R

pyoof.zernike.R(n, m, rho)

Radial Zernike polynomials generator (\(R^m_n(\varrho)\) from Born & Wolf definition). The \(m\), \(n\) are integers, \(n\geqslant 0\) and \(n - m\) even. Only used to compute the general expression for the Zernike circle polynomials, U.

Parameters:

n : int

It is \(n \geqslant 0\). Order of the radial component.

m : int

Positive number, relative to the angle component.

rho : ndarray

Values for the radial component, \(\varrho = \sqrt{x^2 + y^2}\).

Returns:

radial_poly : ndarray

Radial Zernike polynomial already evaluated, \(R^m_n(\varrho)\).

Notes

The original generating formula for the radial polynomials is given by,

\[R^{\pm m}_n (\varrho) = \frac{1}{\left(\frac{n-m}{2}\right)!\cdot \varrho^m}\left\{\frac{\mathrm{d}}{\mathrm{d}\left(\varrho^2 \right)} \right\}^{\frac{n-m}{2}} \left\{ \left( \varrho^2 \right)^{\frac{n+m}{2}} \cdot \left( \varrho^2 -1 \right)^{\frac{n-m}{2}} \right\},\]

Which can also be expressed as a polynomial sum.

Examples

To start using the radial polynomials simply call the package.

>>> import numpy as np
>>> from pyoof import zernike
>>> r = np.linspace(-1, 1, 5)  # only orthogonal under unitary circle
>>> zernike.R(n=4, m=2, rho=r)
array([ 1. , -0.5,  0. , -0.5,  1. ])