"""Helper function for optimize_tilt."""

#The Jacobian, i.e. the derivatives of the model with respect to tilt

#complex_jac = [uv[0]*mod_ft*np.exp(1j*(p[0]*uv[0] + p[1]*uv[1])),\

# uv[1]*mod_ft*np.exp(1j*(p[0]*uv[1] + p[1]*uv[1]))]

#retarray = np.array([complex_jac[0].imag,-complex_jac[0].real,complex_jac[1].imag,-complex_jac[1].real])

complex_jac = [1j*uv[0]*mod_ft*np.exp(1j*(p[0]*uv[0] + p[1]*uv[1])),\

1j*uv[1]*mod_ft*np.exp(1j*(p[0]*uv[1] + p[1]*uv[1]))]

retarray = np.array([complex_jac[0].real,complex_jac[0].imag,complex_jac[1].real,complex_jac[1].imag])

"""Helper function for optimize_tilt. This is used as a function to

input into leastsq

Parameters

----------

p: numpy array (2): [xtilt,ytilt]

The wavefront tilt image-plane pixels.

mod_ft: array

Model Fourier transform that we want to fit to the image

im_ft: array

Image Fourier transform

uv: array

Sampling points for mod_ft and im_ft

Returns

-------

resid: numpy array

Array of residuals to the fit.

"""

new_model = mod_ft*np.exp(1j*(p[0]*uv[0] + p[1]*uv[1]))

if return_model:

return new_model

retval_complex = im_ft - new_model

"""Given a PSF and an image Fourier transform sampled at points u and

v, tilt and scale the model so that it matches the image.

We do this fitting in the Fourier domain rather than the image-plane, because

sub-pixel tilts can be more precise in the Fourier domain. A shift is a convolution

with a delta-function centered at the pixel (xshift, yshift). So we can shift

by multiplying the Fourier transform of the image by the Fourier transform of this

shift operator, which we can call the shift kernel.

Parameters

----------

u,v: float array

Cycles per pixel times 2 np.pi.

Returns

-------

The shifted model Fourier transform."""

if scale_flux:

im_ft_out = im_ft/im_ft[0]

else:

im_ft_out = im_ft

#retvals = op.leastsq(optimize_tilt_function, [0,0], args=(mod_ft,im_ft_out,uv), \

# Dfun=optimize_tilt_derivative, col_deriv=True, xtol=1e-3, ftol=1e-6)

retvals = op.leastsq(optimize_tilt_function, [0,0], args=(mod_ft,im_ft_out,uv), \

xtol=1e-4, ftol=1e-7)

if check_fit:

pdb.set_trace()

if retvals[1] <= 0:

print("Error in finding tilts!")

raise UserWarning

else:

"""A helper function to shift an image, optimize its tilt and subtract the

background.

Parameters

----------

im: numpy array

Image that we want to prepare.

ref_ft: numpy array

Fourier transform of the reference image, that defines a "centered" position.

uv, sampled_uv: numpy array

The (u,v) coordinates and pixel values in the UV plane.

corner_pix: numpy array

The image corner pixels, which defined the background.

center_ft: (optional) bool

Do we return a Fourier transform with sub-pixel sampling?

Returns

-------

(a_psf, a_psf_ft):

Roughly centered image, and precisely centered imaged Fourier transform (unless

center_ft is False)

"""

sz = im.shape[0]

maxpix = np.unravel_index(np.argmax(im), im.shape)

a_psf = np.roll(np.roll(im, sz//2 - maxpix[0], axis=0), sz//2 - maxpix[1],axis=1)

a_psf -= np.median(a_psf[corner_pix])

if scale == 1.0:

mod_ft = np.fft.rfft2(a_psf)[sampled_uv]

else:

if scale > 1:

scaled_psf = zoom(a_psf, scale)[a_psf.shape[0]//2-sz//2:a_psf.shape[0]//2+sz//2,\

a_psf.shape[1]//2-sz//2:a_psf.shape[1]//2+sz//2]

else:

scaled_psf = np.zeros_like(a_psf)

scaled_psf[sz//2-a_psf.shape[0]//2:sz//2+a_psf.shape[0]//2,\

sz//2-a_psf.shape[1]//2:sz//2+a_psf.shape[1]//2] = zoom(a_psf, scale)

mod_ft = np.fft.rfft2(scaled_psf)[sampled_uv]

if center_ft:

tilt = optimize_tilt(mod_ft/mod_ft[0], ref_ft/ref_ft[0], uv)

a_psf_ft = optimize_tilt_function(tilt, mod_ft, ref_ft, uv, return_model=True)

else:

a_psf_ft = mod_ft

"""A set of reference PSFs, which creates the space to marginalise over"""

def __init__(self, psfs=[], psf_files=[], wave=3.5e-6, diam=10.0,pscale=0.01, \

cubefile=None, cube_extn=1, hyperparams=[], subtract_outer_median=True, scale=1.0):

"""Initialise the reference PSFs. This includes reading them in, cleaning

and shifting to the origin. Cleaning here includes Fourier filtering, and

shifting to the origin is done in a least squares sense, i.e. a sub-pixel shift

that refers PSFs to a master (i.e. mean) PSF.

Given that the PSFs have limited support in the Fourier domain, we will store them

as complex Fourier component vectors on this support. Then the process of fitting

to a linear combination of PSFs is just making a linear combination on this support.

Note that the (uv) co-ordinates are stored in a way most convenient for the

tilt_function, going from 0 to 2pi over the full Fourier domain.

Parameters

----------

wave: float

Wavelength in m

diam: float

Telescope diameter in m

pscale: float

Pixel scale in arcsec.

"""

self.ndim = 0 #0 until we embed the PSFs in a lower dimensional space!

self.use_this_psf = 0 #i.e. just use the first PSF until we're told otherwise.

if cubefile:

psfs = pyfits.getdata(cubefile,cube_extn)

else:

print("Not implemented quite yet...")

raise UserWarning

sz = psfs.shape[1]

self.sz = sz

self.npsfs = len(psfs)

outer_pix = (1-ot.circle(sz,2*sz/3)) > 0

for i in range(len(psfs)):

if subtract_outer_median:

psfs[i] -= np.median(psfs[i][outer_pix])

psfs[i] /= np.sum(psfs[i])

uv = np.meshgrid(2*np.pi*np.arange(sz//2 + 1)/float(sz),

2*np.pi*(((np.arange(sz) + sz//2) % sz) - sz//2)/float(sz))

#A variable that is 2*pi for 1 cycle per pixel.

rr = np.sqrt(uv[0]**2 + uv[1]**2)

sampled_uv = np.where(rr < 2*np.pi*diam/wave*np.radians(pscale/3600.))

#While sampled_uv is an integer array of uv pixel coordinates, uv is an

#array of Fourier frequency in inverse pixel units

self.sampled_uv = sampled_uv

self.uv = np.array([uv[0][sampled_uv],uv[1][sampled_uv]])

psf_mn = np.sum(psfs,0)/psfs.shape[0]

psf_mn_ft = np.fft.rfft2(psf_mn)[sampled_uv]

psf_fts = []

#NB This should probably be run twice - once to get a better psf_mn_ft.

corner_pix = np.where(1 - ot.circle(self.sz, self.sz))

for i in range(len(psfs)):

centered_psf, a_psf_ft = prepare_im(psfs[i], psf_mn_ft, self.uv, self.sampled_uv, corner_pix, scale=scale)

psf_fts.append(a_psf_ft)

self.psf_fts = np.array(psf_fts)

self.psf_fts_vect = np.array([np.append(psf_ft.real, psf_ft.imag) for psf_ft in psf_fts])

self.psf_mn_ft = psf_mn_ft

def psf_im(self,ix):

"""Helper function to return a point-spread function from a library MTF

Parameters

----------

ix: int

Index of the point spread function to return

"""

if (ix >= len(self.psf_fts)):

print("ERROR: index out of range")

raise UserWarning

else:

return self.im_from_ft(self.psf_fts[ix])

def im_from_ft(self,im_ft_sampled):

"""Return a full image based on the subsampled Fourier plane."""

im_ft = np.zeros( (self.sz,self.sz//2+1), dtype=np.complex)

im_ft[self.sampled_uv] = im_ft_sampled

return np.fft.irfft2(im_ft)

def lle(self,ndim=2,nk=None, length_nsigma=2.0):

"""Embed the PSFs onto an ndim dimensional space

Parameters

----------

ndim: int

Number of LLE dimensions

nk: int

Number of nearest neighbors to check.

"""

self.ndim=ndim

#The following

if not nk:

nk = int(1.5*np.sqrt(self.psf_fts_vect.shape[0]))

self.nk = nk

self.lle_proj = mdp.nodes.LLENode(nk,output_dim=ndim,verbose=True)(self.psf_fts_vect)

lengths = np.empty(ndim)

for d in range(ndim):

one_sigmas = np.percentile(self.lle_proj[:,d],[16,84])

lengths[d] = length_nsigma*(one_sigmas[1]-one_sigmas[0])

print("Axis lengths: " + str(lengths) )

self.h_density = (np.prod(lengths)/self.lle_proj.shape[0])**(1.0/ndim)

self.tri = Delaunay(self.lle_proj)

self.point_lengths = np.sum(self.tri.points**2,1)

def display_lle_space(self, nsamp=100, return_density=False):

"""Display the space of PSFs as a density in LLE space plus the points

from which it was constructed.

Parameters

----------

nsamp: int (optional)

number of samples in each dimension in the 2D image

Returns

-------

density: numpy array

The probability density that is plotted, if return_dentiy=True.

extent: numpy array

The extent of the plot, if return_dentiy=True.

"""

if self.ndim != 2:

print("A 2D display only works for ndim=2")

sz = self.sz #Short-hand

uv = np.meshgrid(2*np.pi*np.arange(sz//2 + 1)/float(sz),

2*np.pi*(((np.arange(sz) + sz//2) % sz) - sz//2)/float(sz))

x=np.linspace(-0.5,0.5,nsamp)

density = np.empty( (nsamp, nsamp) )

xy = np.meshgrid(x,x)

for i in range(nsamp):

for j in range(nsamp):

density[i,j] = self.lle_density([xy[0][i,j],xy[1][i,j]])

extent = [-0.5,0.5,-0.5,0.5]

density = density[::-1,:]

if return_density:

return density, extent

else:

plt.clf()

plt.imshow(density, extent=extent, cmap=cm.gray)

plt.plot(self.tri.points[:,0], self.tri.points[:,1], '.')

plt.axis(extent)

def augment_zernike(self,naugment=3, amps=np.ones(7)*0.1):

"""Augment the library of reference PSFs by adding zernike's to them

(neglecting tilt)"""

def find_lle_psf(self,x, return_image=True):

"""Return the unique interpolated PSF from ndim+1 library PSFs, by

interpolating within the smallest enclosing simplex where all

angles. If outside the convex hull, find the nearest edge/faces and

the simplex that includes one of these and (if possible) extends the furthest

in the opposite direction. """

#If we havnen't embedded out PSFs into some abstract space, this is simple!

if self.ndim==0:

if return_image:

return self.psf_im(self.use_this_psf)

else:

return self.psf_fts[self.use_this_psf]

#Otherwise, we have to find nearby LLE co-ordinates (an enclosing simplex) and

#interpolate between PSFs.

x = np.array(x)

enclosing_simplex = self.tri.find_simplex(x)

if enclosing_simplex<0:

#Distances between x and the points

dists = self.point_lengths - 2*np.dot(self.tri.points,x) + np.sum(x**2)

nearest = np.argmin(dists)

possible_simplices = np.where(np.sum(self.tri.simplices==nearest,axis=1))[0]

#Given a simplex and reference vertex r we can find c such that.

# T . c = x-v2

#... then x = v2 + c0*(v0-v2) + c1*(v1-v2)

# = c0*v0 + c1*v1 + (1-c0-c1)*v2

min_coeffs=[]

for simplex in possible_simplices:

coeffs = np.dot(self.tri.transform[simplex][:self.ndim,:self.ndim], \

x - self.tri.transform[simplex][-1])

coeffs = np.append(coeffs, 1-np.sum(coeffs))

min_coeffs.append(np.min(coeffs))

#The best simplex is the one with the least negative coefficient.

simplex = possible_simplices[np.argmax(min_coeffs)]

else:

simplex = enclosing_simplex

#Now that we know which simplex to use, get the coefficients and find the PSF

coeffs = np.dot(self.tri.transform[simplex][:self.ndim,:self.ndim], \

x - self.tri.transform[simplex][-1])

coeffs = np.append(coeffs, 1-np.sum(coeffs))

interp_psf_ft = np.dot(coeffs,self.psf_fts[self.tri.simplices[simplex]])

if return_image:

return self.im_from_ft(interp_psf_ft)

else:

return interp_psf_ft

def trunc_gauss(self, q):

""" Compute the truncated Gaussian probability density"""

wl = np.where( (q>0) * (q<0.5) )[0]

wh = np.where( (q>0.5) * (q<1) )[0]

the_sum = (np.sum(1-6*q[wl]**2+6*q[wl]**3) + \

np.sum(2*(1-q[wh])**3))

if self.ndim==2:

return 40/np.pi/7*the_sum

else:

return 8/np.pi*the_sum

def lle_density(self,x):

"""Return the local probability density of a given LLE coordinate

Normalise to a total integral of 1.0 over all LLE parameter space."""

#Brute force here... KDTree will help as we only need to consider

#12-ish nearest neighbors in 2D and 33-ish nearest neighbors in 3D.

if self.ndim==0:

print("Density zero - must compute the LLE first!")

return 0

dists = np.array([np.sqrt(np.sum((x - y)**2)) for y in self.lle_proj])

ww = np.where(dists < 2*self.h_density)[0]

if len(ww)==0:

return 0.0

else:

return self.trunc_gauss(dists[ww]/2.0/self.h_density)/len(self.lle_proj)/(2*self.h_density)**2

def mcmc_explore(self,niter=30,stepsize=0.5):

"""Explore the space of PSFs."""

current_pos = self.lle_proj[0]

current_density = self.lle_density(current_pos)

print("Computing overall background density for plotting...")

density, extent = self.display_lle_space(return_density=True)

for i in range(niter):

plt.clf()

plt.subplot(121)

jump = np.random.normal(size=self.ndim)*stepsize*self.h_density

trial = current_pos + jump

new_density = self.lle_density(trial)

if new_density/current_density > np.random.random():

current_density = new_density

current_pos = trial

psf = self.find_lle_psf(current_pos, return_image=True)

plt.imshow(np.arcsinh(psf/np.max(psf)/0.01),interpolation='nearest', cmap=cm.cubehelix)

plt.title("lnprob: {0:5.1f}".format(current_density))

plt.subplot(122)

plt.imshow(density,extent=extent,cmap=cm.gray)

plt.plot(self.tri.points[:,0],self.tri.points[:,1],'b.')

plt.title("Pos: {0:5.2f} {1:5.2f}".format(current_pos[0], current_pos[1]))

plt.plot(current_pos[0], current_pos[1], 'ro')

plt.axis(extent)

plt.draw()

#!!! Problem: Current matplotlib does not draw here. !!!

dummy = plt.ginput(1)

return None

def hyperparam_prob(self,x, hyperparams=None):

"""Return the hyperparameter probability for a given set of LLE coordinates

and hyperparameters. Uses the same density kernel as lle_density."""

def __init__(self,initp = []):

"""A model for the object on sky, consisting of a single point source.

Other objects can inherit this. Generally, there will be some fixed parameters

and some variable parameters. The model parameters are *not* imaging parameters,

i.e. do not include x, y, flux variables.

"""

self.p = initp

self.np = len(initp)

def model_uv(self, p_in, uv):

"""Return a model of the Fourier transform of the object given a set of

points in the uv plane

Parameters

----------

p_in: array-like

model parameters. Can be None if if the model has no parameters!

uv: array-like

Coordinates in the uv plane """

def __init__(self,initp = [], infits=''):

"""A model for the object on sky, consisting of an input fits files.

Parameters

----------

initp: array-like

Unused: The input parameters

infits: string

A filename for an input model fits files.

"""

if len(infits)==0:

raise ValueError("Must set keyword infits to a filename!")

self.p = initp

self.np = len(initp)

#Read in the fits file.

im = pyfits.getdata(infits)

if im.shape[0] != im.shape[1]:

raise ValueError("Model Image must be square")

#Take the Fourier transform and make sure we have coordinate arrays ready

#for interpolation

self.mod_ft = np.fft.fftshift(np.fft.fft2(np.fft.fftshift(im)))

uv_coord = 2*np.pi*(np.arange(im.shape[0]) - im.shape[0]//2)/float(im.shape[0])

#x = np.arange(im.shape[0]) - im.shape[0]//2 #XXX

#y = np.arange(im.shape[1]) - im.shape[1]//2

#self.mod_ft_func = RectBivariateSpline(uv_coord, uv_coord, self.mod_ft, kx=1, ky=1)

self.mod_ft_realfunc = RectBivariateSpline(uv_coord, uv_coord, self.mod_ft.real, kx=1, ky=1)

self.mod_ft_imagfunc = RectBivariateSpline(uv_coord, uv_coord, self.mod_ft.imag, kx=1, ky=1)

#self.yx = np.meshgrid(x, y)

def model_uv(self, p_in, uv):

"""Return a model of the Fourier transform of the object given a set of

points in the uv plane

Parameters

----------

p_in: array-like

model parameters. Can be None if if the model has no parameters!

uv: array-like

Coordinates in the uv plane """

ret_array = self.mod_ft_realfunc(uv[0], uv[1], grid=False).astype(np.complex)

ret_array += 1j*self.mod_ft_imagfunc(uv[0], uv[1], grid=False)

def __init__(self,initp = [], resid_in=None, psf_in=None):

"""A model for the object on sky, consisting of a point source and a

map that has been convolved with the PSF map.

The idea is that, iteratively, the fit residuals can be added to to the input

residuals to last model residuals, and the final problem is a standard

deconvolution problem with a known PSF.

Parameters

----------

initp: array-like

A single parameter, the relative flux of the resolved part of the image.

resid_in: numpy array

Residuals from the previous iteration. Same size and format as the input

image, but with N down and E left when displayed with imshow.

psf_in: numpy array

The PSF that should be used for the residuals, weighted in the same way.

"""

self.p = initp

self.np = len(initp)

#Normalise the input image and PSF

im = resid_in.copy()

im /= np.sum(im)

mean_psf = psf_in.copy()

mean_psf /= np.sum(mean_psf)

#Error checking

if im.shape[0] != im.shape[1]:

raise ValueError("Model Image must be square")

if im.shape != mean_psf.shape:

raise ValueError("PSF must have the same shape as input residuals.")

#Take the Fourier transform and make sure we have coordinate arrays ready

#for interpolation. The line below could have a divide by zero - but not where

#the Fourier transform has non-zero support.

self.mod_ft = np.fft.fftshift(np.fft.fft2(np.fft.fftshift(im))) / \

np.fft.fftshift(np.fft.fft2(np.fft.fftshift(mean_psf)))

uv_coord = 2*np.pi*(np.arange(im.shape[0]) - im.shape[0]//2)/float(im.shape[0])

self.mod_ft_realfunc = RectBivariateSpline(uv_coord, uv_coord, self.mod_ft.real, kx=1, ky=1)

self.mod_ft_imagfunc = RectBivariateSpline(uv_coord, uv_coord, self.mod_ft.imag, kx=1, ky=1)

def model_uv(self, p_in, uv):

"""Return a model of the Fourier transform of the object given a set of

points in the uv plane

Parameters

----------

p_in: array-like

model parameters. Can be None if if the model has no parameters!

uv: array-like

Coordinates in the uv plane """

ret_array = self.mod_ft_realfunc(uv[0], uv[1], grid=False).astype(np.complex)

ret_array += 1j*self.mod_ft_imagfunc(uv[0], uv[1], grid=False)

def __init__(self, initp=[]):

"""A Model with two point-sources

Parameters

----------

init_p: numpy array(3)

North Separation in pix, East separation in pix, contrast secondary/primary

"""

super(BinaryObject, self).__init__(initp)

def model_uv(self, p_in, uv):

"""Return a model of the Fourier transform of a binary given a set of points

in the uv plane

Parameters

----------

p_in: array-like

North Separation in pix, East Separation in pix, Contrast secondary/primary

uv: array-like

Coordinates in the uv plane """

#The Fourier transform of 2 delta functions is the sum of the Fourier transform

#of each delta function. Lets make the primary star be at co-ordinate (0,0)

ft_object = np.ones(uv.shape[1]) + p_in[2]*np.exp(1j*(p_in[0]*uv[0] + p_in[1]*uv[1]))

ft_object /= 1+p_in[2]

"""A set of target images"""

def __init__(self,psfs,object_model,ims=[], im_files=[],hyperparams=[],

cubefile=None, cube_extn=0, pas_extn=2, pas=[], gain=4.0):

"""Initialise the reference PSFs. This includes reading them in, cleaning,

shifting to the origin, normalising, creating variance arrays, and chopping

out the uv component.

Parameters

----------

psfs: Psfs instance

The PSF library to go with the target

object: PtsrcObject instance

The object model

ims: float numpy array

A set of target images

Notes

-----

The input gain should come from pynrm!!! (or better: be set to 1 by scaling)

"""

self.psfs= psfs

self.object = object_model

if cubefile:

ims = pyfits.getdata(cubefile,cube_extn)

self.pas = pyfits.getdata(cubefile,pas_extn)['pa']

else:

print("Not implemented quite yet...")

raise UserWarning

self.n_ims = len(ims)

self.sz = ims.shape[1]

if self.sz != self.psfs.sz:

print("Error: PSFs and target images must be the same size")

raise UserWarning

self.tgt_uv = np.empty( (self.n_ims, self.psfs.uv.shape[0], self.psfs.uv.shape[1]) )

#!!! NB Sign of rotation not checked below !!!

for i in range(self.n_ims):

self.tgt_uv[i,0] = self.psfs.uv[0]*np.cos(np.radians(self.pas[i])) + \

self.psfs.uv[1]*np.sin(np.radians(self.pas[i]))

self.tgt_uv[i,1] = self.psfs.uv[1]*np.cos(np.radians(self.pas[i])) - \

self.psfs.uv[0]*np.sin(np.radians(self.pas[i]))

self.corner_pix = np.where(1 - ot.circle(self.sz, self.sz))

self.read_var = 0.0

#Assumption: all target images have the same readout and/or background

#variance !!!

#!!! Bad pixels have to be added in separately here !!!

#!!! There should be read noise and variance from pynrm !!!

im_fts = []

for i in range(len(ims)):

centered_im, a_psf_ft = prepare_im(ims[i], self.psfs.psf_mn_ft, \

self.psfs.uv, self.psfs.sampled_uv, self.corner_pix, center_ft = False)

ims[i] = centered_im

self.read_var += np.var(ims[i][self.corner_pix])

im_fts.append(a_psf_ft)

self.ims = ims

self.im_fts = np.array(im_fts)

self.read_var /= len(ims)

self.ivar = 1.0/(self.read_var + ims/gain)

def lnprob(self, x, tgt_use=[], return_mod_ims=False, return_mod_psfs=False):

"""Compute the log probability of a model.

Parameters

----------

x: numpy array

The input LLE coordinates, followed by the model parameters, in the order:

[lle[0],lle[1],,,,lle[len(tgt_use)-1],p_in[0],,,,p_in[n_params-1], where

each of lle[0] etc is a list of length psfs.ndim.

p_use: numpy int array (optional)

The list of model parameters to use, e.g. [0,1,3], in case we want to fix

some of them.

tgt_use: numpy int array (optional)

The list of target PSFs to fit to, in case we don't want to fit to all of them.

return_mod_ims: bool (optional)

Optionally return the model images rather than the log probability.

"""

x = np.array(x)

#If tgt_use is not given, use all targets.

if len(tgt_use)==0:

tgt_use = np.arange(self.n_ims)

if self.psfs.ndim > 0:

x_lle = x[:self.psfs.ndim * len(tgt_use)].reshape( (len(tgt_use), self.psfs.ndim) )

else:

x_lle=[]

x_p = x[self.psfs.ndim * self.psfs.npsfs:]

prior_prob=1.0

chi2 = 0.0

mod_ims = []

mod_psfs = []

#Loop through the image and add to the chi-squared

for i in range(len(tgt_use)):

prior_prob *= self.psfs.hyperparam_prob(x_lle[i])

#What is our object model?

obj_ft = self.object.model_uv(x_p, self.tgt_uv[tgt_use[i]])

#Convolve the object with the PSF model to form an image model.

psf_ft = self.psfs.find_lle_psf(x_lle[i], return_image=False)

if return_mod_psfs:

mod_psfs.append(self.psfs.im_from_ft(psf_ft))

mod_ft = obj_ft * psf_ft

#Find the tilt that best matches the image model, and form an image model,

#and scale the image to match the total flux.

scale_factor = self.im_fts[tgt_use[i]][0].real/mod_ft[0].real

tilt = optimize_tilt(mod_ft, self.im_fts[tgt_use[i]]/scale_factor, self.psfs.uv)#, check_fit=True)

mod_ft = optimize_tilt_function(tilt, mod_ft*scale_factor, self.im_fts[i], self.psfs.uv, return_model=True)

mod_im = self.psfs.im_from_ft(mod_ft)

#Do we want to return the image?

if return_mod_ims:

mod_ims.append(mod_im)

#Compute chi-squared

chi2 += np.sum((mod_im - self.ims[tgt_use[i]])**2*self.ivar[tgt_use[i]])

#Returning multiple things is a little messy, but it saves code duplication, or

#un-necessary computation.

if return_mod_ims and return_mod_psfs:

return np.array(mod_ims), np.array(mod_psfs)

elif return_mod_ims:

return np.array(mod_ims)

elif return_mod_psfs:

return np.array(mod_psfs)

if prior_prob==0:

return -np.inf

else:

return np.log(prior_prob) - chi2/2.0

def lle_simplex_interp(self,x):

"""!!!What is this ???"""

return None

def marginalise(self,init_par=[],walker_sdev=[],nchain=1000, use_threads=True, start_one_at_a_time=True):

"""Use the affine invariant Monte-Carlo Markov chain technique to marginalise

over all PSFs. We cheat a little by not marginalising over the model parameters

simultaneously - the parameters are expected to have Gaussian errors

that come out of a least squares process that fits to PSFs from a point source fit

(at least this is what I think I meant).

WARNING: This doesn't actually marginalise over the model parameters yet, it only

marginalises over the LLE parameters for the point source model.

"""

if len(init_par) != len(walker_sdev):

raise UserWarning("Require same number of parameters (init_par) as walker standard deviations (walker_sdev)!")

threads = multiprocessing.cpu_count()

if start_one_at_a_time:

#Try optimising one image at at time... (no parameters)

ndim = self.psfs.ndim

#Make an even number of walkers.

nwalkers = (3*ndim//2)*2

#Initialise the chain to random psfs.

p0 = np.empty( (nwalkers, ndim) )

init_lle_par = []

for i in range(nwalkers):

p0[i,:] = self.psfs.tri.points[int(np.random.random()*self.psfs.npsfs)]

for j in range(self.n_ims):

kwargs = {"tgt_use":[j]}

if use_threads:

sampler = emcee.EnsembleSampler(nwalkers, ndim, self.lnprob, threads=threads, kwargs=kwargs)

else:

sampler = emcee.EnsembleSampler(nwalkers, ndim, self.lnprob, kwargs=kwargs)

sampler.run_mcmc(p0,nchain)

init_lle_par.append(sampler.flatchain[np.argmax(sampler.flatlnprobability)])

print("Done initial model for chain {0:d}".format(j))

init_lle_par = np.array(init_lle_par).flatten()

#Minimum number of walkers

ndim = self.psfs.ndim * self.n_ims + len(init_par)

nwalkers = 2*ndim

if use_threads:

sampler = emcee.EnsembleSampler(nwalkers, ndim, self.lnprob, threads=threads, kwargs=kwargs)

else:

sampler = emcee.EnsembleSampler(nwalkers, ndim, self.lnprob, kwargs=kwargs)

#Initialise the chain to random psfs.

p0 = np.empty( (nwalkers, ndim) )

if start_one_at_a_time:

for i in range(nwalkers):

p0[i, :self.psfs.ndim * self.n_ims] = init_lle_par + 0.01*np.random.normal(size=ndim)*self.psfs.h_density

#Add in a Gaussian distribution of model parameters.

p0[i, self.psfs.ndim * self.n_ims:] = init_par + np.random.normal(size=len(init_par))*walker_sdev

else:

for i in range(nwalkers):

for j in range(self.n_ims):

p0[i,j*self.psfs.ndim:(j+1)*self.psfs.ndim] = \

self.psfs.tri.points[int(np.random.random()*self.psfs.npsfs)]

#Add in a Gaussian distribution of model parameters.

p0[i, self.psfs.ndim * self.n_ims:] = init_par + np.random.normal(size=len(init_par))*walker_sdev

sampler.run_mcmc(p0,nchain)

print("Best lnprob: {0:5.2f}".format(np.max(sampler.lnprobability)))

best_x = sampler.flatchain[np.argmax(sampler.flatlnprobability)]

return best_x, sampler

#Uncomment the following line for FunnelWeb line_profile.

#kernprof -l best_psf_binary

#python -m line_profiler best_psf_binary.lprof

# @profile

def find_best_psfs(self, p_fix, return_lnprob=False):

"""Make a simple fit of every target image, for fixed model parameters.

Parameters

----------

p_fix: array-like

Model parameters, that are fixed when finding the best PSF. Note that for a

point source model, this should be [].

"""

best_fit_ims = np.empty( (self.n_ims, self.ims[0].shape[0], self.ims[0].shape[1]) )

best_ixs = np.empty( self.n_ims, dtype=np.int )

chi2_total = 0.0

for i in range(self.n_ims):

#What is our object model?

obj_ft = self.object.model_uv(p_fix, self.tgt_uv[i])

#Chi-squared

chi2s = np.empty(self.psfs.npsfs)

mod_ims = np.empty( (self.psfs.npsfs, self.ims[0].shape[0], self.ims[0].shape[1]) )

for j in range(self.psfs.npsfs):

#Convolve the object with the PSF model to form an image model.

mod_ft = obj_ft * self.psfs.psf_fts[j]

#Find the tilt that best matches the image model, and form an image model,

#and scale the image to match the total flux.

#NB This is copied from lnprob, which is a little messy. They should be

#consolidated!

scale_factor = self.im_fts[i][0].real/mod_ft[0].real

tilt = optimize_tilt(mod_ft, self.im_fts[i]/scale_factor, self.psfs.uv)

mod_ft = optimize_tilt_function(tilt, mod_ft*scale_factor, self.im_fts[i], self.psfs.uv, return_model=True)

mod_ims[j] = self.psfs.im_from_ft(mod_ft)

#Find the mean square uncertainty of this fit

chi2s[j] = np.sum( (mod_ims[j] - self.ims[i])**2 * self.ivar[i] )

#pdb.set_trace()

#Find the best image

best_ixs[i] = np.argmin(chi2s)

chi2_total += np.min(chi2s)

best_fit_ims[i] = mod_ims[best_ixs[i]]

if return_lnprob:

return -chi2_total/2.0

else:

return best_ixs, best_fit_ims

def marginalise_best_psf(self, init_par=[],walker_sdev=[],nchain=100, nburnin=50, use_threads=True):

"""Use the affine invariant Monte-Carlo Markov chain technique to marginalise

over all PSFs.

This brute force algorithm takes a long time, because for every parameter it fits,

it runs a monte-carlo chain which requires nwalkers * nchain evaluations of

find_best_psfs, which in turn requires N_psfs * N_target_frames evaluations of

optimize_tilt.

e.g. if running this over a 100 x 100 grid, fitting for 1 parameter with 6 walkers

and a chain length of 100, 100 psfs and 50 target frames, this is 3 x 10^10

evaluations of optimize_tilt.

An alternative to this would be to just add 2 model parameters per target image,

i.e. the tilt of each image, which e.g. could be 50 parameters for 50 target

images. The problem with this is that it would then require nwalkers to be

50 times larger in the case of fitting to only 1 parameter (e.g. contrast).

Parameters

----------

"""

threads = multiprocessing.cpu_count()

#Minimum number of walkers

ndim = len(init_par)

nwalkers = 2*ndim

kwargs = {"return_lnprob":True}

if use_threads:

sampler = emcee.EnsembleSampler(nwalkers, ndim, self.find_best_psfs, threads=threads, kwargs=kwargs)

else:

sampler = emcee.EnsembleSampler(nwalkers, ndim, self.find_best_psfs, kwargs=kwargs)

#Initialise the chain to random parameters.

p0 = np.empty( (nwalkers, ndim) )

for i in range(nwalkers):

p0[i] = init_par + np.random.normal(size=len(init_par))*walker_sdev

import time

then = time.time()

best_ixs, best_fit_ims = self.find_best_psfs(p0[0])

now = time.time()

print(now-then)

print("Running Chain... (burn in)")

pos, prob, state = sampler.run_mcmc(p0,nburnin)

sampler.reset()

print("Running Chain... ")

sampler.run_mcmc(pos,nchain)

print("Best lnprob: {0:5.2f}".format(np.max(sampler.lnprobability)))

best_x = sampler.flatchain[np.argmax(sampler.flatlnprobability)]