J. Eur. Opt. Society-Rapid Publ. 22, 31( 2026) 311
Table 1. A standard 6-row Pascal triangle and its right half vectors.
8 T 0 ¼ 1
>< T 1 ¼ 1
>: T n6 ¼ t fmg n
T n�2
0
: ð7Þ
Table 2. A left-aligned 6-row Pascal triangle and its antidiagonal vectors.
Note that both the extracted weight vector wn fkg and the deweighted factor vector t fmg n are closely related to an n-row Pascal triangle. In particular, t fmg n corresponds to the right half of the nth row vector in a standard Pascal triangle( Table 1), i. e., here we have t fmg
6 ¼½20 15 6 1Š T, t fmg 5 ¼½10 5 1Š T, t fmg
4 ¼½6 4 1Š T, t fmg
3 ¼½3 1Š T, t fmg 2 ¼½2 1Š T, t fmg
1 ¼ 1andt fmg
0 ¼ 1 for radial polynomials of orders n 6; On the other hand, all extracted weight vectors w fkg n can be found as corresponding antidiagonal vectors in the left-aligned Pascal triangle( Table 2),
as w fkg
0; 1 ¼ 1, w fkg 2 ¼½1 �1 Š, w fkg
3 ¼½1 �2 Š, w fkg
4 ¼
½ 1 �3 1Š, w fkg
5 ¼½1 �4 3Š, w fkg
6 ¼½1 �5 6� 1Š,
where all odd elements( numbered from 0 onwards) are additionally assigned a negative sign.
To our knowledge, the above-mentioned discoveries are rarely discussed in the literature. Therefore, it is necessary to prove that these discoveries, i. e., the separation of coordinate-relevant and-irrelevant computations( equation( 4)), the extraction of order-specific weight vectors( equation( 5)), the block-wise recurrence within de-weighted factor matrices( equation( 7)), and the close relationship to an n-row Pascal triangle( Tables 1 and 2), hold for Zernike radial polynomials without order limit( i. e., " n > 6); if possible and expected, this also holds for complete Zernike polynomials( i. e., radial azimuth polynomials)
2.2 From canonical to block-wise Zernike calculation
The canonical definition of Zernike radial polynomials, equation( 2), has its binomial form: ðn�mÞ
R m n ¼ X= 2
ð�1Þ k n � k n � 2k q n�2k: ð8Þ n�m k � k
2 k¼0
If we swap the terms m and k in the second binomial term, we obtain ðn�mÞ
R m n ¼ X= 2
ð�1Þ k n � k n � 2k q n�2k: ð9Þ k k¼0
n�2k 2
� m 2
n � k In this equation, the term explicitly denotes the k anti-diagonal
vectors in a left-aligned Pascal triangle, and n � 2k
the term n�2k
� m denotes the left and right halves of
2 2 the( n � 2k) th-row in a standard Pascal triangle, since Pascal’ s triangle has its mirror symmetry with respect to n � 2k its center position.
2
By Combining de Moivre’ s formula( cosmu + isinmu) =( cosu + isinu) m with the radial polynomials and converting the coordinate system from polar to Cartesian coordinates, we obtain a direct block-wise transformation method for calculating Zernike basis functionshin the
Cartesian coordinate i system. If we define X fqg j ¼ x 2j y 0
T x 2j�2 y 2 x 0 y 2j, and set w k n
¼ ð�1Þ k
n � k and
k fl¼m = 2g n � 2k
tn ¼
, where { l } denotes the index vector n�2k
� fmg
2 2 as flg ¼½0 1 n = 2 Š T for even orders or flg ¼½0 1 ðn � 1Þ = 2 Š T for odd orders, then such a direct block-wise transformation is represented as a transformation of homogeneous bivariate( xy) polynomials into complete Zernike polynomials, as the following: 8
><
>: z �f2lg 2j
z þf2lg 2j
z �f2lþ1g 2jþ1
z þf2lþ1g 2jþ1
¼ Pj w k 2j k¼0
¼ Pj w k 2j k¼0
¼ Pj w k 2jþ1 k¼0
¼ Pj w k 2jþ1 k¼0
t flg 2ðj�kÞ B� 2ðj�kÞ
t flg 2ðj�kÞ Bþ 2ðj�kÞ
X fqg ðj�kÞ x�1 y
X fqg ðj�kÞ
tflg 2ðj�kÞþ1 B� 2ðj�kÞþ1
tflg 2ðj�kÞþ1 Bþ 2ðj�kÞþ1
X fqg ðj�kÞ y
X fqg ðj�kÞ x
; ð10Þ