im, # In electron counts/s.

psf, # Normalised.

scale_factor,

t_exp,

final_sz,

tt = np.array([0, 0]),

im_star = None, # In electron counts/s.

noise_frame_gain_multiplied = 0, # Noise injected into the system that is multiplied up by the detector gain after conversion to counts via a Poisson distribution, e.g. sky background, emission from telescope, etc. Must have shape final_sz. It is assumed that this noise frame has already been multiplied up by the detector gain!

noise_frame_post_gain = 0, # Noise injected into the system after gain multiplication, e.g. read noise. Must have shape final_sz.

gain = 1, # Detector gain.

detector_saturation=np.inf, # Detector saturation.

plate_scale_as_px_conv = 1, # Only used for plotting.

plate_scale_as_px = 1, # Only used for plotting.

plotit=False):

"""

This function can be used to generate a short-exposure 'lucky' image that can be input to the Lucky Imaging algorithms.

Input: one 'raw' countrate image of a galaxy; one PSF with which to convolve it (at the same plate scale)

Output: a 'Lucky' exposure.

Process: convolve with PSF --> resize to detector --> add tip and tilt (from a premade vector of tip/tilt values) --> convert to counts --> add noise --> subtract the master sky/dark current.

"""

# Convolve with PSF.

im_raw = im

im_convolved = obssim.convolve_psf(im_raw, psf)

# Add a star to the field. We need to add the star at the convolution plate scale BEFORE we resize down because of the tip-tilt adding step!

if is_numlike(im_star):

if im_star.shape != im_convolved.shape:

print("ERROR: the input image of the star MUST have the same size and plate scale as the image of the galaxy after convolution!")

raise UserWarning

im_convolved += im_star

# Resize to detector (+ edge buffer).

im_resized = imutils.fourier_resize(

im = im_convolved,

scale_factor = scale_factor,

conserve_pixel_sum = True)

# Add tip and tilt. To avoid edge effects, max(tt) should be less than or equal to the edge buffer.

edge_buffer_px = (im.shape[0] - final_sz[0]) / 2

if edge_buffer_px > 0 and max(tt) > edge_buffer_px:

print("WARNING: the edge buffer is less than the supplied tip and tilt by a margin of {:.2f} pixels! Shifted image will be clipped.".format(np.abs(edge_buffer_px - max(tt))))

im_tt = obssim.add_tt(image = im_resized, tt_idxs = tt)[0]

# Crop back down to the detector size.

if edge_buffer_px > 0:

im_tt = imutils.centre_crop(im_tt, final_sz)

# Convert to counts. Note that we apply the gain AFTER we convert to integer

# counts.

im_counts = etcutils.expected_count_to_count(im_tt, t_exp = t_exp) * gain

# Add the pre-gain noise. Here, we assume that the noise frame has already

# been multiplied by the gain before being passed into this function.

im_noisy = im_counts + noise_frame_gain_multiplied

# Add the post-gain noise (i.e. read noise)

im_noisy += noise_frame_post_gain

# Account for detector saturation

im_noisy = np.clip(im_noisy, a_min=0, a_max=detector_saturation)

if plotit:

plate_scale_as_px = plate_scale_as_px_conv * scale_factor

# Plotting

mu.newfigure(1,3)

plt.suptitle('Convolving input image with PSF and resizing to detector')

mu.astroimshow(im=im_raw,

title='Truth image (electrons/s)',

plate_scale_as_px = plate_scale_as_px_conv,

colorbar_on=True,

subplot=131)

mu.astroimshow(im=psf,

title='Point spread function (normalised)',

plate_scale_as_px = plate_scale_as_px_conv,

colorbar_on=True,

subplot=132)

# mu.astroimshow(im=im_convolved,

# title='Star added, convolved with PSF (electrons/s)',

# plate_scale_as_px = plate_scale_as_px_conv,

# colorbar_on=True,

# subplot=143)

mu.astroimshow(im=im_resized,

title='Resized to detector plate scale (electrons/s)',

plate_scale_as_px=plate_scale_as_px,

colorbar_on=True,

subplot=133)

# Zooming in on the galaxy

# mu.newfigure(1,4)

# plt.suptitle('Convolving input image with PSF and resizing to detector')

# mu.astroimshow(im=imutils.centre_crop(im=im_raw, units='arcsec', plate_scale_as_px=plate_scale_as_px_conv, sz_final=(6, 6)), title='Raw input image (electrons/s)', plate_scale_as_px = plate_scale_as_px_conv, colorbar_on=True, subplot=141)

# mu.astroimshow(im=psf, title='Point spread function (normalised)', plate_scale_as_px = plate_scale_as_px_conv, colorbar_on=True, subplot=142)

# mu.astroimshow(im=imutils.centre_crop(im=im_convolved, units='arcsec', plate_scale_as_px=plate_scale_as_px_conv, sz_final=(6, 6)), title='Star added, convolved with PSF (electrons/s)', plate_scale_as_px = plate_scale_as_px_conv, colorbar_on=True, subplot=143)

# mu.astroimshow(im=imutils.centre_crop(im=im_resized, units='arcsec', plate_scale_as_px=plate_scale_as_px, sz_final=(6, 6)), title='Resized to detector plate scale (electrons/s)', plate_scale_as_px=plate_scale_as_px, colorbar_on=True, subplot=144)

mu.newfigure(1,3)

plt.suptitle('Adding tip and tilt, converting to integer counts and adding noise')

mu.astroimshow(im=im_tt,

title='Atmospheric tip and tilt added (electrons/s)',

plate_scale_as_px=plate_scale_as_px,

colorbar_on=True,

subplot=131)

mu.astroimshow(im=im_counts,

title=r'Converted to integer counts and gain-multiplied by %d (electrons)' % gain,

plate_scale_as_px=plate_scale_as_px,

colorbar_on=True,

subplot=132)

mu.astroimshow(im=im_noisy,

title='Noise added (electrons)',

plate_scale_as_px=plate_scale_as_px,

colorbar_on=True,

subplot=133)

# plt.subplot(1,4,4)

plt.figure()

x = np.linspace(-im_tt.shape[0]/2, +im_tt.shape[0]/2, im_tt.shape[0]) * plate_scale_as_px

plt.plot(x, im_tt[:, im_tt.shape[1]/2], 'g', label='Electron count rate')

plt.plot(x, im_counts[:, im_tt.shape[1]/2], 'b', label='Converted to integer counts ($t_{exp} = %.2f$ s)' % t_exp)

plt.plot(x, im_noisy[:, im_tt.shape[1]/2], 'r', label='Noise added')

plt.xlabel('arcsec')

plt.ylabel('Pixel value (electrons)')

plt.title('Linear profiles')

plt.axis('tight')

plt.legend(loc='lower left')

mu.show_plot()

if type(image) == list:

image = np.array(image)

# Search in the bid area of the input image for the peak pixel coordinates.

if bid_area:

sub_image = imutils.rotateAndCrop(image_in_array = images, cropArg = bid_area)

else:

sub_image = image

img_peak_idx = np.asarray(np.unravel_index(np.argmax(sub_image), sub_image.shape))

# Shift the image by the relative amount.

rel_shift_idx = (img_ref_peak_idx - img_peak_idx)

image_shifted = scipy.ndimage.interpolation.shift(image, rel_shift_idx)

peak_pixel_val = max(sub_image.flatten()) # Maximum pixel value (for now, not used)

if type(image) == list:

image = np.array(image)

# # Thresholding the image

image_subtracted_bg = np.copy(image)

# image_subtracted_bg[image<1.5*np.mean(image.flatten())] = 0

# image -= min(image.flatten())

image_subtracted_bg[image < centroid_threshold * max(image.flatten())] = 0

img_peak_idx = _centroid(image_subtracted_bg)

# Shift the image by the relative amount.

rel_shift_idx = (img_ref_peak_idx - img_peak_idx)

image_shifted = scipy.ndimage.interpolation.shift(image, rel_shift_idx)

if type(image) == list:

image = np.array(image)

# Subtracting the mean of each image

image_subtracted_bg = image - np.mean(image.flatten())

image_ref_subtracted_bg = image_ref - np.mean(image_ref.flatten())

height, width = image.shape

# if fftwconvolve.NTHREADS==0:

corr = scipy.signal.fftconvolve(image_ref_subtracted_bg, image_subtracted_bg[::-1,::-1], 'same')

# else:

# corr = fftwconvolve.fftconvolve(image_ref_subtracted_bg, image_subtracted_bg[::-1,::-1], 'same')

corr /= max(corr.flatten()) # The fitting here does not work if the pixels have large values!

if sub_pixel_shift:

# Fitting a Gaussian.

Y, X = np.mgrid[-(height-2*buff_xcorr)/2:(height-2*buff_xcorr)/2, -(width-2*buff_xcorr)/2:(width-2*buff_xcorr)/2]

x_peak, y_peak = np.unravel_index(np.argmax(corr), corr.shape)

try:

p_init = astropy.modeling.models.Gaussian2D(x_mean=X[x_peak,y_peak],y_mean=Y[x_peak,y_peak],x_stddev=5.,y_stddev=5.,amplitude=np.max(corr.flatten()))

except:

p_init = astropy.modeling.models.Gaussian2D(x_mean=x_peak,y_mean=y_peak,x_stddev=1.,y_stddev=1.,amplitude=1.)

fit_p = astropy.modeling.fitting.LevMarLSQFitter()

p_fit = fit_p(p_init, X, Y, corr[buff_xcorr:height-buff_xcorr, buff_xcorr:width-buff_xcorr])

rel_shift_idx = (p_fit.y_mean.value, p_fit.x_mean.value) # NOTE: the indices have to be swapped around here for some reason!

else:

rel_shift_idx = np.unravel_index(np.argmax(corr), corr.shape)

rel_shift_idx = (rel_shift_idx[0] - height/2, rel_shift_idx[1] - width/2)

# mu.newfigure(2,2)

# mu.astroimshow(im=image, title='Input image', subplot=221)

# mu.astroimshow(im=image_ref, title='Reference image', subplot=222)

# mu.astroimshow(im=corr, title='Cross-correlation', subplot=223)

# mu.astroimshow(im=p_fit(X,Y), title='Gaussian fit', subplot=224)

# mu.show_plot()

image_shifted = scipy.ndimage.interpolation.shift(image, rel_shift_idx)

if type(image) == list:

image = np.array(image)

# Subtracting the mean of the input image

image_subtracted_bg = image - np.mean(image.flatten())

# Fitting a Gaussian to the mean-subtracted image.

peak_idx = _gaussfit_peak(image_subtracted_bg)

rel_shift_idx = -(peak_idx - img_ref_peak_idx)

image_shifted = scipy.ndimage.interpolation.shift(image, rel_shift_idx)

mode = 'serial', # whether or not to process images in parallel

image_ref = None, # reference image

fsr = 1, # for peak pixel/FAS method

bid_area = None, # for peak pixel method

N = None,

centroid_threshold = 0.25, # for centroiding method

sub_pixel_shift = True, # for xcorr/FAS method

buff_xcorr = 25, # for xcorr/FAS method

buff_fas = 32, # for FAS method (edge ramp buffer)

cutoff_freq_frac = 1, # for FAS method

sigma_kernel = 0, # for FAS method (sigma of Gaussian filter)

use_vals_outside_cutoff_freq = True, # for FAS method

stacking_method = 'average',

timeit = True

):

"""

Apply a Lucky Imaging (LI) technique to a sequence of images stored in the input array images.

The type of LI technique used is specified by input string type and any additional arguments which may be required are given in vararg.

"""

tic = time.time()

images, image_ref, N = _li_error_check(images, image_ref, N)

if not timeit:

print("Applying Lucky Imaging technique '{}' to input series of {:d} images...".format(li_method, N))

# For each of these functions, the output must be of the form

# image_shifted, rel_shift_idxs

li_method = li_method.lower()

if li_method == 'cross-correlation':

shift_fun = partial(shift_xcorr, image_ref=image_ref, buff_xcorr=buff_xcorr, sub_pixel_shift=sub_pixel_shift)

elif li_method == 'gaussian fit':

img_ref_peak_idx = _gaussfit_peak(image_ref - np.mean(image_ref.flatten()))

shift_fun = partial(shift_gaussfit,

img_ref_peak_idx=img_ref_peak_idx)

elif li_method == 'peak pixel':

# Determining the reference coordinates.

if bid_area:

sub_image_ref = imutils.centre_crop(image_ref, bid_area)

else:

sub_image_ref = image_ref

img_ref_peak_idx = np.asarray(np.unravel_index(np.argmax(sub_image_ref), sub_image_ref.shape))

shift_fun = partial(shift_pp,

img_ref_peak_idx=img_ref_peak_idx,

bid_area=bid_area,

fsr=fsr)

elif li_method == 'centroid':

image_ref_subtracted_bg = np.copy(image_ref)

image_ref_subtracted_bg[image_ref < centroid_threshold * max(image_ref.flatten())] = 0

img_ref_peak_idx = _centroid(image_ref_subtracted_bg)

shift_fun = partial(shift_centroid,

img_ref_peak_idx=img_ref_peak_idx,

centroid_threshold=centroid_threshold)

elif li_method == 'blind stack':

if stacking_method == 'median combine':

arr = np.ndarray((1, image_ref.shape[0], image_ref.shape[1]))

arr[0,:] = image_ref

image_ref = arr

image_stacked = obssim.median_combine(np.concatenate((image_ref, images)))

elif stacking_method == 'average':

image_stacked = (image_ref + np.sum(images, axis=0)) / (N + 1)

rel_shift_idxs = np.zeros( (N, 2) )

return image_stacked, rel_shift_idxs

elif li_method == 'fourier amplitude selection' or li_method =='fas':

# Need to shift each image.

shift_fun = partial(shift_xcorr,

image_ref=image_ref,

buff_xcorr=buff_xcorr,

sub_pixel_shift=sub_pixel_shift)

# THEN we need to apply the Fourier amplitude selection technique.

else:

print("ERROR: invalid Lucky Imaging method '{}' specified; must be 'cross-correlation', 'peak pixel', 'centroid' or 'Gaussian fit' for now...".format(li_method))

raise UserWarning

# In here, want to parallelise the processing for *each image*. So make

# shift functions that work on a single image and return the shifted image,

# then stack it out here.

if mode == 'parallel':

# Setting up to execute in parallel.

images = images.tolist() # Need to convert the image array to a list.

# Executing in parallel.

pool = ProcPool()

results = pool.map(shift_fun, images, 1)

pool.close()

pool.join()

# Extracting the output arguments.

images_shifted = np.array(zip(*results)[0])

rel_shift_idxs = np.array(zip(*results)[1])

if li_method == 'peak pixel' and fsr < 1:

peak_pixel_vals = np.array(zip(*results)[2])

elif mode == 'serial':

# Loop through each image individually.

images_shifted = np.zeros( (N, image_ref.shape[0], image_ref.shape[1]) )

rel_shift_idxs = np.zeros( (N, 2) )

for k in range(N):

if li_method == 'peak pixel':

if k == 0:

peak_pixel_vals = np.zeros(N)

images_shifted[k], rel_shift_idxs[k], peak_pixel_vals[k] = shift_fun(image=images[k])

else:

images_shifted[k], rel_shift_idxs[k] = shift_fun(image=images[k])

else:

print("ERROR: mode must be either parallel or serial!")

raise UserWarning

# If we're using an FSR < 1 in the peak pixel method, then we must do the following:

# 1. Get our method to return a list of peak pixel values.

# 2. Sort that list in descending order and get the indices of the corresponding images in the range [0, FSR * N)

# 3. Add these images together.

if li_method == 'peak pixel' and fsr < 1:

sorted_idx = np.argsort(peak_pixel_vals)[::-1] # Array holding indices of images

N = np.ceil(fsr * N)

# Is averaging the best way to do this? Probably not...

if stacking_method == 'median combine':

arr = np.ndarray((1, image_ref.shape[0], image_ref.shape[1]))

arr[0,:] = image_ref

image_ref = arr

image_stacked = obssim.median_combine(

np.concatenate((image_ref, images_shifted[sorted_idx[:N]])))

elif stacking_method == 'average':

image_stacked = (image_ref + \

np.sum(images_shifted[sorted_idx[:N]], 0)) / (N + 1)

elif li_method == 'fourier amplitude selection' or li_method =='fas':

# From Mackay 2013: within the cutoff spatial frequency radius, we

# select (u,v) pixels by using the Fourier amplitude. Outside this

# cutoff frequency, we select (u,v) pixels by using the peak pixel value

# in the image.

# Linearly ramp the images to zero.

images_shifted = edge_ramp(images_shifted, buff_fas)

h, w = image_ref.shape

N_frames_to_keep = max(1, int(np.round(fsr * N)))

images_fft = pyfftw.interfaces.numpy_fft.fftshift(

pyfftw.interfaces.numpy_fft.fft2(images_shifted),

axes=(1,2))

cutoff_freq_px = int(np.round(cutoff_freq_frac * min(h,w)))

U,V = np.meshgrid(np.linspace(-w/2,w/2-1,w),np.linspace(-h/2,h/2-1,h))

uv_map = np.zeros( (h,w) )

uv_map[np.sqrt(U**2 + V**2) < cutoff_freq_px]=1

fft_sum = np.zeros( (h, w), dtype=complex )

# OUTSIDE THE CUTOFF FREQUENCY

if use_vals_outside_cutoff_freq:

# Step 1. Sort the images in order of peak pixel value.

max_pixel_vals = np.max(images_shifted,axis=(1,2))

idxs_to_keep = np.argsort(max_pixel_vals)[-N_frames_to_keep:]

# Step 2.

vals_to_keep=np.zeros( (h, w, N_frames_to_keep), dtype=complex )

for u, v in zip(V[uv_map==0],U[uv_map==0]):

# For these coordinates, grab the indices of the N highest

# values in the data cube.

try:

vals_to_keep[u+h/2,v+w/2] = images_fft[

idxs_to_keep,

u+h/2,

v+w/2]

except:

ipdb.set_trace()

fft_sum += np.sum(vals_to_keep, axis=2)

# WITHIN THE CUTOFF FREQUENCY

images_fft_amp = np.abs(images_fft)

# Smoothing with a Gaussian kernel

if sigma_kernel != 0:

for k in range(images_fft_amp.shape[0]):

images_fft_amp[k] = imutils.gaussian_smooth(

im=images_fft_amp[k],

sigma=sigma_kernel

)

# For now, don't worry about parallelisation.

vals_to_keep=np.zeros( (h, w, N_frames_to_keep), dtype=complex )

idxs_to_keep=np.zeros( (h, w, N_frames_to_keep), dtype=int) # Indices along the zeroth axis of the datacube indicating which frames' data we want to keep for the whole image

for u, v in zip(V[uv_map==1],U[uv_map==1]):

# For these coordinates, grab the indices of the N highest values in the data cube.

idxs_to_keep[u+h/2,v+w/2] = np.argsort(

images_fft_amp[:,u+h/2,v+w/2])[-N_frames_to_keep:]

vals_to_keep[u+h/2,v+w/2] = images_fft[

idxs_to_keep[u+h/2,v+w/2],

u+h/2,

v+w/2]

# Take the sum along the zeroth axis and IFFT to get the reassembled image.

fft_sum += np.sum(vals_to_keep, axis=2)

image_stacked = np.abs(pyfftw.interfaces.numpy_fft.ifft2(

pyfftw.interfaces.numpy_fft.fftshift(fft_sum/N_frames_to_keep)))

else:

# Now, stacking the images. Need to change N if FSR < 1.

if stacking_method == 'median combine':

arr = np.ndarray((1, image_ref.shape[0], image_ref.shape[1]))

arr[0,:] = image_ref

image_ref = arr

image_stacked = obssim.median_combine(np.concatenate(

(image_ref, images_shifted)))

elif stacking_method == 'average':

image_stacked = (image_ref + np.sum(images_shifted, 0)) / (N + 1)

toc = time.time()

if timeit:

print("APPLYING LUCKY IMAGING TECHNIQUE {}: Elapsed time for {:d} {}-by-{} images in {} mode: {:.5f}".format(li_method, N, image_ref.shape[0], image_ref.shape[1], mode, (toc-tic)))

li_method='',

plotHist=True,

verbose=True):

"""

Compute the alignment errors arising in the Lucky Imaging shifting-and-stacking process given an input array of tip and tilt coordinates applied to the input images and the coordinates of the shifts applied in the shifting-and-stacking process.

"""

N = in_idxs.shape[0]

errs_as = np.zeros( (N, 3) )

for k in range(N):

errs_as[k, 0] = (in_idxs[k,0] - out_idxs[k,0]) * opticalsystem.plate_scale_as_px

errs_as[k, 1] = (in_idxs[k,1] - out_idxs[k,1]) * opticalsystem.plate_scale_as_px

errs_as[k, 2] = np.sqrt(errs_as[k, 0]**2 + errs_as[k, 1]**2)

errs_px = errs_as / opticalsystem.plate_scale_as_px

# Print the alignment errors to screen.

if verbose:

print('------------------------------------------------')

print('Tip/tilt coordinates\nInput\t\tOutput\t\tError\tError (arcsec)')

print('------------------------------------------------')

for k in range(N):

print('(%6.2f,%6.2f)\t(%6.2f,%6.2f)\t%4.2f\t%4.2f' % (in_idxs[k,0],

in_idxs[k,1],out_idxs[k,0],out_idxs[k,1],errs_px[k,2],

errs_as[k,2]))

print('------------------------------------------------')

print('\t\t\tMean\t%4.2f' % np.mean(errs_as))

if plotHist:

plot_alignment_err_histogram(errs_as, li_method)

li_method=''):

x_errs_as = errs_as[:,0]

y_errs_as = errs_as[:,1]

# Plot a pretty histogram showing the distribution of the alignment errors, and fit a Gaussian to them.

range_as = 2 * max(max(np.abs(y_errs_as)), max(np.abs(x_errs_as)))

nbins = int(errs_as.shape[0] / 100)

mu.newfigure(1.5,1)

plt.suptitle('{} Lucky Imaging shifting-and-stacking alignment errors'.format(li_method))

plt.subplot(211)

if nbins > 5:

plt.hist(x_errs_as, bins=nbins, range=(-range_as/2,+range_as/2), normed=True)

else:

plt.hist(x_errs_as, range=(-range_as/2,+range_as/2), normed=True)

mean_x = np.mean(x_errs_as)

sigma_x = np.sqrt(np.var(x_errs_as))

x = np.linspace(-range_as/2, range_as/2, 100)

plt.plot(x, normpdf(x,mean_x,sigma_x), 'r', label=r'$\sigma_x$ = %.4f"' % (sigma_x))

plt.title(r'$x$ alignment error')

plt.xlabel('arcsec')

plt.legend()

plt.subplot(212)

if nbins > 5:

plt.hist(y_errs_as, bins=nbins, range=(-range_as/2,+range_as/2), normed=True)

else:

plt.hist(y_errs_as, range=(-range_as/2,+range_as/2), normed=True)

mean_y = np.mean(y_errs_as)

sigma_y = np.sqrt(np.var(y_errs_as))

y = np.linspace(-range_as/2, range_as/2, 100)

plt.plot(y, normpdf(y,mean_y,sigma_y), 'r', label=r'$\sigma_y$ = %.4f"' % (sigma_y))

plt.title(r'$y$ alignment error')

plt.xlabel('arcsec')

plt.legend()

image_ref = None,

N = None):

"""

A private method to be used to check the inputs to the Lucky Imaging methods.

"""

# Need to convert to float if necessary.

if type(images.flatten()[0]) != np.float64:

images = images.astype(np.float64)

if image_ref is not None and type(image_ref.flatten()[0]) != np.float64:

image_ref = image_ref.astype(np.float64)

# Checking image dimensions.

if len(images.shape) > 4:

print("WARNING: for now, please only input a 3D array of images to shift and stack! I'm only going to operate on the first set of images...")

images = np.squeeze(images[0])

if len(images.shape) == 3:

if N and N > images.shape[0]:

print("ERROR: if specified, N must be equal to or less than the length of the first dimension of the images array.")

raise UserWarning

if image_ref is None:

# If the reference image is not specified, we use the first image in the array as the reference:

# i.e. we align all other images to images[0].

if not N:

N = images.shape[0]-1

image_ref = np.copy(images[0])

images = np.copy(images[1:]) # Only need to go through images 1:N-1.

else:

if image_ref.shape != images[0].shape:

print("ERROR: if specified, reference image shape must be equal to input image stack shape.")

raise UserWarning

if not N:

N = images.shape[0]

else:

# Error: cannot shift and stack a single image!

print("ERROR: cannot shift and stack a single image! Input array must have N > 1.")

raise UserWarning

""" Returns the centroid coordinates of an image. """

height = image.shape[0]

width = image.shape[1]

x = np.arange(height)

y = np.arange(width)

X, Y = np.meshgrid(y,x)

M_10 = np.sum((X * image).flatten())

M_01 = np.sum((Y * image).flatten())

M_00 = np.sum(image.flatten())

centroid = np.asarray([M_01 / M_00, M_10 / M_00])

""" Returns the coordinates of the peak of a 2D Gaussian fitted to the

image. """

height, width = image.shape

# Fitting a Gaussian.

Y, X = np.mgrid[-(height)/2:(height)/2, -(width)/2:(width)/2]

try:

p_init = astropy.modeling.models.Gaussian2D(x_stddev=1.,y_stddev=1.)

except:

p_init = astropy.modeling.models.Gaussian2D(x_mean=1.,y_mean=1.,x_stddev=1.,y_stddev=1.,amplitue=1.)

fit_p = astropy.modeling.fitting.LevMarLSQFitter()

p_fit = fit_p(p_init, X, Y, image)

peak_idx = np.array([p_fit.y_mean.value, p_fit.x_mean.value]) # NOTE: the indices have to be swapped around here for some reason!

""" Linearly ramps the values of an image to zero over a buffer with width

buff at the edges of the each image. """

buff = int(buff)

# Crop the images by an amount buff_xcorr.

if len(im.shape) == 3:

_, h, w = im.shape

else:

h, w = im.shape

im = imutils.centre_crop(im, (h-2*buff,w-2*buff))

# Pad.

if len(im.shape) == 3:

im = np.pad(im, ( (0,0), (buff,buff), (buff,buff) ), mode='linear_ramp')

else:

im = np.pad(im, buff, mode='linear_ramp')