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

  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},
  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. 
Zernike coefficients of a scaled pupil.
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.
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?
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
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.
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.
  • G. Dai
  • Physics, Medicine
    Journal 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
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.
Validation of Mahajan’s formula for scaling ocular higher-order aberrations by pupil size
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
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.
Analysis of the robustness of the lens GRIN profile in a schematic model eye
In this work, an improvement of a previously-published human eye model with aging is presented. The previous eye model showed the overestimated performance of an averaged MTF at low spatial


Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula.
  • G. Dai
  • Physics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2006
A more intuitive derivation of a simpler, nonrecursive formula, which is used to calculate the instantaneous refractive power, is described.
Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils.
  • L. Lundström, P. Unsbo
  • Physics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2007
This work presents the first complete theory to transform Zernike coefficients analytically with regard to concentric scaling, translation of pupil center, and rotation for circular and elliptical pupils.
Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils.
This work describes an equivalent algebraic approach that allows for the conversion of aberration coefficients in a single step and in a straightforward way to handle a wide range of pupil transformations, including, but not restricted to, anisotropic scalings.
Orthonormal polynomials in wavefront analysis: analytical solution
Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent
Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations
An analytical method to convert the set of Zernike coefficients that fits the wavefront aberration for a pupil into another corresponding to a contracted and horizontally translated pupil is
Scaling Zernike expansion coefficients to different pupil sizes.
  • J. Schwiegerling
  • Education
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2002
This investigation derives relationships between the Zernike expansion coefficients for two different pupil sizes by using a technique for converting Zernke expansion coefficients from one pupil size to another.
General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes.
A general method is proposed for establishing the relationship between two sets of Zernike coefficients computed with different aperture sizes and for developing a technique for converting the Zernik coefficients obtained from one aperture size to another size.
Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed.
  • C. Campbell
  • Physics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2003
A matrix method is developed that allows a new set of Zernike coefficients that describe a surface or wave front appropriate for a new aperture size to be found from an original set of Zernike
Wavefront Optics for Vision Correction
This book addresses some of the issues in visual optics with a functional analysis of ocular aberrations, especially for the purpose of vision correction. The basis is the analytical representation