Fitting high-order Zernike polynomials to finite data

Benjamin Lewis, James H. Burge

Research output: Chapter in Book/Report/Conference proceedingConference contribution


While the use of Zernike polynomials to represent simulated or measured data on a grid of points is common, the accuracy of the coefficients can be limited by the non-orthogonality of the functions over the pixelated domains. The Zernike polynomials are defined to be analytically orthogonal over a circular domain, but this breaks down for discrete data. A simple correction is presented that uses a weighted scalar product to determine coefficients. This method preserves the meaning of the Zernike polynomials and allows efficient calculations using an inner product. The algorithm for defining the weighting function is provided, and simulations are included that demonstrate nearly an order of magnitude improvement in accuracy when the new weighted scalar product is used.

Original languageEnglish (US)
Title of host publicationInterferometry XVI
Subtitle of host publicationTechniques and Analysis
StatePublished - 2012
EventInterferometry XVI: Techniques and Analysis - San Diego, CA, United States
Duration: Aug 13 2012Aug 15 2012

Publication series

NameProceedings of SPIE - The International Society for Optical Engineering
ISSN (Print)0277-786X


OtherInterferometry XVI: Techniques and Analysis
Country/TerritoryUnited States
CitySan Diego, CA


  • Edge effects
  • Edge weighting
  • Finite data
  • Fitting
  • Gram-schmidt
  • Orthogonal
  • Weight mapping
  • Zernike polynomials

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics
  • Computer Science Applications
  • Applied Mathematics
  • Electrical and Electronic Engineering


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