Calculation of Zernike polynomial value, their sums and basic plotting of their values in polar coordinates
Project description
zernpy - package for calculation Zernike polynomials
This project is intended for calculation of Zernike polynomials parameters / values / properties using exact (analytical) and recursive equations,
the last ones were supposed to be the faster way to calculate values of high order polynomials in comparison to usage of their exact
definition (that used the sum of factorials, see the Wiki article for details).
However, it's turned out that usage of sum of factorials (calculated from math.factorial method) even for high orders provides
sufficiently fast calculation of radial polynomials. On other hand, I found that for polynomials with radial orders higher than 46th order
the exact equation with factorials starts providing ambiguous results due to high integers produced by factorials. Thus, it turns out that
only possible stable way to get polynomial value for high orders is to use the recursive equations for retain stability of calculation along
with the drawback of fast decreasing the performance with increasing of the radial order.
Also, I hope that someone also would find useful some of developed methods for usage in any further project.
The recursive and tabular equations are taken from the articles: [1],
[2] and [3].
Several useful transformations (e.g., from OSA / ANSI index to Noll one) are implemented as the methods of the main class.
Setup instructions
Basic installation
For installation of this package, use the command: pip install zernpy
Running tests for the code from the repository
Using the library pytest just run in the root folder for the folder, containing the package: pytest
It should collect 10 tests and automatically run them.
Requirements
For installation the numpy and matplotlib libraries are required.
For running tests, the pytest library is required for automatic recognition of tests stored in package folders.
A few examples of the library features usage
Initialization of base class instance
The useful calculation methods are written as the instance and static methods. The first ones are accessible after initialization of a class instance by providing characteristic orders (see the Zernike polynomial definition, e.g. in Wiki):
from zernpy import ZernPol
zp = ZernPol(m=-2, n=2)
Alternative initializations using other indices: ZernPol(osa_index=3), ZernPol(noll_index=5), ZernPol(fringe_index=6)
For details, please, refer to the API Dictionary provided on the GitHub page (see "Documentation" tab on pypi).
Some useful class instance methods:
- For getting all characteristic indices for the initialized polynomial: zp.get_indices()
This method returns the following tuple: ((azimuthal order, radial order), OSA index, Noll index, Fringe index) - For getting the string name of the initialized polynomial (up to 7th order): zp.get_polynomial_name()
- For calculating polynomial value for polar coordinates (r, theta): zp.polynomial_value(r, theta)
Note that r and theta are accepted as float numbers or numpy.ndarrays with the equal shape, it is also applicable for functions below 4. - 7. - For calculating radial polynomial value for radius (radii) r: zp.radial(r)
- For calculating derivative of radial polynomial value for radius (radii) r: zp.radial_dr(r)
- For calculating triangular function value for angle theta: zp.triangular(theta)
- For calculating derivative of triangular function value for angle theta: zp.triangular_dtheta(theta)
- For calculating normalization factor (N): zp.normf()
Some useful static methods of ZernPol class:
- For getting tuple as (azimuthal order, radial order) for OSA index i: ZernPol.index2orders(osa_index=i)
Same for Fringe and Noll indices: ZernPol.index2orders(noll_index=i) or ZernPol.index2orders(fringe_index=i) - Conversion between indices: ZernPol.osa2noll(osa_index), with similar signature: noll2osa(...), osa2fringe(...), osa2fringe(...), fringe2osa(...)
- Calculation of Zernike polynomials sum: ZernPol.sum_zernikes(coefficients, polynomials, r, theta, get_surface)
It calculates the sum of initialized Zernike polynomials (ZernPol) using coefficients and (r, theta) polar coordinates. The variable get_surface allows returning for vector polar coordinates with different shapes the values as for mesh of these coordinates. The details of acceptable values - see the docstring of this method or the API Dictionary. - Plotting the initialized Zernike polynomial (ZernPol): ZernPol.plot_zernike_polynomial(polynomial)
It plots the Zernike polynomial on unit circle polar coordinates (blocked non-interactive call of matplotlib.pyplot.show()) - Plotting Zernike polynomials sum: ZernPol.plot_sum_zernikes_on_fig(...) - check the list of parameters in the docstring. By using only default parameters, this method will plot sum of Zernike polynomials specified in the list with their coefficients on the provided figure (expected as an instance of the class matplotlib.pyplot.Figure).
Fitting Zernike polynomials to a 2D image with phases
Random generated set of Zernike polynomials plotted on an image - as the sample for testing the fitting procedure:
This image is assumed to contain phases wrapped in a circular aperture, used function for generation:
generate_random_phases(...) from the main zernikepol module.
Below is profile made by calculation of fitted Zernike polynomials:
The function used for fitting: fit_polynomials(...) from the main zernikepol module.
This function could be useful for making approximation of any image containing phases recorded by the optical system
to the sum of Zernike polynomials. Check the detailed description of functions in the API dictionary, available on
the separate tab on the GitHub page of this repository.
Import statement for using the scripts the mentioned functions:
from zernpy import generate_polynomials, fit_polynomials, generate_random_phases
Or for importing all available functions and base class in one statement:
from zernpy import *
Note that the function generate_polynomials(...) returns tuple with OSA indexed polynomials as instances of ZernPol class.
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