#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Mon May 27 17:31:18 2019
@author: rfetick
"""
import numpy as np
from scipy.optimize import least_squares
from astropy.io import fits
import time
from numpy.fft import fft2, fftshift, ifft2
from functools import lru_cache
from paompy.config import _EPSILON
from paompy.utils import binning
#%% FITTING FUNCTION
def lsq_flux_bck(model, data, weights, background=True, positive_bck=False):
"""Compute the analytical least-square solution for flux and background
LS = SUM_pixels { weights*(flux*model + bck - data)² }
Parameters
----------
model: numpy.ndarray
data: numpy.ndarray
weights: numpy.ndarray
Keywords
--------
background: bool
Activate/inactivate background (activated by default:True)
positive_bck : bool
Makes background positive (default:False)
"""
ws = np.sum(weights)
mws = np.sum(model * weights)
mwds = np.sum(model * weights * data)
m2ws = np.sum(weights * (model ** 2))
wds = np.sum(weights * data)
if background:
delta = mws ** 2 - ws * m2ws
amp = 1. / delta * (mws * wds - ws * mwds)
bck = 1. / delta * (-m2ws * wds + mws * mwds)
else:
amp = mwds / m2ws
bck = 0.0
if bck<0 and positive_bck: #re-implement above equation
amp = mwds / m2ws
bck = 0.0
return amp, bck
def psffit(psf,Model,x0,weights=None,dxdy=(0.,0.),flux_bck=(True,True),
positive_bck=False,fixed=None,**kwargs):
"""Fit a PSF with a parametric model solving the least-square problem
epsilon(x) = SUM_pixel { weights * (amp * Model(x) + bck - psf)² }
Parameters
----------
psf : numpy.ndarray
The experimental image to be fitted
Model : class
The class representing the fitting model
x0 : tuple, list, numpy.ndarray
Initial guess for parameters
weights : numpy.ndarray
Least-square weighting matrix (same size as `psf`)
Default: uniform weighting
dxdy : tuple of two floats
Eventual guess on PSF shifting
flux_bck : tuple of two bool
Only background can be activate/inactivated
Flux is always activated (sorry!)
positive_bck : bool
Force background to be positive or null
fixed : numpy.ndarray
Fix some parameters to their initial value (default: None)
**kwargs :
All keywords used to instantiate your `Model`
Returns
-------
out.x : numpy.array
Parameters at optimum
.dxdy : tuple of 2 floats
PSF shift at optimum
.flux_bck : tuple of two floats
Estimated image flux and background
.psf : numpy.ndarray (dim=2)
Image of the PSF model at optimum
.success : bool
Minimization success
.status : int
Minimization status (see scipy doc)
.message : string
Human readable minimization status
.active_mask : numpy.array
Saturated bounds
.nfev : int
Number of function evaluations
.cost : float
Value of cost function at optimum
"""
Model_inst = Model(np.shape(psf),**kwargs)
if weights is None:
weights = np.ones_like(psf)
elif len(psf)!=len(weights):
raise ValueError("Keyword `weights` must have same number of elements as `psf`")
sqW = np.sqrt(weights)
class CostClass(object):
def __init__(self):
self.iter = 0
def __call__(self,y):
if (self.iter%3) == 0:
print("-",end="")
self.iter += 1
x, dxdy = mini2input(y)
m = Model_inst(x,dx=dxdy[0],dy=dxdy[1])
amp, bck = lsq_flux_bck(m, psf, weights, background=flux_bck[1], positive_bck=positive_bck)
return np.reshape(sqW * (amp * m + bck - psf), np.size(psf))
cost = CostClass()
if fixed is not None:
if len(fixed)!=len(x0):
raise ValueError("When defined, `fixed` must be same size as `x0`")
FREE = [not fixed[i] for i in range(len(fixed))]
INDFREE = np.where(FREE)[0]
def input2mini(x,dxdy):
# Transform user parameters to minimizer parameters
if fixed is None:
xfree = x
else:
xfree = np.take(x,INDFREE)
return np.concatenate((xfree,dxdy))
def mini2input(y):
# Transform minimizer parameters to user parameters
if fixed is None:
xall = y[0:-2]
else:
xall = np.copy(x0)
for i in range(len(y)-2):
xall[INDFREE[i]] = y[i]
return (xall,y[-2:])
def get_bound(inst):
b_low = inst.bounds[0]
if fixed is not None:
b_low = np.take(b_low,INDFREE)
b_low = np.concatenate((b_low,[-np.inf,-np.inf]))
b_up = inst.bounds[1]
if fixed is not None:
b_up = np.take(b_up,INDFREE)
b_up = np.concatenate((b_up,[np.inf,np.inf]))
return (b_low,b_up)
result = least_squares(cost, input2mini(x0,dxdy), bounds=get_bound(Model_inst))
print("") #finish line of "-"
result.x, result.dxdy = mini2input(result.x) #split output between x and dxdy
m = Model_inst(result.x,dx=result.dxdy[0],dy=result.dxdy[1])
amp, bck = lsq_flux_bck(m, psf, weights, background=flux_bck[1], positive_bck=positive_bck)
result.flux_bck = (amp,bck)
result.psf = m
return result
#%% CLASS PARAMETRIC PSF AND ITS SUBCLASSES
class ParametricPSF(object):
"""Super-class defining parametric PSFs
Not to be instantiated, only serves as a referent for subclasses
"""
def __init__(self,Npix):
"""
Parameters
----------
Npix : tuple of two elements
Model X and Y pixel size when called
"""
if type(Npix)!=tuple:
raise TypeError("Argument `Npix` must be a tuple")
self.Npix = Npix
self.bounds = (-np.inf,np.inf)
def __repr__(self):
return "ParametricPSF of size (%u,%u)"%self.Npix
def __call__(self,*args,**kwargs):
raise ValueError("ParametricPSF is not made to be instantiated. Better use its subclasses")
def otf(self,*args,**kwargs):
"""Return the Optical Transfer Function (OTF)"""
psf = self.__call__(args,kwargs)
return fftshift(fft2(psf))
def mtf(self,*args,**kwargs):
"""Return the Modulation Transfer Function (MTF)"""
return np.abs(self.otf(args,kwargs))
def tofits(self,param,filename,*args,keys=None,keys_comment=None,**kwargs):
psf = self.__call__(param,*args,**kwargs)
hdr = self._getfitshdr(param,keys=keys,keys_comment=keys_comment)
hdu = fits.PrimaryHDU(psf, hdr)
hdu.writeto(filename, overwrite=True)
def _getfitshdr(self,param,keys=None,keys_comment=None):
if keys is None:
keys = ["PARAM %u"%(i+1) for i in range(len(param))]
if len(keys)!=len(param):
raise ValueError("When defined, `keys` must be same size as `param`")
if keys_comment is not None:
if len(keys_comment)!=len(param):
raise ValueError("When defined, `keys_comment` must be same size as `param`")
hdr = fits.Header()
hdr["HIERARCH ORIGIN"] = "PAOMPY automatic header"
hdr["HIERARCH CREATION"] = (time.ctime(),"Date of file creation")
for i in range(len(param)):
if keys_comment is None:
hdr["HIERARCH PARAM "+keys[i]] = param[i]
else:
hdr["HIERARCH PARAM "+keys[i]] = (param[i],keys_comment[i])
return hdr
class ConstantPSF(ParametricPSF):
"""Create a constant PSF, given as a 2D image, using ParametricPSF formalism
With such a formalism, a constant PSF is just a particular case of a parametric PSF
"""
def __init__(self,image_psf):
super().__init__(np.shape(image_psf))
self.image_psf = image_psf
self.bounds = ()
def __call__(self,*args,**kwargs):
return self.image_psf
def moffat(XY, param, norm=None):
"""
Compute a Moffat function on a meshgrid
moff = A * Enorm * (1+u)^(-beta)
with `u` the quadratic coordinates in the shifted and rotated frame
Parameters
----------
XY : numpy.ndarray (dim=2)
The (X,Y) meshgrid with X = XY[0] and Y = XY[1]
param : list, tuple, numpy.ndarray (len=7)
param[0] - Amplitude
param[1] - Alpha X
param[2] - Alpha Y
param[3] - Theta
param[4] - Beta
param[5] - Center X
param[6] - Center Y
Keywords
--------
norm : None, np.inf, float (>0), int (>0)
Radius for energy normalization
None - No energy normalization
Enorm = 1.0
np.inf - Total energy normalization (on the whole X-Y plane)
Enorm = (beta-1)/(pi*ax*ay)
float,int - Energy normalization up to the radius defined by this value
Enorm = (beta-1)/(pi*ax*ay)*(1-(1+(R**2)/(ax*ay))**(1-beta))
Returns
-------
The 2D Moffat array
"""
if len(param)!=7:
raise ValueError("Parameter `param` must contain exactly 7 elements, but input has %u elements"%len(param))
c = np.cos(param[3])
s = np.sin(param[3])
s2 = np.sin(2.0 * param[3])
Rxx = (c / param[1]) ** 2 + (s / param[2]) ** 2
Ryy = (c / param[2]) ** 2 + (s / param[1]) ** 2
Rxy = s2 / param[2] ** 2 - s2 / param[1] ** 2
u = Rxx * (XY[0]-param[5])**2 + Rxy * (XY[0]-param[5]) * (XY[1]-param[6]) + Ryy * (XY[1]-param[6])**2
if norm is None:
Enorm = 1
elif norm == np.inf:
if param[4]<=1:
raise ValueError("Cannot compute Moffat energy for param[4]<=1")
Enorm = (param[4]-1) / (np.pi*param[1]*param[2])
else:
if param[4]==1:
raise ValueError("Energy computation for param[4]=1.0 not implemented yet. Sorry!")
Enorm = (param[4]-1) / (np.pi*param[1]*param[2])
Enorm = Enorm / (1 - (1 + (norm**2) / (param[1]*param[2]))**(1-param[4]))
return Enorm * param[0] * (1. + u) ** (-param[4])
class Moffat(ParametricPSF):
def __init__(self,Npix,norm=np.inf):
#super(ParametricPSF,self).__init__(Npix)
self.Npix = Npix
self.norm = norm
bounds_down = [_EPSILON,_EPSILON,-np.inf,1+_EPSILON]
bounds_up = [np.inf for i in range(4)]
self.bounds = (bounds_down,bounds_up)
@lru_cache(maxsize=5)
def _XY(self,Npix):
YX = np.mgrid[0:Npix[0],0:Npix[1]]
YX[1] = YX[1] - Npix[0]/2
YX[0] = YX[0] - Npix[1]/2
return YX
def __call__(self,x,dx=0,dy=0):
"""
Parameters
----------
x : list, tuple, numpy.ndarray (len=4)
x[0] - Alpha X
x[1] - Alpha Y
x[2] - Theta
x[3] - Beta
"""
y = np.concatenate(([1],x,[dx,dy]))
return moffat(self._XY(self.Npix),y,norm=self.norm)
def tofits(self,param,filename,*args,keys=["ALPHA_X","ALPHA_Y","THETA","BETA"],**kwargs):
super(Moffat,self).tofits(param,filename,*args,keys=keys,**kwargs)
class Psfao(ParametricPSF):
"""PSF model based on a parametrization of the phase PSD
See documentation of methods "__init__" and "__call__"
"""
def __init__(self,Npix,system=None,Lext=10.,samp=None,symmetric=False,diffotf=True):
"""
Parameters
----------
Npix : tuple
Size of output PSF
system : OpticalSystem
Optical system for this PSF
samp : float
Sampling at the observation wavelength
Lext : float
Von-Karman external scale (default = 10 m)
Useless if Fao >> 1/Lext
diffotf : bool
Enable/disable diffraction OTF for PSF computation
(default=True)
"""
#super(ParametricPSF,self).__init__(Npix)
if not (type(Npix) in [tuple,list,np.ndarray]):
raise ValueError("Npix must be a tuple, list or numpy.ndarray")
if len(Npix)!=2:
raise ValueError("Npix must be of length = 2")
if (Npix[0]%2) or (Npix[1]%2):
raise ValueError("Each Npix component must be even")
self.Npix = Npix
if system is None:
raise ValueError("Keyword `system` must be defined")
if samp is None:
raise ValueError("Keyword `samp` must be defined")
self.system = system
self.Lext = Lext
self.samp = samp
self.symmetric = symmetric
self.diffotf = diffotf
@property
def symmetric(self):
return self._symmetric
@symmetric.setter
def symmetric(self,value):
self._symmetric = value
if not value:
bounds_down = [_EPSILON,0,_EPSILON,_EPSILON,_EPSILON,-np.inf,1+_EPSILON]
bounds_up = [np.inf for i in range(7)]
else:
bounds_down = [_EPSILON,0,_EPSILON,_EPSILON,1+_EPSILON]
bounds_up = [np.inf for i in range(5)]
self.bounds = (bounds_down,bounds_up)
@property
def samp(self):
return self._samp
@samp.setter
def samp(self,value):
# Manage cases of undersampling
self._samp = value
if value >=2:
self._samp_num = value
self._k = 1
else:
self._k = int(np.ceil(2.0/value))
self._samp_num = self._k * value
@lru_cache(maxsize=2)
def _freq_array(self,Nx,Ny,samp,D):
"""
Returns
-------
tab - numpy.array (dim=3)
3D array of frequencies [1/m]
"""
pix2freq = 1.0/(D*samp)
f2D = np.mgrid[0:Nx, 0:Ny].astype(float)
#null frequency at [Nx//2,Ny//2] according to numpy fft convention
f2D[0] -= Nx//2
f2D[1] -= Ny//2
return f2D * pix2freq
@lru_cache(maxsize=2)
def _shift_array(self,Nx,Ny):
Y, X = np.mgrid[0:Nx,0:Ny].astype(float)
X = (X-Nx/2) * 2*np.pi*1j/Nx
Y = (Y-Ny/2) * 2*np.pi*1j/Ny
return X, Y
@lru_cache(maxsize=5)
def _dlFTO(self,Nx,Ny,pupfct,samp):
# samp as a tuple is not ready yet, don't use it
if type(samp)==tuple:
dlFTO = np.zeros((Nx,Ny,len(samp)))
for i in range(len(samp)):
dlFTO[...,i] = self._dlFTO(Nx,Ny,pupfct, samp[i])
return np.mean(dlFTO,axis=2)
NpupX = np.ceil(Nx/samp)
NpupY = np.ceil(Ny/samp)
tab = np.zeros((Nx, Ny), dtype=np.complex)
tab[0:int(NpupX), 0:int(NpupY)] = pupfct((NpupX,NpupY),samp=samp)
return fftshift(abs(ifft2(abs(fft2(tab)) ** 2)) / np.sum(tab))
def psd(self,x0):
"""Compute the PSD model from parameters
PSD is given in [rad²/f²] = [rad² m²]
Parameters
----------
x0 : numpy.array (dim=1), tuple, list
See __call__ for more details
Returns
-------
psd : numpy.array (dim=2)
"""
if len(x0)==7:
x = x0
elif len(x0)==5:
x = np.concatenate((x0[0:4],[x0[3],0],x0[4:]))
else:
raise ValueError("Wrong size of x0")
f2D = self._freq_array(self.Npix[0]*self._k,self.Npix[1]*self._k,self._samp_num,self.system.D)
F2 = f2D[0] ** 2. + f2D[1] ** 2.
Fao = self.system.Nact/(2.0*self.system.D)
PSD = 0.0229* x[0]**(-5./3.) * ((1. / self.Lext**2.) + F2)**(-11./6.)
PSD *= (F2 >= Fao**2.)
param = np.concatenate((x[2:],[0,0]))
PSD += (F2 < Fao**2.) * np.abs(x[1] + moffat(f2D,param,norm=Fao))
# Set PSD = 0 at null frequency (according to numpy fft convention)
PSD[self.Npix[0]//2,self.Npix[1]//2] = 0.0
return PSD
def otf(self,x0,dx=0,dy=0,_caller='user'):
"""
See __call__ for input arguments
Warning: result of otf will be unconsistent if undersampled!!!
This issue is solved with oversampling + binning in __call__ but not here
For the moment, the `_caller` keyword prevents user to misuse otf
"""
if (self._k > 1) and (_caller != 'self'):
raise ValueError("Cannot call `Psfao.otf(...)` when undersampled (functionality not implemented yet)")
PSD = self.psd(x0)
L = self.system.D * self._samp_num
Bg = fft2(fftshift(PSD)) / L ** 2
Dphi = fftshift(np.real(2 * (Bg[0, 0] - Bg)))
if self.diffotf:
dlFTO = self._dlFTO(self.Npix[0]*self._k,self.Npix[1]*self._k,
self.system.pupil, self._samp_num)
else:
dlFTO = 1.
X, Y = self._shift_array(self.Npix[0]*self._k,self.Npix[1]*self._k)
return np.exp(-Dphi/2.)*dlFTO*np.exp(X*dx + Y*dy)
def __call__(self,x0,dx=0,dy=0):
"""
Parameters
----------
x0 : numpy.array (dim=1), tuple, list
x[0] - Fried parameter r0 [m]
x[1] - PSD corrected area background C [rad² m²]
x[2] - PSD corrected area phase variance A [rad²]
x[3] - PSD alpha X [1/m]
x[4] - PSD alpha Y [1/m] (not defined in symmetric case)
x[5] - PSD theta [rad] (not defined in symmetric case)
x[6] - PSD beta power law (becomes x[4] in symmetric case)
dx : float
PSF X shifting [pix] (default = 0)
dy : float
PSF Y shifting [pix] (default = 0)
Returns
-------
tab : numpy.ndarray (dim=2)
The PSF computed for the given parameters
Note
----
The PSD integral on the corrected area is x[2]+x[1]*PI*fao²
"""
out = np.real(fftshift(ifft2(fftshift(self.otf(x0,dx=dx,dy=dy,_caller='self')))))
out = out/out.sum() # ensure unit energy on the field of view
if self._k==1:
return out
else:
return binning(out,int(self._k))
def tofits(self,param,filename,*args,keys=None,**kwargs):
if keys is None:
if len(param)==5:
keys = ["R0","CST","SIGMA2","ALPHA","BETA"]
keys_comment = ["Fried parameter [m]",
"PSD AO area constant C [rad2]",
"PSD AO area Moffat variance A [rad2]",
"PSD AO area Moffat alpha [1/m]",
"PSD AO area Moffat beta"]
else: # if not 5, then equals 7
keys = ["R0","CST","SIGMA2","ALPHA_X","ALPHA_Y","THETA","BETA"]
keys_comment = ["Fried parameter [m]",
"PSD AO area constant C [rad2]",
"PSD AO area Moffat variance A [rad2]",
"PSD AO area Moffat alpha X [1/m]",
"PSD AO area Moffat alpha Y [1/m]",
"PSD AO area Moffat theta [rad]",
"PSD AO area Moffat beta"]
# redefine tofits() because extra hdr is required
psf = self.__call__(param,*args,**kwargs)
hdr = self._getfitshdr(param,keys=keys,keys_comment=keys_comment)
hdr["HIERARCH SYSTEM"] = (self.system.name,"System name")
hdr["HIERARCH SAMP"] = (self.samp,"Sampling (eg. 2 for Shannon)")
hdr["HIERARCH LEXT"] = (self.Lext,"Von-Karman outer scale")
hdr["HIERARCH DIFFOTF"] = (self.diffotf,"Is diffraction OTF enabled")
hdu = fits.PrimaryHDU(psf, hdr)
hdu.writeto(filename, overwrite=True)