320
J. Eur. Opt. Society-Rapid Publ. 22, 31( 2026)
Further, the radial coordinate-dependent vector in equation( A. 7) can be represented in matrix form as follows:
2 3 ðq 2 Þ j
ðq 2 Þ j�1. 6 7 4 ðq 2 Þ 5 1 ðjþ1Þ1
|
2 b fqg j
0
¼
.
6
4
0
|
0 b fqg j�1
.
0
|
.
..
|
0
0
.
b fqg
1
|
0
0
.
0
|
3
7
5
|
0 |
0 |
|
0 |
b fqg
0
|
|
ðjþ1Þ ðjþ1Þðjþ2Þ 2
2
6 4
X fqg j
X fqg j�1
.
X fqg 1
X fqg 0
3
7 5 ðjþ1Þðjþ2Þ
2 1
;
ðA: 9Þ
which can be iteratively represented in submatrix form as follows: 2 3 ðq 2 Þ j
ðq 2 Þ j�1.. 6 7 4 ðq 2 Þ 5 1 ðjþ1Þ1
¼ B j X j ¼
" # bfqg j 0
0 B j�1 ðjþ1Þ ðjþ1Þðjþ2Þ 2
" # X fqg j
X j�1 ðjþ1Þðjþ2Þ
2 1
ðA: 10Þ
where B j denotes the complete matrix with( j + 1) rowsand ðjþ1Þðjþ2Þ columns in equation( A. 9), B
2 j�1 is its right / below submatrix with j rows and ðjÞðjþ1Þ columns, and the same applies to X
2 j and its sub – vector X j�1 fortherightcolumnvectorcontainingall
X fqg j s. Subsequentially equation( A. 7) can be written as 8
>< R f2lg
2j ¼ T 2j diag w fkg 2j B j X j
ðA: 11Þ
>: R f2lþ1g
2jþ1 ¼ T 2jþ1 diag w fkg 2jþ1 B j X j q
inside which we have two cases of coordinate-irrelevant block-wise recurrence, as follows:
8
R f2lg 2j ¼ t >< f2lg
2j
>:
R f2lþ1g 2jþ1 ¼ t f2lþ1g
2jþ1
T 2j�2
0
" #
diag w fkg
2j
bfqg j 0 0 B j�1
T 2j�1
0
diag w fkg
2jþ1
" # X fqg
j
X j�1 " # " #:
bfqg j 0 X fqg
j
q
0 B j�1 X j�1 ðA: 12Þ
For weight factors, w 0 n ¼ 1 always applies, and subsequentially wn fkg fk > 0g
¼ 1 wn, wherew fk > 0g n denotes the sub-vector of wn fkg without its first element. Equation( A. 12) can be written as follows:
8 ><
>:
R f2lg 2j ¼ t f2lg
2j
R f2lþ1g 2jþ1 ¼ t f2lþ1g
2jþ1
T 2j�2
0
T 2j�1
0
" 1 0 # " # " #
fk > 0g bfqg j 0
X fqg
j
0 diag w2j 0 B j�1 X j�1
" 1 0 # " # " #
fk > 0g bfqg j 0
X fqg
j
q 0 diag w2jþ1
0 B j�1 X j�1 ðA: 13Þ
where all coordinate-irrelevant calculations, i. e., calculations of and between t, T, w, b and B, are fully supported by one n-row Pascal triangle.
A. 3 Complete Zernike by one Pascal triangle
If we further define J kl ¼ j � k � l and combine de Moivre’ sformula with equation( A. 2), we obtain
See the Equation( A. 14) bottom of the page
which is equivalent to the following equation in matrix form:
See the Equation( A. 15) top of the next page
This is an extension of equation( A. 13), where we have distinguished four submatrices B e� j�1; Beþ j�1; Bo� j�1 and Boþ j�1 for the sin and cos components of even and odd orders, respectively, as well as the submatrix with index ðl; kÞ introduced by the intermediate variable J kl ¼ j � k � l, as follows:
B þ 2j ¼ bf2pg 2l
2 b h i flgfkg b fqg j�l ¼ 6 4 f2pg 0 b fqg j b f2pg 2 b fqg j�1
.. b f2pg 2j b fqg 0
3
7 5 ðjþ1Þðjþ1Þ
:
ðA: 16Þ
Each submatrix in the matrix above corresponds to a convolution of two vectors as follows:
b f2pg 2l b fqg j�l ¼
02
3 ð�1Þ 0 b 0 T1 2l ð�1Þ 1 b 2
2l
6
. B 7 C @ 4. 5 A ð�1Þ l b 2l
2l 1ðlþ1Þ
|
8
><
>:
z þ2l 2j
z �2l 2j
z þð2lþ1Þ 2jþ1
z �ð2lþ1Þ 2jþ1
¼ Pj�l k¼0
¼ x �1 y Pj�l
ð�1Þ k w e k te lk
k¼0
ð�1Þ k w e k te lk
¼ x Pj�l ð�1Þ k w o k to lk
k¼0
¼ y Pj�l ð�1Þ k w o k to lk
k¼0
|
P J kl
q¼0
P J kl
q¼0
P J kl
q¼0
P J kl
q¼0
! b q J kl ðx 2 Þ q ðy 2 Þ J kl �q
!
b q J kl ðx 2 Þ q ðy 2 Þ J kl �q
!
b q J kl ðx 2 Þ q ðy 2 Þ J kl �q
!
b q J kl ðx 2 Þ q ðy 2 Þ J kl �q
|
Pl
p¼0
Pl�1
p¼0
Pl
p¼0
Pl
p¼0
ð�1Þ p b 2p
2l ðx2 Þ l�p ðy 2 Þ p!!
ð�1Þ p b 2pþ1
2l ðx 2 Þ l�p ðy 2 Þ p!!
ð�1Þ p b 2p
2lþ1 ðx2 Þ l�p ðy 2 Þ p!!
ð�1Þ p b 2pþ1
2lþ1 ðx2 Þ l�p ðy 2 Þ p!!
|
; ðA: 14Þ |