| Issue |
J. Eur. Opt. Society-Rapid Publ.
Volume 22, Number 1, 2026
EOSAM 2025
|
|
|---|---|---|
| Article Number | 31 | |
| Number of page(s) | 15 | |
| DOI | https://doi.org/10.1051/jeos/2026023 | |
| Published online | 05 May 2026 | |
Research Article
Zernike by one Pascal triangle: A high performance, low memory cost and flexible computation scheme for Zernike polynomials
Carl Zeiss Meditec AG, Göschwitzer Str. 51–52, 07745 Jena, Germany
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
21
December
2025
Accepted:
4
March
2026
Abstract
This work uncovers two hidden cases of block-wise recurrence in Zernike computations. Based on these findings, a new computation scheme for Zernike polynomials is proposed. It uses one Pascal’s triangle for all internal factors, thus avoiding the computation of factorials, cos/sin functions, and matrix inversions. It offers both a direct transformation method and a block-wise recursive method for calculating Zernike basis functions, thereby fulfilling the requirements for high accuracy, high speed, low memory footprint, and great application flexibility.
Key words: Zernike polynomials / Pascal triangle / Homogeneous bivariate polynomials / Block-wise recurrence / Recursive method / Surface/wavefront reconstruction
© The Author(s), published by EDP Sciences, 2026
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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