Chebyshev gradient polynomials for high resolution surface and wavefront reconstruction

Maham Aftab, James H. Burge, Greg A. Smith, Logan Graves, Chang Jin Oh, Dae Wook Kim

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

7 Scopus citations


A new data processing method based on orthonormal rectangular gradient polynomials is introduced in this work. This methodology is capable of effectively reconstructing surfaces or wavefronts with data obtained from deflectometry systems, especially during fabrication and metrology of high resolution and freeform surfaces. First, we derived a complete and computationally efficient vector polynomial set, called G polynomials. These polynomials are obtained from gradients of Chebyshev polynomials of the first kind-a basis set with many qualities that are useful for modal fitting. In our approach both the scalar and vector polynomials, that are defined and manipulated easily, have a straightforward relationship due to which the polynomial coefficients of both sets are the same. This makes conversion between the two sets highly convenient. Another powerful attribute of this technique is the ability to quickly generate a very large number of polynomial terms, with high numerical efficiency. Since tens of thousands of polynomials can be generated, mid-To-high spatial frequencies of surfaces can be reconstructed from high-resolution metrology data. We will establish the strengths of our approach with examples involving simulations as well as real metrology data from the Daniel K. Inouye Solar Telescope (DKIST) primary mirror.

Original languageEnglish (US)
Title of host publicationOptical Manufacturing and Testing XII
EditorsRay Williamson, Dae Wook Kim, Rolf Rascher
ISBN (Electronic)9781510620551
StatePublished - 2018
EventOptical Manufacturing and Testing XII 2018 - San Diego, United States
Duration: Aug 20 2018Aug 22 2018

Publication series

NameProceedings of SPIE - The International Society for Optical Engineering
ISSN (Print)0277-786X
ISSN (Electronic)1996-756X


OtherOptical Manufacturing and Testing XII 2018
Country/TerritoryUnited States
CitySan Diego


  • Deflectometry
  • Information processing
  • Modal fitting
  • Numerical approximation and analysis
  • Optical metrology
  • Surface measurements
  • Surface reconstruction
  • Testing

ASJC Scopus subject areas

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


Dive into the research topics of 'Chebyshev gradient polynomials for high resolution surface and wavefront reconstruction'. Together they form a unique fingerprint.

Cite this