Source code for photutils.isophote.harmonics

# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""
This module provides tools for computing and fitting harmonic functions.
"""

import numpy as np

__all__ = ['first_and_second_harmonic_function',
           'fit_first_and_second_harmonics', 'fit_upper_harmonic']


def _least_squares_fit(optimize_func, parameters):
    # call the least squares fitting
    # function and handle the result.
    from scipy.optimize import leastsq
    solution = leastsq(optimize_func, parameters, full_output=True)

    if solution[4] > 4:
        raise RuntimeError("Error in least squares fit: " + solution[3])

    # return coefficients and covariance matrix
    return (solution[0], solution[1])


[docs]def first_and_second_harmonic_function(phi, c): r""" Compute the harmonic function value used to calculate the corrections for ellipse fitting. This function includes simultaneously both the first and second order harmonics: .. math:: f(phi) = c[0] + c[1]*\sin(phi) + c[2]*\cos(phi) + c[3]*\sin(2*phi) + c[4]*\cos(2*phi) Parameters ---------- phi : float or `~numpy.ndarray` The angle(s) along the elliptical path, going towards the positive y axis, starting coincident with the position angle. That is, the angles are defined from the semimajor axis that lies in the positive x quadrant. c : `~numpy.ndarray` of shape (5,) Array containing the five harmonic coefficients. Returns ------- result : float or `~numpy.ndarray` The function value(s) at the given input angle(s). """ return (c[0] + c[1]*np.sin(phi) + c[2]*np.cos(phi) + c[3]*np.sin(2*phi) + c[4]*np.cos(2*phi))
[docs]def fit_first_and_second_harmonics(phi, intensities): r""" Fit the first and second harmonic function values to a set of (angle, intensity) pairs. This function is used to compute corrections for ellipse fitting: .. math:: f(phi) = y0 + a1*\sin(phi) + b1*\cos(phi) + a2*\sin(2*phi) + b2*\cos(2*phi) Parameters ---------- phi : float or `~numpy.ndarray` The angle(s) along the elliptical path, going towards the positive y axis, starting coincident with the position angle. That is, the angles are defined from the semimajor axis that lies in the positive x quadrant. intensities : `~numpy.ndarray` The intensities measured along the elliptical path, at the angles defined by the ``phi`` parameter. Returns ------- y0, a1, b1, a2, b2 : float The fitted harmonic coefficient values. """ a1 = b1 = a2 = b2 = 1. def optimize_func(x): return first_and_second_harmonic_function( phi, np.array([x[0], x[1], x[2], x[3], x[4]])) - intensities return _least_squares_fit(optimize_func, [np.mean(intensities), a1, b1, a2, b2])
[docs]def fit_upper_harmonic(phi, intensities, order): r""" Fit upper harmonic function to a set of (angle, intensity) pairs. With ``order`` set to 3 or 4, the resulting amplitudes, divided by the semimajor axis length and local gradient, measure the deviations from perfect ellipticity. The harmonic function that is fit is: .. math:: y(phi, order) = y0 + An*\sin(order*phi) + Bn*\cos(order*phi) Parameters ---------- phi : float or `~numpy.ndarray` The angle(s) along the elliptical path, going towards the positive y axis, starting coincident with the position angle. That is, the angles are defined from the semimajor axis that lies in the positive x quadrant. intensities : `~numpy.ndarray` The intensities measured along the elliptical path, at the angles defined by the ``phi`` parameter. order : int The order of the harmonic to be fitted. Returns ------- y0, An, Bn : float The fitted harmonic values. """ an = bn = 1. def optimize_func(x): return (x[0] + x[1]*np.sin(order*phi) + x[2]*np.cos(order*phi) - intensities) return _least_squares_fit(optimize_func, [np.mean(intensities), an, bn])