zernpy - Python package for calculation real-valued Zernike polynomials and associated 2D PSF kernels

Project description and references

This project is intended for calculation of Zernike polynomials' parameters / real values / properties using exact (analytical) and recursive equations.
The recursive and tabular equations, as well as the valuable information, are taken from the articles: [1], [2] and [3].
The possibly useful functionality for further usage include: conversion between common indexing schemes (OSA / Noll / Fringe), radial / angular / polynomial real values calculation along with their derivatives, plotting the circular 2D or 3D profile of single or sum of polynomials, fitting the arbitrary phase profile specified on the circle aperture by the set of polynomials (see the description below). For the complete list of implemented methods for Zernike polynomial, please refer to ZernPol API documentation.
For the implemented 2D PSF calculation as the class, please refer to [ZernPol API documentation] (https://sklykov.github.io/zernpy/api/zernpy/zernpsf.html).

Notes about implementation

It was supposed at the project start, that the recursive equations would be the faster way to calculate real 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 during the development and tests, that sum of factorials computation (based on Python built-in math.factorial method) even for high orders (> 10th order) 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 the big integer coefficients (exceeding the size of Python built-in int type) produced by factorials. Thus, it turns out that the stable way to get polynomial (radial) values for these orders is to use the recursive equations for retain stability of calculation along with the drawback of fast decreasing of the computational performance with the increased radial order.

Setup instructions

Basic installation

For installation of this package, use the command: pip install zernpy
For updating already installed package: pip install --upgrade zernpy or pip install -U zernpy

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.

Running tests

Simply run the command pytest in the command line from the project folder root. It should collect 10 tests and automatically run them.

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:

1. 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)
2. For getting the string name of the initialized polynomial (up to 7th order): zp.get_polynomial_name()
3. 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.
4. For calculating radial polynomial value for radius (radii) r: zp.radial(r)
5. For calculating derivative of radial polynomial value for radius (radii) r: zp.radial_dr(r)
6. For calculating triangular function value for angle theta: zp.triangular(theta)
7. For calculating derivative of triangular function value for angle theta: zp.triangular_dtheta(theta)
8. For calculating normalization factor (N): zp.normf()

Some useful static methods of the ZernPol class:

1. 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)
2. Conversion between indices: ZernPol.osa2noll(osa_index), with similar signature: noll2osa(...), osa2fringe(...), osa2fringe(...), fringe2osa(...)
3. 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.
4. Plotting the initialized Zernike polynomial (ZernPol) with default parameters for coordinates: ZernPol.plot_profile(polynomial)
   It plots the Zernike polynomial on unit circle using polar coordinates - on "2D" projection (blocked non-interactive call of matplotlib.pyplot.show()).
   For "3D" projection the polar coordinates are converted to the cartesian ones, as demanded by plotting method.
   (!): Note that the plotting method name has been changed from plot_profile(...) for ver. <= 0.0.10, the new version of the package will be simply sequentially incremented, without minor version changing, since there is no packages dependent on this one.
5. 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.
The function fit_polynomials_vectors(...) allows to fit composed in vectors (arrays with single dimension) phases recorded in polar coordinates (provided separately also in vectors) to the provided set of Zernike polynomials. This is analogous to the procedure described above, but this function doesn't perform any cropping or phases pre-selection.
Import statement for using the scripts the mentioned functions:

from zernpy import generate_polynomials, fit_polynomials, generate_random_phases, generate_phases_image, fit_polynomials_vectors

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 the ZernPol class, starting from the 'Piston' polynomial.

2D PSF kernel calculation

The 2D PSF kernel is calculated from the diffraction integral over the round pupil plane and described as Zernike polynomial phase distribution for the focal point (no Z-axis dependency). The used references are listed in the docstring of the calculate_psf_kernel() method.
The sample of calculated PSF for Vertical Trefoil:

Initialization and usage of the class instance (basic usage with default calculation parameters, such as the kernel size):

from zernpy import ZernPSF, ZernPol
zpsf = ZernPSF(ZernPol(m=1, n=3))  # horizontal coma
NA = 0.95; wavelength = 0.55; expansion_coeff = -0.26; pixel_physical_size = 0.2*wavelength   # example of physical properties
zpsf.set_physical_properties(NA, wavelength, expansion_coeff, pixel_physical_size)  # provide physical properties of the system
kernel = zpsf.calculate_psf_kernel(normalized=True)  # get the kernel as the square normalized matrix

Check the API documentation for other available methods.

PSF kernel for several polynomials

Similarly to the code above, it's possible to calculate the PSF associated with the sum profile of several polynomials:

from zernpy import ZernPSF, ZernPol 
zp1 = ZernPol(m=-1, n=3); zp2 = ZernPol(m=2, n=4); zp3 = ZernPol(m=0, n=4); pols = (zp1, zp2, zp3); coeffs = (0.5, 0.21, 0.15)
zpsf_pic = ZernPSF(pols); zpsf_pic.set_physical_props(NA=0.65, wavelength=0.6, expansion_coeff=coeffs, pixel_physical_size=0.6/5.0)
zpsf_pic.calculate_psf_kernel(); zpsf_pic.plot_kernel("Sum of Polynomials Profile")

The resulting profile is:

Acceleration of kernel calculation by numba

It's possible to accelerate the calculation of a kernel by installing the numba library in the same Python environment and providing the appropriate flags in a calculation method, similar to the following code snippet:

from zernpy import *
force_get_psf_compilation()  # optional precompilation of calculation methods for further using of their compiled forms 
NA = 0.95; wavelength = 0.55; pixel_size = wavelength / 4.6; ampl = -0.16
zp = ZernPol(m=0, n=2); zpsf = ZernPSF(zp) 
zpsf.set_physical_props(NA, wavelength, ampl, pixel_size)
zpsf.calculate_psf_kernel(accelerated=True)