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with their close relationship to an n-row Pascal’ s triangle; subsequently Section 3 explores a new computation scheme for Zernike polynomials, which covers basis function computation, coefficient determination, polar / Cartesian coordinate conversion, and application specific derivative analysis, among others; further technical discussions can be found in Section 4, where a preliminary performance evaluation is conducted in term of computational complexity, accuracy stability, memory requirements, and application flexibility; finally, Section 5 concludes this paper with an outlook on future works.
2 Zernike polynomials supported by one Pascal’ s triangle
2.1 Pascal triangle supported block-wise recurrence
The Zernike radial polynomials of the first n 6orderscan be expressed in matrix form, where all radial polynomials of an order are obtained by multiplying a coordinate value irrelevant factor matrix by a power vector of the radial coordinate values, as: 8
><
R 0
0ðqÞ ¼1 ðqÞ ¼q
R 1
1
R 0
2 ðqÞ ¼2q2 � 1 ðqÞ ¼q2
3 ðqÞ ¼3q3 � 2q ðqÞ ¼q3
R 2
2
R 1
R 3
3
R 0
4 ðqÞ ¼6q4 � 6q 2 þ 1
R 2 4 ðqÞ ¼4q4 � 3q 2 ðqÞ ¼q4
R 4
4
R 1
5 ðqÞ ¼10q5 � 12q 3 þ 3q
R 3 5 ðqÞ ¼5q5 � 4q 3 ðqÞ ¼q5
R 5
5
R 0
6 ðqÞ ¼20q6 � 30q 4 þ 12q 2 � 1
R 2
6 ðqÞ ¼15q6 � 20q 4 þ 6q 2
R 4
6 ðqÞ ¼6q6 � 5q
>: 4
R 6
6ðqÞ ¼q6
8
R fmg 0 ¼ 1 q 0
R fmg 1 ¼ 1 q 1
R fmg
2 ¼ 2 �1
q2
1 0 q 0
R fmg
3 ¼ 3 �2
q3
2 1 0
3 q 1
2 3 ><
6 �6 1 q 4
¼)
R fmg 4 ¼ 4 4 �3 05 4 q 2 5 2
1 0 0
3 q
2 0
3 10 �12 3 q 5
R fmg 5 ¼ 4 5 �4 05 4 q 3 5 2
1 0 0 q
3 1
2 3 20 �30 12 �1 q 6
R fmg 15 �20 6 0
6 ¼ 6
7 4 6 �5 0 0 5 q 4 6 7 4 q 2 5
>: 1 0 0 0 q 0 ð4Þ
where R fmg n denotes a column vector R 0
2j
R 2 2j
R 2j h for even orders( n = 2j) or R 1
2jþ1
R 3 2jþ1
R 2jþ1
2jþ1
2jŠ T i T for odd orders( n = 2j + 1).
From the above equation seven weight vectors, fw f n kg jn ¼ 0; 1;:::; 6g with 0 k j, can be extracted from the corresponding order-specific factor matrices( equation( 5)). A block-wise recurrence among the de-weighted factor matrices({ T n | n = 0,1,..., 6 }) can be observed: the 0th order de-weighted factor matrix appears in the 2nd order de-weighted factor matrix, while the later subsequentially appears in the 4th order de-weighted factor matrix, which in turn appears in the 6th order de-weighted factor matrix. The same applies to odd orders from the 1st order to the 5th order, as follows: 8
R fmg 0 ¼ 1 diagð1Þq 0
R fmg 1 ¼ 1 diagð1Þq 1
R fmg 2 ¼ 2 1 T
! 1
diag
q2
1 0 �1 q 0
R fmg 3 ¼ 3 1 T
! 1
diag
q3
1 0
�2 q 1 2 3 02
3T1 2 3
6 2 1
1 q
>< 4
R fmg 6 7 B6
7 C 6
4 ¼ 4 4 1 05 diag@
4 �3 5 A q 2 7 4 5
1 0 0 1 q 0
2 3 02 3T1
2 3 10 3 1
1 q 5
R fmg 6 7 B6
7 C 6
5 ¼ 4 5 1 05 diag@
4 �4 5 A q 3 7 4 5
1 0 0 3 q 1
2 3 02
3 1
T 2 20 6 2 1
1
R fmg 15 4 1 0
6 ¼ 6
7 4 6 1 0 05 diag
�5
6 7 B @ 4 6 5 C
A
6 4
>: 1 0 0 0
�1
¼) R fmg n6 ¼ T n diagðw fkg n Þ
6 4
2 q n
q n�2
.. q ðnmod2Þ q 6 q 4 q 2 q 0
3
7 5
3
7 5; ð5Þ where diagðw fkg n
Þ denotes the diagonalization of a vector to a diagonal matrix, as
2 w 0 3 n
0 0
0 w 1 diagðw fkg n Þ¼ n
0 6..... 7 4.
5: ð6Þ
0 0 w j n
It appears that for order n 2, the de-weighted factor matrix T n can be interpreted as an extended T n�2, with zero-padding at the bottom and a factor vector t fmg n attached to its left side, as described in the following equation: