Add Psfao model

paompy/instrument.py

@@ -55,6 +55,8 @@ class Instrument(object):
         self.ron = ron
         self.binning = 1
         
+        self.name = "(unamed)"
+        
     def __repr__(self):
         s  = "PAOMPY Instrument\n"
         s += "----------------------------\n" 
@@ -111,6 +113,8 @@ class Instrument(object):
 ZIMPOL = Instrument(D=8.,occ=0.14,res=30*1e-6/1768.,gain=10.5,ron=20.,Nact=40)
 ZIMPOL.filters["V"] = (554*1e-9, 80.6*1e-9)
 ZIMPOL.filters["N_R"] = (645.9*1e-9, 56.7*1e-9)
+ZIMPOL.name = "VLT SPHERE/ZIMPOL"
 
 
 MUSE = Instrument(D=8.,occ=0.14,res=237.15*1e-6/1980.,gain=5.,ron=15.,Nact=39)
+MUSE.name = "VLT MUSE"

paompy/psfmodel.py

@@ -6,3 +6,573 @@ 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)