Transformation of Zernike coefficients: a Fourier-based method for scaled, translated, and rotated wavefront apertures.

  title={Transformation of Zernike coefficients: a Fourier-based method for scaled, translated, and rotated wavefront apertures.},
  author={Eric Tatulli},
  journal={Journal of the Optical Society of America. A, Optics, image science, and vision},
  volume={30 4},
  • E. Tatulli
  • Published 28 February 2013
  • Mathematics
  • Journal of the Optical Society of America. A, Optics, image science, and vision
This paper studies the effects on Zernike coefficients of aperture scaling, translation, and rotation, when a given aberrated wavefront is described on the Zernike polynomial basis. It proposes an analytical method for computing the matrix that enables the building of transformed Zernike coefficients from the original ones. The technique is based on the properties of Zernike polynomials and Fourier transform, and, in the case of a full aperture without central obstruction, the coefficients of… 
9 Citations

Figures from this paper

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.
Hyperbolic Transformations of Zernike Functions and Coefficients
The mathematical background of the recent approach to correct certain types of corneal shape measurement errors is presented, which is based on constructions of systems of orthonormal functions based on the well-known Zernike polynomials.
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.
Correlation between the phase and the log-amplitude of a wave through vertical atmospheric propagation.
  • G. Molodij
  • Physics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2015
Expressions of the correlation between the log-amplitude and the phase of a wavefront propagating through atmospheric turbulence are presented to evaluate the feasibility of proposed methods to increase the confidence level of the detection of faint transient astronomical objects.
Synthetic aperture wavefront sensing
The numerical results indicate that the synthetic aperture wavefront sensing is affected by less bias and noise propagation than the averaging method, providing smaller mean-squared estimation error.
Laser tomography adaptive optics: a performance study.
We present an analytical derivation of the on-axis performance of adaptive optics systems using a given number of guide stars of arbitrary altitude, distributed at arbitrary angular positions in the
Zernike power spectra of clear and cloudy light-polluted urban night skies
The Zernike power spectra of the all-sky night brightness distributions of clear and cloudy nights are computed using a modal projection approach. The results obtained in the B, V, and R
The usability of the optical parametric amplification of light for high-angular-resolution imaging and fast astrometry
High-angular-resolution imaging is crucial for many applications in modern astronomy and astrophysics. The fundamental diffraction limit constrains the resolving power of both ground-based and
Computer Aided Systems Theory – EUROCAST 2019: 17th International Conference, Las Palmas de Gran Canaria, Spain, February 17–22, 2019, Revised Selected Papers, Part II
This paper proposes an algorithm based on the generalized expectation maximization (GEM) method to obtain estimators and shows the performance of the proposed algorithm using both synthetic data and real financial data.


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.
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
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.
Concise formula for the Zernike coefficients of scaled pupils
Modern steppers and scanners have a projection lens whose numerical aperture (NA) can be varied so as to optimize the image performance for certain lithographic features. Thus a variable fraction of
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.
Wavefront propagation in turbulence: an unified approach to the derivation of angular correlation functions.
  • G. Molodij
  • Physics, Mathematics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2011
The formalism is able to give the coefficients of phase-expansion functions orthogonal over an arbitrary circularly symmetric weighting function for an isotropic turbulence spectrum, as well as to describe the effect of the finite outer and inner scales of the turbulence.
Statistics of a Geometric Representation of Wavefront Distortion: Errata
A precise statistical definition is established for the geometric “shape” of a randomly distorted wavefront. Relationships between the phase-structure function and the statistics governing the shape
Statistical variation of aberration structure and image quality in a normal population of healthy eyes.
It is inferred that subjective best focus occurs when the area of the central, aberration-free region of the pupil is maximized, and that correction of the 12 largest principal components, or 14 largest Zernike modes, would be required to achieve diffraction-limited performance on average for a 6-mm pupil.
Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations
Mellin transform methods have been used to obtain Taylor series or asymptotic approximations to definite integrals whose integrands consist of the product of two generalized hypergeometric functions.