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where the terms of B � 2j; Bþ 2j; B� 2jþ1 and Bþ 2jþ1 denote four coordinate transformation matrices from the polar coordinate system to the Cartesian coordinate system for corresponding cos / sin components in Zernike basis functions of even and odd order. These four transformation matrices form the second block-wise recurrence in Zernike computations, since the following recursion relation holds: 2
8
B � 0 ¼ 0; Bþ 0 ¼ 1; B� 1 ¼ 1; Bþ 1 " # ¼ 1
B � 2j ¼ 0 0 B þ 2j�1 þ B� 2j�1
0 >< 2
B þ
B þ 2j ¼ 2j�1
0 þ 0 B
�
3
2j�1
4 h i h i5; ð11Þ
B þðj�1Þ 2j�1 0 � 0 B �ðj�1Þ
2j�1
B � 2jþ1 ¼ Bþ 2j þ B� 2j h i >:
B þ 2jþ1 ¼ Bþ 2j
� 0 B�f0kj�1g 2j
which is fully supported by the recursive nature of Pascal’ s |
|
triangle, as are w fkg n and t flg n
. By expanding the expressions w fkg n
; tflg n and B n in equation( 10) with their corresponding recursive forms, we obtain a block-wise recursive computation
|
method for Zernike basis functions: |
|
8
><
|
z � 0
¼ 0; zþ 0
¼ 1; z� 1
¼ y; zþ 1
"
#
"
¼ x
#
0
0 z � 2j
¼ x
O þ ðz � 2j�1 Þ
� y
O � ðz þ 2j�1 Þ
� z� 2j�2
0
"
#
"
#
2z þ0 z þ
2j�1
2z �0
2j�1
2j
¼ x
O þ ðz þ 2j�1 Þ þ y
O � ðz � 2j�1 Þ
� zþ 2j�2
0
z � 2jþ1 ¼ xO þðz � 2j Þ�yO �ðz þ 2j Þ�
|
; |
>: z þ 2jþ1 ¼ xO þðz þ 2j ÞþyO �ðz � 2j Þ� z� 2j�1
0 zþ 2j�1
0
ð12Þ where z 2j z f0ljg
2j z fm¼2lg n¼2j denotes a column vector of the Zernike basis function of order n = 2j with j + 1elementsatposition( x, y), and analogously z 2jþ1 for an odd order 2j + 1. The two vector operators O þ and O � are defined as follows:
8 " #
O þ ðz n Þ¼ zf n < g þ z n
0 " #
>< O � ðz n Þ¼ zf n < g � z n
0; ð13Þ
z f n < g ¼ z 0 n z 1 n z 2 n
z j T n
¼ z 1 n z 2 n
z j T n
>:
2 The mathematical derivation in this section briefly proves the
two previously mentioned cases of block – wise recurrence in Zernike computations without order restriction. Further details can be found in Appendix A. for performing offset addition and offset subtraction on a given vector.
3 New computation scheme
Zernike calculations in optical applications include, among others, the following topics: basis function calculation, surface( or wavefront) reconstruction, coefficients evaluation / transformation, derivative analysis, etc.
3.1 Zernike basis functions
Equation( 10) can be rewritten as a z m n
¼ X i 1 ðnþ1Þðnþ2Þ n T
2
1 ðnþ1Þðnþ2Þ 2
2 z 0 3T
2 3
0 x 0 y 0 T z �1 1 x 1 y 0 z 1 1 x 0 y 1 z �2 2 x 2 y 0 z 0 2 x 1 y 1 as z 2 2
�! x. ¼ 0 y 2
. T; ð14Þ
. z �n n x n y 0 zn
�nþ2 x n�1 y 1
6 4
. 7 6 5 4
. 7
5 z n x 0 y n n
where X i n ¼ xn�i y i denotes the set of basis functions of homogeneous bivariate( x, y) polynomials with orders from 0ton, z m n denotes the set of basis functions of Zernike polynomials with the same orders, and T denotes the transformation matrix between these two sets of basis function. It is convenient to arrange xy basis functions in a naturally increasing order( equation( 14)), and the same applies to Zernike basis functions, i. e., with the notation of m 3.
ThereistheinversetransformationmatrixT �1, with which we have X i n ¼ z
m n T �1: ð15Þ
Although T �1 can be obtained by matrix inversion of T, itis recommended to construct it recursively on the basis of the two block-wise recurrences supported by Pascal’ s triangle. 4
The direct transformation method for calculating Zernike basis functions can be performed by first calculating the basis functions of homogeneous xy polynomials and then transforming them by multiplying the sparse matrix T. Theoretically, the calculation of the xy-basis functions can be carried out efficiently and accurately according to
8
½ x n y 0 x n�1 y 1 x 1 y n�1 Š >< ¼ x ½ x n�1 y 0 x n�2 y 1 x 0 y n�1 Š
: ð16Þ ½ x n y 0 x n�1 y 1 x 1 y n�1 x 0 y n Š
>: ¼ ½½ x n y 0 x n�1 y 1 x 1 y n�1 Š y n Š
3 This arrangement corresponds to the OSA / ANSI standard indices for Zernike polynomials [ 3 ]. 4 Table B1 in Appendix B gives T, Table B2 its inverse T �1,
each up to order 6.