# Zernike coefficients for concentric, circular scaled pupils: an equivalent expression

@article{Diaz2009ZernikeCF, title={Zernike coefficients for concentric, circular scaled pupils: an equivalent expression}, author={Jose A. Diaz and Jos{\'e} Fern{\'a}ndez-Dorado and Carles Pizarro and Josep Arasa}, journal={Journal of Modern Optics}, year={2009}, volume={56}, pages={131 - 137} }

We present an alternative formal calculation of the scaled Zernike coefficient expansion by means of the inner product of the Zernike polynomials and the wavefront error corresponding to the scaled pupil. The relationship exhibited by the radial polynomials and Bessel functions leads to a general expression in terms of the Gauss hypergeometric function. Direct properties and index selection rules are established, and easy derivation of the non-normalized coefficients is also straightforward.

## 19 Citations

Zernike coefficients of a scaled pupil.

- EducationApplied optics
- 2010

A simple derivation is given for determining the Zernike coefficients of an aberration function of a scaled pupil in terms of their values for a corresponding unscaled pupil.

Unified analytical method for Zernike coefficient transformation of scaled, rotated, and translated pupils based on Shack's vector multiplication.

- PhysicsJournal of the Optical Society of America. A, Optics, image science, and vision
- 2021

A novel method based on Shack's vector multiplication is first proposed to derive the transformation relation of Zernike polynomials, providing a generalized methodology to analyze the relationship between weighted coefficients for different description basis sets.

Estimating the eye aberration coefficients in resized pupils: is it better to refit or to rescale?

- EducationJournal of the Optical Society of America. A, Optics, image science, and vision
- 2014

The results for normal eye populations show that in case of reducing the pupil size it is better to rescale the original coefficients than to refit them using the measurements contained within the smaller pupil.

Corneal aberrations in keratoconic eyes: influence of pupil size and centering

- Physics
- 2011

Ocular aberrations vary among subjects and under different conditions and are commonly analyzed expanding the wavefront aberration function in Zernike polynomials. In previous articles, explicit…

A formula for the mean human optical modulation transfer function as a function of pupil size.

- PhysicsJournal of vision
- 2013

We have constructed an analytic formula for the mean radial modulation transfer function of the best-corrected human eye as a function of pupil diameter, based on previously collected wave front…

Validity of scaling zernike coefficients to a larger diameter for refractive surgery.

- Physics, MedicineJournal of refractive surgery
- 2011

Rcaling Zernike coefficients from a smaller diameter to a larger one has practical applications in optical zone extension for wavefront-guided refractive surgery and no significant difference was found for the variability for different pupil sizes.

Pupil Scaling for the Estimation of Aberrations in Natural Pupils

- Education, MedicineOptometry and vision science : official publication of the American Academy of Optometry
- 2014

Estimation of ocular wavefront aberration coefficients either scaling down from large to smaller pupils or scaling up from smaller to large pupils provides estimates that are not significantly different from clinically measured values, however, when scaling up to a larger pupil size, the estimates are more variable.

A Theoretical Comparison among Recursive Algorithms for Fast Computation of Zernike Moments Using the Concept of Time Complexity

- Computer ScienceAmerican Journal of Computational Mathematics
- 2021

The purpose of this research is to study several methods among the most popular recursive methods for fast Zernike computation and compare them together by a global theoretical evaluation system called worst-case time complexity.

Validation of Mahajan’s formula for scaling ocular higher-order aberrations by pupil size

- MedicineIndian journal of ophthalmology
- 2020

Pupil scaling enables accurate comparison of individual higher order aberrations in clinical research for situations involving different pupil sizes, and is validated as accurate.

New strategy for misalignment calculation in optical systems using artificial neural networks

- Physics
- 2013

The method uses the wavefront information in the exit pupil in the form of Zernike coefficients and a function that relates them to the misalignment values to solve the problem of element misalignments in optical systems.

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