Rational Zernike Functions Capture the Rotations of the Eye-Ball

  • Zoltán FazekasEmail author
  • Levente Lócsi
  • Alexandros Soumelidis
  • Ferenc Schipp
  • Zsolt Németh
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 30)


Measurement and mathematical description of the corneal surface and of the optical properties of the human eye are actively researched topics. To enhance the mathematical tools used in the field, a novel set of orthogonal functions—called rational Zernike functions—are presented in the paper; these functions are of great promise for correcting certain types of measurement errors that adversely affect the quality of corneal maps. Such errors arise e.g., due to unintended eye-movements, or spontaneous rotations of the eye-ball. The rational Zernike functions can be derived from the well-known Zernike polynomials—the latter polynomials are used widely in eye-related measurements and ophthalmology—via an argument transformation with a Blaschke function. This transformation is a congruent transformation in the Poincaré disk model of the Bolyai-Lobachevsky hyperbolic geometry.



This research was supported in the frame of contract EFOP-3.6.3-VEKOP-16-2017-00001: Talent Management in Autonomous Vehicle Control Technologies. The Project is supported by the Hungarian Government and co-financed by the European Social Fund. This research was also supported by Research Funds (OTKA) in the frame of the research contract No. K115804.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Zoltán Fazekas
    • 1
    Email author
  • Levente Lócsi
    • 2
  • Alexandros Soumelidis
    • 1
  • Ferenc Schipp
    • 2
  • Zsolt Németh
    • 2
  1. 1.Institute for Computer Science and Control (MTA SZTAKI)BudapestHungary
  2. 2.Department of Numerical Analysis, Faculty of InformaticsELTE Eötvös Loránd UniversityBudapestHungary

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