Source code for vip_hci.fm.negfc_simplex

#! /usr/bin/env python

"""
Module with simplex (Nelder-Mead) optimization for defining the flux and
position of a companion using the Negative Fake Companion.
     
"""


import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import minimize
from .negfc_fmerit import chisquare, get_mu_and_sigma
from ..psfsub import pca_annulus
from ..var import frame_center
from ..config import time_ini, timing
from ..config.utils_conf import sep


__author__ = 'O. Wertz, C. A. Gomez Gonzalez, V. Christiaens'
__all__ = ['firstguess',
           'firstguess_from_coord']


[docs]def firstguess_from_coord(planet, center, cube, angs, psfn, fwhm, annulus_width, aperture_radius, ncomp, cube_ref=None, svd_mode='lapack', scaling=None, fmerit='sum', imlib='vip-fft', interpolation='lanczos4', collapse='median', algo=pca_annulus, delta_rot=1, algo_options={}, f_range=None, transmission=None, mu_sigma=(0, 1), weights=None, plot=False, verbose=True, save=False, debug=False, full_output=False): """ Determine a first guess for the flux of a companion at a given position in the cube by doing a simple grid search evaluating the reduced chi2. Parameters ---------- planet: numpy.array The (x,y) position of the planet in the pca processed cube. center: numpy.array The (x,y) position of the cube center. cube: 3d or 4d numpy ndarray Input ADI or ADI+IFS cube. angs: numpy.array The parallactic angle fits image expressed as a numpy.array. psfn: numpy 2D or 3D array Normalised PSF template used for negative fake companion injection. The PSF must be centered and the flux in a 1xFWHM aperture must equal 1 (use ``vip_hci.metrics.normalize_psf``). If the input cube is 3D and a 3D array is provided, the first dimension must match for both cubes. This can be useful if the star was unsaturated and conditions were variable. If the input cube is 4D, psfn must be either 3D or 4D. In either cases, the first dimension(s) must match those of the input cube. fwhm : float The FHWM in pixels. annulus_width: int, optional The width in pixels of the annulus on which the PCA is done. aperture_radius: int, optional The radius of the circular aperture in terms of the FWHM. ncomp: int The number of principal components. cube_ref : numpy ndarray, 3d, optional Reference library cube. For Reference Star Differential Imaging. svd_mode : {'lapack', 'randsvd', 'eigen', 'arpack'}, str optional Switch for different ways of computing the SVD and selected PCs. scaling : {'temp-mean', 'temp-standard'} or None, optional With None, no scaling is performed on the input data before SVD. With "temp-mean" then temporal px-wise mean subtraction is done and with "temp-standard" temporal mean centering plus scaling to unit variance is done. fmerit : {'sum', 'stddev'}, string optional Figure of merit to be used, if mu_sigma is set to None. imlib : str, optional See the documentation of the ``vip_hci.preproc.frame_rotate`` function. interpolation : str, optional See the documentation of the ``vip_hci.preproc.frame_rotate`` function. collapse : {'median', 'mean', 'sum', 'trimmean', None}, str or None, optional Sets the way of collapsing the frames for producing a final image. If None then the cube of residuals is used when measuring the function of merit (instead of a single final frame). algo: python routine, opt {pca_annulus, pca_annular, pca, custom} Routine to be used to model and subtract the stellar PSF. From an input cube, derotation angles, and optional arguments, it should return a post-processed frame. delta_rot: float, optional If algo is set to pca_annular, delta_rot is the angular threshold used to select frames in the PCA library (see description of pca_annular). algo_options: dict, opt Dictionary with additional parameters for the pca algorithm (e.g. tol, min_frames_lib, max_frames_lib). Note: arguments such as svd_mode, scaling imlib, interpolation or collapse can also be included in this dict (the latter are also kept as function arguments for compatibility with older versions of vip). f_range: numpy.array, optional The range of tested flux values. If None, 20 values between 0 and 5000 are tested. transmission: numpy array, optional Array with 2 columns. First column is the radial separation in pixels. Second column is the off-axis transmission (between 0 and 1) at the radial separation given in column 1. mu_sigma: tuple of 2 floats or None, opt If set to None: not used, and falls back to original version of the algorithm, using fmerit. Otherwise, should be a tuple of 2 elements, containing the mean and standard deviation of pixel intensities in an annulus centered on the location of the companion, excluding the area directly adjacent to the companion. weights : 1d array, optional If provided, the negative fake companion fluxes will be scaled according to these weights before injection in the cube. Can reflect changes in the observing conditions throughout the sequence. plot: boolean, optional If True, the figure chi2 vs. flux is displayed. verbose: boolean If True, display intermediate info in the shell. save: boolean, optional If True, the figure chi2 vs. flux is saved as .pdf if plot is also True debug: bool, optional Whether to print details of the grid search full_output : bool, optional Whether to also return the range of fluxes tested and their chi2r values. Returns ------- res : tuple The polar coordinates and the flux(es) of the companion. f_range: 1d numpy.array [full_output=True] The range of tested flux values. chi2r: 1d numpy.array [full_output=True] The chi2r values corresponding to tested flux values. """ def _grid_search_f(r0, theta0, ch, cube, angs, psfn, fwhm, annulus_width, aperture_radius, ncomp, cube_ref=None, svd_mode='lapack', scaling=None, fmerit='sum', imlib='vip-fft', interpolation='lanczos4', collapse='median', algo=pca_annulus, delta_rot=1, algo_options={}, f_range=np.geomspace(1e-1, 1e4, 30), transmission=None, mu_sigma=None, weights=None, verbose=True, debug=False): chi2r = [] if verbose: print('Step | flux | chi2r') counter = 0 for j, f_guess in enumerate(f_range): if cube.ndim == 3: params = (r0, theta0, f_guess) elif ch is not None and cube.ndim == 4: params = [r0, theta0] fluxes = [0]*cube.shape[0] fluxes[ch] = f_guess params = tuple(params+fluxes) else: raise TypeError("If cube is 4d, channel index must be provided") chi2r.append(chisquare(params, cube, angs, psfn, fwhm, annulus_width, aperture_radius, (r0, theta0), ncomp, cube_ref, svd_mode, scaling, fmerit, collapse, algo, delta_rot, imlib, interpolation, algo_options, transmission, mu_sigma, weights, debug)) if chi2r[j] > chi2r[j-1]: counter += 1 if counter == 4: break if verbose: print('{}/{} {:.3f} {:.3f}'.format(j + 1, n, f_guess, chi2r[j])) return chi2r xy = planet-center r0 = np.sqrt(xy[0]**2 + xy[1]**2) theta0 = np.mod(np.arctan2(xy[1], xy[0]) / np.pi*180, 360) if f_range is not None: n = f_range.shape[0] else: n = 30 f_range = np.geomspace(1e-1, 1e4, n) if cube.ndim == 3: chi2r = _grid_search_f(r0, theta0, None, cube, angs, psfn, fwhm, annulus_width, aperture_radius, ncomp, cube_ref=cube_ref, svd_mode=svd_mode, scaling=scaling, fmerit=fmerit, imlib=imlib, interpolation=interpolation, collapse=collapse, algo=algo, delta_rot=delta_rot, algo_options=algo_options, f_range=f_range, transmission=transmission, mu_sigma=mu_sigma, weights=weights, verbose=verbose, debug=debug) chi2r = np.array(chi2r) f0 = f_range[chi2r.argmin()] if plot: plt.figure(figsize=(8, 4)) plt.title('$\\chi^2_{r}$ vs flux') plt.xlim(f_range[0], f_range[:chi2r.shape[0]].max()) plt.ylim(chi2r.min()*0.9, chi2r.max()*1.1) plt.plot(f_range[:chi2r.shape[0]], chi2r, linestyle='-', color='gray', marker='.', markerfacecolor='r', markeredgecolor='r') plt.xlabel('flux') plt.ylabel(r'$\chi^2_r$') plt.grid('on') if save and plot: plt.savefig('chi2rVSflux.pdf') if plot: plt.show() res = (r0, theta0, f0) else: f0 = [] chi2r = [] if plot: plt.figure(figsize=(8, 4)) plt.title('$\\chi^2_{r}$ vs flux') plt.xlabel('flux') plt.ylabel(r'$\chi^2_{r}$') plt.grid('on') for i in range(cube.shape[0]): if verbose: print('Processing spectral channel {}...'.format(i)) chi2r_tmp = _grid_search_f(r0, theta0, i, cube, angs, psfn, fwhm, annulus_width, aperture_radius, ncomp, cube_ref=cube_ref, svd_mode=svd_mode, scaling=scaling, fmerit=fmerit, imlib=imlib, interpolation=interpolation, collapse=collapse, algo=algo, delta_rot=delta_rot, algo_options=algo_options, f_range=f_range, transmission=transmission, mu_sigma=mu_sigma, weights=weights, verbose=False, debug=False) chi2r.append(chi2r_tmp) chi2r_tmp = np.array(chi2r_tmp) f0.append(f_range[chi2r_tmp.argmin()]) if verbose: msg = r'... optimal grid flux: {:.3f} ($\chi^2_r$ = {:.1f})' print(msg.format(f0[i], np.amin(chi2r_tmp))) if i == 0: min_chi2r = chi2r_tmp.min() max_chi2r = chi2r_tmp.max() fmax = f0[i] else: if min_chi2r > chi2r_tmp.min(): min_chi2r = chi2r_tmp.min() if max_chi2r < chi2r_tmp.max(): max_chi2r = chi2r_tmp.max() if fmax < f0[i]: fmax = f0[i] if plot: plt.plot(f_range[:chi2r_tmp.shape[0]], chi2r_tmp, linestyle='-', marker='.', markerfacecolor='r', markeredgecolor='r', label='ch. {}'.format(i)) if plot: plt.xlim(f_range[0], f_range[:chi2r_tmp.shape[0]].max()) plt.ylim(min_chi2r*0.9, max_chi2r*1.1) plt.legend() if save and plot: plt.savefig('chi2rVSflux.pdf') if plot: plt.show() res = tuple([r0, theta0]+f0) if full_output: return res, f_range, chi2r else: return res
def firstguess_simplex(p, cube, angs, psfn, ncomp, fwhm, annulus_width, aperture_radius, cube_ref=None, svd_mode='lapack', scaling=None, fmerit='sum', imlib='vip-fft', interpolation='lanczos4', collapse='median', algo=pca_annulus, delta_rot=1, algo_options={}, p_ini=None, transmission=None, mu_sigma=(0, 1), weights=None, force_rPA=False, options=None, verbose=False, **kwargs): """ Determine the position of a companion using the negative fake companion technique and a standard minimization algorithm (Default=Nelder-Mead) . Parameters ---------- p : np.array Estimate of the candidate position. cube: 3d or 4d numpy ndarray Input ADI or ADI+IFS cube. angs: numpy.array The parallactic angle fits image expressed as a numpy.array. psfn: numpy 2D or 3D array Normalised PSF template used for negative fake companion injection. The PSF must be centered and the flux in a 1xFWHM aperture must equal 1 (use ``vip_hci.metrics.normalize_psf``). If the input cube is 3D and a 3D array is provided, the first dimension must match for both cubes. This can be useful if the star was unsaturated and conditions were variable. If the input cube is 4D, psfn must be either 3D or 4D. In either cases, the first dimension(s) must match those of the input cube. ncomp: int or None The number of principal components. fwhm : float The FHWM in pixels. annulus_width: int, optional The width in pixels of the annulus on which the PCA is done. aperture_radius: int, optional The radius of the circular aperture in terms of the FWHM. cube_ref : numpy ndarray, 3d, optional Reference library cube. For Reference Star Differential Imaging. svd_mode : {'lapack', 'randsvd', 'eigen', 'arpack'}, str optional Switch for different ways of computing the SVD and selected PCs. scaling : {'temp-mean', 'temp-standard'} or None, optional With None, no scaling is performed on the input data before SVD. With "temp-mean" then temporal px-wise mean subtraction is done and with "temp-standard" temporal mean centering plus scaling to unit variance is done. fmerit : {'sum', 'stddev'}, string optional Figure of merit to be used, if mu_sigma is set to None. imlib : str, optional See the documentation of the ``vip_hci.preproc.frame_rotate`` function. interpolation : str, optional See the documentation of the ``vip_hci.preproc.frame_rotate`` function. collapse : {'median', 'mean', 'sum', 'trimmean', None}, str or None, optional Sets the way of collapsing the frames for producing a final image. If None then the cube of residuals is used when measuring the function of merit (instead of a single final frame). algo: python routine, opt {pca_annulus, pca_annular, pca, custom} Routine to be used to model and subtract the stellar PSF. From an input cube, derotation angles, and optional arguments, it should return a post-processed frame. delta_rot: float, optional If algo is set to pca_annular, delta_rot is the angular threshold used to select frames in the PCA library (see description of pca_annular). algo_options: dict, opt Dictionary with additional parameters for the pca algorithm (e.g. tol, min_frames_lib, max_frames_lib). Note: arguments such as svd_mode, scaling imlib, interpolation or collapse can also be included in this dict (the latter are also kept as function arguments for compatibility with older versions of vip). p_ini : np.array Position (r, theta) of the circular aperture center. transmission: numpy array, optional Array with 2 columns. First column is the radial separation in pixels. Second column is the off-axis transmission (between 0 and 1) at the radial separation given in column 1. mu_sigma: tuple of 2 floats or None, opt If set to None: not used, and falls back to original version of the algorithm, using fmerit. Otherwise, should be a tuple of 2 elements, containing the mean and standard deviation of pixel intensities in an annulus centered on the location of the companion, excluding the area directly adjacent to the companion. weights : 1d array, optional If provided, the negative fake companion fluxes will be scaled according to these weights before injection in the cube. Can reflect changes in the observing conditions throughout the sequence. force_rPA: bool, optional Whether to only search for optimal flux, provided (r,PA). options: dict, optional The scipy.optimize.minimize options. verbose : boolean, optional If True, additional information is printed out. **kwargs: optional Optional arguments to the scipy.optimize.minimize function Returns ------- solu : scipy.optimize.minimize solution object The solution of the minimization algorithm. """ if verbose: print('\nNelder-Mead minimization is running...') if p_ini is None: p_ini = p if force_rPA: p_t = p[2:] p_ini = (p[0], p[1]) else: p_t = p solu = minimize(chisquare, p_t, args=(cube, angs, psfn, fwhm, annulus_width, aperture_radius, p_ini, ncomp, cube_ref, svd_mode, scaling, fmerit, collapse, algo, delta_rot, imlib, interpolation, algo_options, transmission, mu_sigma, weights, force_rPA), method='Nelder-Mead', options=options, **kwargs) if verbose: print(solu) return solu
[docs]def firstguess(cube, angs, psfn, ncomp, planets_xy_coord, fwhm=4, annulus_width=4, aperture_radius=1, cube_ref=None, svd_mode='lapack', scaling=None, fmerit='sum', imlib='vip-fft', interpolation='lanczos4', collapse='median', algo=pca_annulus, delta_rot=1, p_ini=None, f_range=None, transmission=None, mu_sigma=True, wedge=None, weights=None, force_rPA=False, algo_options={}, simplex=True, simplex_options=None, plot=False, verbose=True, save=False): """ Determines a first guess for the position and the flux of a planet, as explained in [WER17]_. We process the cube without injecting any negative fake companion. This leads to the visual detection of the planet(s). For each of them, one can estimate the (x,y) coordinates in pixel for the position of the star, as well as the planet(s). From the (x,y) coordinates in pixels for the star and planet(s), we can estimate a preliminary guess for the position and flux for each planet by using the method "firstguess_from_coord". The argument "f_range" allows to indicate prior limits for the flux (optional, default: None). This step can be reiterate to refine the preliminary guess for the flux. We can go a step further by using a Simplex Nelder_Mead minimization to estimate the first guess based on the preliminary guess. Parameters ---------- cube: 3d or 4d numpy ndarray Input ADI or ADI+IFS cube. angs: numpy.array The parallactic angle fits image expressed as a numpy.array. psfn: numpy 2D or 3D array Normalised PSF template used for negative fake companion injection. The PSF must be centered and the flux in a 1xFWHM aperture must equal 1 (use ``vip_hci.metrics.normalize_psf``). If the input cube is 3D and a 3D array is provided, the first dimension must match for both cubes. This can be useful if the star was unsaturated and conditions were variable. If the input cube is 4D, psfn must be either 3D or 4D. In either cases, the first dimension(s) must match those of the input cube. ncomp : int or 1d numpy array of int The number of principal components. If cube is a 4D cube, ncomp can be a list of integers, with length matching the first dimension of the cube. planets_xy_coord: array or list The list of (x,y) positions of the planets. plsc: float, optional The platescale, in arcsec per pixel. fwhm : float, optional The FHWM in pixels. annulus_width: int, optional The width in pixels of the annulus on which the PCA is done. aperture_radius: int, optional The radius of the circular aperture in terms of the FWHM. cube_ref : numpy ndarray, 3d, optional Reference library cube. For Reference Star Differential Imaging. svd_mode : {'lapack', 'randsvd', 'eigen', 'arpack'}, str optional Switch for different ways of computing the SVD and selected PCs. scaling : {'temp-mean', 'temp-standard'} or None, optional With None, no scaling is performed on the input data before SVD. With "temp-mean" then temporal px-wise mean subtraction is done and with "temp-standard" temporal mean centering plus scaling to unit variance is done. fmerit : {'sum', 'stddev'}, string optional Figure of merit to be used, if mu_sigma is set to None. imlib : str, optional See the documentation of the ``vip_hci.preproc.frame_rotate`` function. interpolation : str, optional See the documentation of the ``vip_hci.preproc.frame_rotate`` function. collapse : {'median', 'mean', 'sum', 'trimmean', None}, str or None, opt Sets the way of collapsing the frames for producing a final image. If None then the cube of residuals is used when measuring the function of merit (instead of a single final frame). algo: python routine, opt {pca_annulus, pca_annular, pca, custom} Routine to be used to model and subtract the stellar PSF. From an input cube, derotation angles, and optional arguments, it should return a post-processed frame. delta_rot: float, optional If algo is set to pca_annular, delta_rot is the angular threshold used to select frames in the PCA library (see description of pca_annular). p_ini: numpy.array Position (r, theta) of the circular aperture center. f_range: numpy.array, optional The range of flux tested values. If None, 20 values between 0 and 5000 are tested. transmission: numpy array, optional Array with 2 columns. First column is the radial separation in pixels. Second column is the off-axis transmission (between 0 and 1) at the radial separation given in column 1. mu_sigma: tuple of 2 floats, bool or None, opt If set to None: not used, and falls back to original version of the algorithm, using fmerit. If a tuple of 2 elements: should be the mean and standard deviation of pixel intensities in an annulus centered on the lcoation of the companion candidate, excluding the area directly adjacent to the CC. If set to anything else, but None/False/tuple: will compute said mean and standard deviation automatically. wedge: tuple, opt Range in theta where the mean and standard deviation are computed in an annulus defined in the PCA image. If None, it will be calculated automatically based on initial guess and derotation angles to avoid. If some disc signal is present elsewhere in the annulus, it is recommended to provide wedge manually. The provided range should be continuous and >0. E.g. provide (270, 370) to consider a PA range between [-90,+10]. weights : 1d array, optional If provided, the negative fake companion fluxes will be scaled according to these weights before injection in the cube. Can reflect changes in the observing conditions throughout the sequence. force_rPA: bool, optional Whether to only search for optimal flux, provided (r,PA). algo_options: dict, opt Dictionary with additional parameters for the pca algorithm (e.g. tol, min_frames_lib, max_frames_lib). Note: arguments such as svd_mode, scaling imlib, interpolation or collapse can also be included in this dict (the latter are also kept as function arguments for compatibility with older versions of vip). simplex: bool, optional If True, the Nelder-Mead minimization is performed after the flux grid search. simplex_options: dict, optional The scipy.optimize.minimize options. plot: boolean, optional If True, the figure chi2 vs. flux is displayed. verbose: bool, optional If True, display intermediate info in the shell. save: bool, optional If True, the figure chi2 vs. flux is saved. Returns ------- out : tuple of 3+ elements The polar coordinates and the flux(es) of the companion. Note ---- Polar angle is not the conventional NORTH-TO-EAST P.A., but the counter-clockwise angle measured from the positive x axis. """ if cube.ndim != 3 and cube.ndim != 4: raise TypeError("Input cube is not 3D nor 4D") if verbose: start_time = time_ini() planets_xy_coord = np.array(planets_xy_coord) n_planet = planets_xy_coord.shape[0] center_xy_coord = np.array(frame_center(cube[0])) r_0 = np.zeros(n_planet) theta_0 = np.zeros_like(r_0) if cube.ndim == 3: f_0 = np.zeros_like(r_0) else: if psfn.ndim < 3: msg = "The normalized PSF should be 3D for a 4D input cube" raise TypeError(msg) f_0 = np.zeros([n_planet, cube.shape[0]]) if weights is not None: if not len(weights) == cube.shape[-3]: msg = "Weights should have same length as temporal cube axis" raise TypeError(msg) norm_weights = weights/np.sum(weights) else: norm_weights = weights for index_planet in range(n_planet): if verbose: print('\n'+sep) print(' Planet {} '.format(index_planet)) print(sep+'\n') msg2 = 'Planet {}: flux estimation at the position [{},{}], ' msg2 += 'running ...' print(msg2.format(index_planet, planets_xy_coord[index_planet, 0], planets_xy_coord[index_planet, 1])) # Measure mu and sigma once in the annulus (instead of each MCMC step) if isinstance(mu_sigma, tuple): if len(mu_sigma) != 2: raise TypeError("If a tuple, mu_sigma must have 2 elements") elif mu_sigma is not None: xy = planets_xy_coord[index_planet]-center_xy_coord r0 = np.sqrt(xy[0]**2 + xy[1]**2) theta0 = np.mod(np.arctan2(xy[1], xy[0]) / np.pi*180, 360) mu_sigma = get_mu_and_sigma(cube, angs, ncomp, annulus_width, aperture_radius, fwhm, r0, theta0, cube_ref=cube_ref, wedge=wedge, svd_mode=svd_mode, scaling=scaling, algo=algo, delta_rot=delta_rot, imlib=imlib, interpolation=interpolation, collapse=collapse, weights=norm_weights, algo_options=algo_options) res_init = firstguess_from_coord(planets_xy_coord[index_planet], center_xy_coord, cube, angs, psfn, fwhm, annulus_width, aperture_radius, ncomp, f_range=f_range, cube_ref=cube_ref, svd_mode=svd_mode, scaling=scaling, fmerit=fmerit, imlib=imlib, collapse=collapse, algo=algo, delta_rot=delta_rot, interpolation=interpolation, algo_options=algo_options, transmission=transmission, mu_sigma=mu_sigma, weights=weights, plot=plot, verbose=verbose, save=save) r_pre = res_init[0] theta_pre = res_init[1] f_pre = res_init[2:] if verbose: msg3a = 'Planet {}: preliminary position guess: (r, theta)=({:.1f}, ' msg3a += '{:.1f})' print(msg3a.format(index_planet, r_pre, theta_pre)) msg3b = 'Planet {}: preliminary flux guess: '.format(index_planet) for z in range(len(f_pre)): msg3b += '{:.1f}'.format(f_pre[z]) if z < len(f_pre)-1: msg3b += ', ' print(msg3b) if simplex or force_rPA: if verbose: msg4 = 'Planet {}: Simplex Nelder-Mead minimization, ' msg4 += 'running ...' print(msg4.format(index_planet)) if simplex_options is None: simplex_options = {'xatol': 1e-6, 'fatol': 1e-6, 'maxiter': 800, 'maxfev': 2000} res = firstguess_simplex(res_init, cube, angs, psfn, ncomp, fwhm, annulus_width, aperture_radius, cube_ref=cube_ref, svd_mode=svd_mode, scaling=scaling, fmerit=fmerit, imlib=imlib, interpolation=interpolation, collapse=collapse, algo=algo, delta_rot=delta_rot, algo_options=algo_options, p_ini=p_ini, transmission=transmission, mu_sigma=mu_sigma, weights=weights, force_rPA=force_rPA, options=simplex_options, verbose=False) if force_rPA: r_0[index_planet], theta_0[index_planet] = (r_pre, theta_pre) f_0[index_planet], = res.x else: r_0[index_planet] = res.x[0] theta_0[index_planet] = res.x[1] if cube.ndim == 3: f_0[index_planet] = res.x[2] else: f_0[index_planet] = res.x[2:] if verbose: msg5 = 'Planet {}: Success: {}, nit: {}, nfev: {}, chi2r: {}' print(msg5.format(index_planet, res.success, res.nit, res.nfev, res.fun)) print('message: {}'.format(res.message)) else: if verbose: msg4bis = 'Planet {}: Simplex Nelder-Mead minimization skipped.' print(msg4bis.format(index_planet)) r_0[index_planet] = r_pre theta_0[index_planet] = theta_pre if cube.ndim == 3: f_0[index_planet] = f_pre[0] else: f_0[index_planet] = f_pre if verbose: centy, centx = frame_center(cube[0]) posy = r_0 * np.sin(np.deg2rad(theta_0[index_planet])) + centy posx = r_0 * np.cos(np.deg2rad(theta_0[index_planet])) + centx msg6 = 'Planet {}: simplex result: (r, theta, '.format(index_planet) if cube.ndim == 3: msg6 += 'f)=({:.3f}, {:.3f}, {:.3f})'.format(r_0[index_planet], theta_0[index_planet], f_0[index_planet]) else: msg6b = '(' for z in range(cube.shape[0]): msg6 += 'f{}'.format(z) msg6b += '{:.3f}'.format(f_0[index_planet, z]) if z < cube.shape[0]-1: msg6 += ', ' msg6b += ', ' msg6 += ')=' msg6b += ')' msg6 += msg6b msg6 += ' at \n (X,Y)=({:.2f}, {:.2f})'.format(posx[0], posy[0]) print(msg6) if verbose: print('\n', sep, '\nDONE !\n', sep) timing(start_time) return r_0, theta_0, f_0