J. Eur. Opt. Society-Rapid Publ. 2026, 22, 31 Ó The Author( s), published by EDP Sciences, 2026 https:// doi. org / 10.1051 / jeos / 2026023 Available online at: https:// jeos. edpsciences. org
EOSAM 2025 Guest editors: Omar El Gawhary, Stefan Witte, Ignacio Moreno
Journal of the European Optical Society-Rapid Publications
RESEARCH ARTICLE
Zernike by one Pascal triangle: A high performance, low memory cost and flexible computation scheme for Zernike polynomials
Wei-Jun Chen * Carl Zeiss Meditec AG, Göschwitzer Str. 51 – 52, 07745 Jena, Germany
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.
Keywords: Zernike polynomials, Pascal triangle, Homogeneous bivariate polynomials, Block-wise recurrence, Recursive method, Surface / wavefront reconstruction.
1 Introduction
Zernike polynomials play a key role in many optical applications [ 1 – 3 ] due to their two main features: 1. the orthogonality between individual polynomial components and 2. the direct correspondence with optical Seidel aberrations. Zernike polynomials are usually defined as
z m n ðq; uÞ ¼R m n ðqÞ cosðmuÞ for þm; ð1Þ sinðmuÞ for �m ðn�mÞ
R m n ¼ X= 2 k¼0 ð�1Þ k ðn � kÞ! � � q
!; n�2k ð2Þ k! nþm � k
2! n�m
� k
2! where R m n is called the Zernike radial polynomial. The above equations appear in the literature [ 4, 5 ] withdifferent notations for the m-term. This work adopts the ± m notation since this notation provides an explicit and convenient way to divide Zernike components of order n into their even part(± m =± 2l), and their odd part ± m =±( 2l + 1), and to subdivide each part into distinct cos and sin subparts. Incidentally, all odd parts of the components for the even order( n = 2j) and all even parts of the components for the odd order( n = 2j + 1) are discarded to avoid non-integer factorials in Zernike radial polynomials( equation( 2)), and all index terms n, m, l, j, k in this work are non-negative integers, and l j and k( j � l).
If the polynomial order is not high, the calculation of Zernike polynomials is usually satisfied by the definition
* Corresponding author: wei-jun. chen @ zeiss. com formulas. For Zernike radial polynomials a three-termrecurrence-relation( TTRR) exists and has been applied component-wise to higher-order recursive Zernike calculations [ 6, 7 ]:
¼ q Rjm�1j n�1 ðqÞþR mþ1 � R m n�2 ðqÞ: ð3Þ
R m n n�1 ðqÞ
On the other hand, Zernike polynomials are essentially one of the orthogonalized versions of the homogeneous bivariate polynomials in a Cartesian coordinate system with a unit circle and a coordinate center. In practice, the computation of Zernike polynomials is often based on a look-up-table( LUT) of the corresponding Cartesian forms of the individual Zernike components [ 3, 8, 9 ], although the complexity of such Cartesian forms increases sharply with polynomial order.
To achieve greater flexibility compared to componentwise recursive computations, and to lower the limits of polynomial order and complexity in Cartesian form based computations, this work uncovers two cases of block-wise recurrence in Zernike computations 1 and offers both a direct block-wise transformation method and a block-wise recursive method for matrix-form Zernike computations. The application of such block-wise methods promises higher computational speed and lower memory footprint without compromising accuracy.
The rest of this paper is organized as follows: The next section presents two hidden block-wise recurrences along
1 To our knowledge, block-wise recurrence in Zernike calculations has hardly been addressed in the literature, although there is a long history of intensive studies on Zernike polynomials.
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