Wavefront expansion basis functions are important in representing ocular aberrations and phase perturbations due to atmospheric turbulence. A general discussion is presented for the conversions of the coefficients between any two sets of basis functions. Several popular sets of basis functions, namely, Zernike polynomials, Fourier series, and Taylor monomials, are discussed and the conversion matrices between any two of these basis functions are derived. Some analytical and numerical examples are given to demonstrate conversion of coefficients of different basis function sets.
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The original Zernike coefficients were converted to Taylor coefficients; the Taylor coefficients were scaled, the original Zernike coefficients were scaled, and the scaled Taylor coefficients were converted to scaled Zernike coefficients. The scaling ratio was .
Tables (6)
Table 1
Fourier Series of the First Four Orders
i
u
v
Fourier Series
0
0
0
1
1
0
1
2
1
1
3
1
0
4
0
2
5
1
2
6
2
2
7
2
1
8
2
0
9
0
3
10
1
3
11
2
3
12
3
3
13
3
2
14
3
1
15
3
0
16
0
4
17
1
4
18
2
4
19
3
4
20
4
4
21
4
3
22
4
2
23
4
1
24
4
0
Table 2
Taylor Monomials of the First Four Orders
i
p
q
Taylor Monomials
0
0
0
1
1
1
0
2
1
1
3
2
0
4
2
1
5
2
2
6
3
0
7
3
1
8
3
2
9
3
3
10
4
0
11
4
1
12
4
2
13
4
3
14
4
4
Table 3
Conversion Matrix from Taylor Monomials to Zernike Polynomials for the First Four Orders
l
1
2
3
4
5
6
7
8
9
10
11
12
13
14
p
1
1
2
2
2
3
3
3
3
4
4
4
4
4
q
0
1
0
1
2
0
1
2
3
0
1
2
3
4
i
n
m
1
1
0
0
0
0
0
0
0
0
0
0
0
2
1
1
0
0
0
0
0
0
0
0
0
0
0
3
2
0
0
0
0
0
0
0
0
0
0
0
4
2
0
0
0
0
0
0
0
0
0
0
5
2
2
0
0
0
0
0
0
0
0
0
0
6
3
0
0
0
0
0
0
0
0
0
0
0
0
7
3
0
0
0
0
0
0
0
0
0
0
0
0
8
3
1
0
0
0
0
0
0
0
0
0
0
0
0
9
3
3
0
0
0
0
0
0
0
0
0
0
0
0
10
4
0
0
0
0
0
0
0
0
0
0
0
0
11
4
0
0
0
0
0
0
0
0
0
0
0
0
12
4
0
0
0
0
0
0
0
0
0
0
0
0
13
4
2
0
0
0
0
0
0
0
0
0
0
0
0
14
4
4
0
0
0
0
0
0
0
0
0
0
0
Table 4
Conversion Matrix from Zernike Polynomials to Taylor Monomials for the First Four Orders
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
n
1
1
2
2
2
3
3
3
3
4
4
4
4
4
m
1
0
2
1
3
0
2
4
l
p
q
1
1
0
2
0
0
0
0
0
0
0
0
0
0
0
0
2
1
1
0
2
0
0
0
0
0
0
0
0
0
0
0
3
2
0
0
0
0
0
0
0
0
0
0
0
4
2
1
0
0
0
0
0
0
0
0
0
0
0
0
5
2
2
0
0
0
0
0
0
0
0
0
0
6
3
0
0
0
0
0
0
0
0
0
0
0
0
0
7
3
1
0
0
0
0
0
0
0
0
0
0
0
0
8
3
2
0
0
0
0
0
0
0
0
0
0
0
0
9
3
3
0
0
0
0
0
0
0
0
0
0
0
0
10
4
0
0
0
0
0
0
0
0
0
0
0
0
11
4
1
0
0
0
0
0
0
0
0
0
0
0
0
12
4
2
0
0
0
0
0
0
0
0
0
0
0
0
13
4
3
0
0
0
0
0
0
0
0
0
0
0
0
14
4
4
0
0
0
0
0
0
0
0
0
0
0
Table 5
Input Taylor Coefficients, Converted Zernike Coefficients, and Output Taylor Coefficients
The original Zernike coefficients were converted to Taylor coefficients; the Taylor coefficients were scaled, the original Zernike coefficients were scaled, and the scaled Taylor coefficients were converted to scaled Zernike coefficients. The scaling ratio was .