Zernike polynomials represent the aberrations inherent in the optics, the engineer can determine whether or not the optics will meet performance requirements and if not, what measures need to be taken to reduce the aberrations. This paper examines the least squares fitting process from a statistical point of view and points out the necessary steps and precautions which must be taken to ensure that the analysis rests on a firm foundation so that the conclusions drawn by the optical engineer are warranted. The fitting process is examined from the standpoint of statistical linear regression theory and the following concepts and ideas are introduced: 1) How to determine whether the fit is a good fit and what to do about outliers. 2) Whether the underlying assumptions of normality and constant variance have been met, and, if not, what should be done. 3) How to compute confidence intervals 4) What do the correlation and covariance matrices tell us. 5) How to look at the information contained in the residuals and how to detect influential observations. Regression analysis has proved to be a powerful tool in other fields and there is every reason to believe it can make a significant contribution to optical engineering.

Citation Download Citation

Louis D. Grey "Regression Analysis Of Zernike Polynomials", Proc. SPIE 0818, Current Developments in Optical Engineering II, (1 January 1987); https://doi.org/10.1117/12.978912

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KEYWORDS

Zernike polynomials

Statistical analysis

Electro-optical engineering

Error analysis

Optical engineering

Antimony

Distance measurement

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