pyzernike.zernike_index_to_order#

pyzernike.zernike_index_to_order(j: Sequence[Integral]) → Tuple[List[int], List[int]][source]#

Convert indices in the OSA/ANSI Zernike polynomial ordering to their corresponding Zernike orders (n, m).

\[j = \frac{n(n + 2) + m}{2}\]

See also

pyzernike.zernike_polynomial() for computing the Zernike polynomial \(Z_{n}^{m}(\rho, \theta)\).
pyzernike.zernike_order_to_index() for converting Zernike orders to indices.

j must be a sequence of non-negative integers.

The first few Zernike polynomials in the OSA/ANSI ordering are:

j	n	m
0	0	0
1	1	-1
2	1	1
3	2	-2
4	2	0
5	2	2
…	…	…

The process to compute the Zernike orders from the index is as follows:

\[n(n+2) = 2j - m \in [2j - n, 2j + n]\]

So :

\[n = \text{int}\left(\frac{-1 + \sqrt{1 + 8j}}{2}\right) \quad \text{and} \quad m = 2j - n(n + 2)\]

Parameters:

j (Sequence[Integral]) – The indices of the Zernike polynomials in the OSA/ANSI ordering.

Returns:

List[int] – A list of radial orders (n) of the Zernike polynomials.
List[int] – A list of azimuthal frequencies (m) of the Zernike polynomials.

Raises:

TypeError – If j is not a sequence of integers.

Examples

from pyzernike import zernike_index_to_order

j = [2, 3, 4]
n, m = zernike_index_to_order(j)
print(n)  # Output: [1, 2, 2]
print(m)  # Output: [1, -2, 0]

pyzernike.zernike_index_to_order#

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