def ft_hologram(self, apodize=True):
"""
`~numpy.ndarray` of the self.hologram FFT
"""
if self._ft_hologram is None:
if apodize == True:
apodized_hologram = self.apodize(self.hologram)
self._ft_hologram = fftshift(fft2(apodized_hologram))
else:
self._ft_hologram = fftshift(fft2(self.hologram))
return self._ft_hologram
def _paganin(self, data):
pci1 = fft.fft2(np.float32(data))
pci2 = fft.fftshift(pci1) / self.filtercomplex
fpci = np.abs(fft.ifft2(pci2))
result = -0.5 * self.parameters['Ratio'] * np.log(
fpci + self.parameters['increment'])
return result
def _paganin(self, data):
pci1 = fft.fft2(np.float32(data))
pci2 = fft.fftshift(pci1) / self.filtercomplex
fpci = np.abs(fft.ifft2(pci2))
result = -0.5 * self.parameters['Ratio'] * np.log(
fpci + self.parameters['increment'])
return result
def _fine_search(self, sino, start_cor, search_radius, search_step, ratio,
drop):
"""
Fine search for finding the rotation center.
"""
# Denoising
(nrow, ncol) = sino.shape
flip_sino = np.fliplr(sino)
search_radius = np.clip(np.abs(search_radius), 1, ncol // 10 - 1)
search_step = np.clip(np.abs(search_step), 0.1, 1.1)
start_cor = np.clip(start_cor, search_radius, ncol - search_radius - 1)
cen_fliplr = (ncol - 1.0) / 2.0
list_cor = start_cor + np.arange(
-search_radius, search_radius + search_step, search_step)
comp_sino = np.flipud(sino) # Used to avoid local minima
list_metric = np.zeros(len(list_cor), dtype=np.float32)
mask = self._create_mask(2 * nrow, ncol, 0.5 * ratio * ncol, drop)
for i, cor in enumerate(list_cor):
shift = 2.0 * (cor - cen_fliplr)
sino_shift = ndi.interpolation.shift(flip_sino, (0, shift),
order=3,
prefilter=True)
if shift >= 0:
shift_int = np.int16(np.ceil(shift))
sino_shift[:, :shift_int] = comp_sino[:, :shift_int]
else:
shift_int = np.int16(np.floor(shift))
sino_shift[:, shift_int:] = comp_sino[:, shift_int:]
mat1 = np.vstack((sino, sino_shift))
list_metric[i] = np.mean(
np.abs(np.fft.fftshift(fft.fft2(mat1))) * mask)
min_pos = np.argmin(list_metric)
cor = list_cor[min_pos]
return cor
def monogenic_filter_one_scale_gray(img, ss=1, minWaveLength=3, mult=2.1, sigmaOnf=0.55):
# float point is very important in order to compute the fft2
if img.dtype not in ['float32', 'float64']:
img = np.float64(img)
imgdtype = 'float64'
else:
imgdtype = img.dtype
# for 3 channel we make one channel computing the mean value
if img.ndim == 3: # hay que cambiar esto
img = img.mean(2)
rows, cols = img.shape
#compute the monnogenic scale in frequency domain
logGabor, logGabor_H1, logGabor_H2 = monogenic_scale(cols,rows, ss, minWaveLength, mult, sigmaOnf)
# FFT2 in the corner
IM = fft2(img) # Discrete Fourier Transform of image
IMF = IM * logGabor # Frequency bandpassed image
f = np.real(ifft2(IMF)) # Spatially bandpassed image
# Bandpassed monogenic filtering, real part of h contains
IMH1=IM*logGabor_H1
IMH2=IM*logGabor_H2
h1= np.real(ifft2(IMH1))
h2= np.real(ifft2(IMH2))
# Amplitude of this scale component
An = np.sqrt(f * f + h1 * h1 + h2 * h2)
#Orientation computation
ori = np.arctan(-h2 / h1)
# Wrap angles between -pi and pi and convert radians to degrees
ori_d = np.fix((ori % np.pi) / np.pi * 180.)
# Feature type (a phase angle between -pi/2 and pi/2)
ft = np.arctan2(f, np.sqrt(h1 * h1 + h2 * h2))
#proyectionin ij plane
fr= np.sqrt(h1 * h1 + h2 * h2)
return An,ori_d,ori,ft,fr,f
def _fine_search(self, sino, raw_cor):
(Nrow, Ncol) = sino.shape
centerfliplr = (Ncol + 1.0)/2.0-1.0
# Use to shift the sino2 to the raw CoR
shiftsino = np.int16(2*(raw_cor-centerfliplr))
sino2 = np.roll(np.fliplr(sino[1:]), shiftsino, axis=1)
lefttake = 0
righttake = Ncol-1
search_rad = self.parameters['search_radius']
if raw_cor <= centerfliplr:
lefttake = np.ceil(search_rad+1)
righttake = np.floor(2*raw_cor-search_rad-1)
else:
lefttake = np.ceil(raw_cor-(Ncol-1-raw_cor)+search_rad+1)
righttake = np.floor(Ncol-1-search_rad-1)
Ncol1 = righttake-lefttake + 1
mask = self._create_mask(2*Nrow-1, Ncol1,
0.5*self.parameters['ratio']*Ncol)
numshift = np.int16((2*search_rad+1.0)/self.parameters['step'])
listshift = np.linspace(-search_rad, search_rad, num=numshift)
listmetric = np.zeros(len(listshift), dtype=np.float32)
num1 = 0
for i in listshift:
logging.debug("list shift %d", i)
sino2a = ndi.interpolation.shift(sino2, (0, i), prefilter=False)
sinojoin = np.vstack((sino, sino2a))
listmetric[num1] = np.sum(np.abs(fft.fftshift(
fft.fft2(sinojoin[:, lefttake:righttake + 1])))*mask)
num1 = num1 + 1
minpos = np.argmin(listmetric)
rotcenter = raw_cor + listshift[minpos]/2.0
return rotcenter, listmetric
def _coarse_search(self, sino):
# search minsearch to maxsearch in 1 pixel steps
smin, smax = self.parameters['search_area']
logging.debug("SMIN and SMAX %d %d", smin, smax)
(Nrow, Ncol) = sino.shape
centre_fliplr = (Ncol - 1.0)/2.0
# check angles here to determine if a sinogram should be chopped off.
# Copy the sinogram and flip left right, the purpose is to make a full
# [0;2Pi] sinogram
sino2 = np.fliplr(sino[1:])
# This image is used for compensating the shift of sino2
compensateimage = np.zeros((Nrow-1, Ncol), dtype=np.float32)
# Start coarse search in which the shift step is 1
compensateimage[:] = sino[-1]
start_shift = self._get_start_shift(centre_fliplr)*2
list_shift = np.arange(smin, smax + 1)*2 - start_shift
logging.debug("%s", list_shift)
list_metric = np.zeros(len(list_shift), dtype=np.float32)
mask = self._create_mask(2*Nrow-1, Ncol,
0.5*self.parameters['ratio']*Ncol)
count = 0
for i in list_shift:
logging.debug("list shift %d", i)
sino2a = np.roll(sino2, i, axis=1)
if i >= 0:
sino2a[:, 0:i] = compensateimage[:, 0:i]
else:
sino2a[:, i:] = compensateimage[:, i:]
list_metric[count] = np.sum(
np.abs(fft.fftshift(fft.fft2(np.vstack((sino, sino2a)))))*mask)
count += 1
minpos = np.argmin(list_metric)
rot_centre = centre_fliplr + list_shift[minpos]/2.0
return rot_centre, list_metric
def psf_correction(self, mat, win, pad_width):
(nrow, ncol) = mat.shape
mat_pad = np.pad(mat, pad_width, mode="reflect")
win_pad = np.pad(win, pad_width, mode="constant", constant_values=1.0)
mat_dec = fft.ifft2(fft.fft2(mat_pad) / fft.ifftshift(win_pad))
return np.abs(mat_dec)[pad_width:pad_width + nrow,
pad_width:pad_width + ncol]
def apply_gaussian_filter(mat, sigmax, sigmay, pad):
"""
Filtering an image using a 2D Gaussian window.
Parameters
----------
mat : array_like
2D array.
sigmax : int
Sigma in the x-direction.
sigmay : int
Sigma in the y-direction.
pad : int
Padding for the Fourier transform.
Returns
-------
ndarray
2D array. Filtered image.
"""
mat_pad = np.pad(mat, ((0, 0), (pad, pad)), mode='edge')
mat_pad = np.pad(mat_pad, ((pad, pad), (0, 0)), mode='mean')
(nrow, ncol) = mat_pad.shape
window = make_2d_gaussian_window(nrow, ncol, sigmax, sigmay)
listx = np.arange(0, ncol)
listy = np.arange(0, nrow)
x, y = np.meshgrid(listx, listy)
mat_sign = np.power(-1.0, x + y)
mat_filt = np.real(
fft.ifft2(fft.fft2(mat_pad * mat_sign) * window) * mat_sign)
return mat_filt[pad:nrow - pad, pad:ncol - pad]
def _coarse_search(self, sino):
# search minsearch to maxsearch in 1 pixel steps
smin, smax = self.parameters['search_area']
(Nrow, Ncol) = sino.shape
centre_fliplr = (Ncol - 1.0) / 2.0
# check angles here to determine if a sinogram should be chopped off.
# Copy the sinogram and flip left right, the purpose is to make a full
# [0;2Pi] sinogram
sino2 = np.fliplr(sino[1:])
# This image is used for compensating the shift of sino2
compensateimage = np.zeros((Nrow - 1, Ncol), dtype=np.float32)
# Start coarse search in which the shift step is 1
compensateimage[:] = np.flipud(sino)[1:]
start_shift = self._get_start_shift(centre_fliplr) * 2
list_shift = np.arange(smin, smax + 1) * 2 - start_shift
list_metric = np.zeros(len(list_shift), dtype=np.float32)
mask = self._create_mask(2 * Nrow - 1, Ncol,
0.5 * self.parameters['ratio'] * Ncol)
count = 0
for i in list_shift:
sino2a = np.roll(sino2, i, axis=1)
if i >= 0:
sino2a[:, 0:i] = compensateimage[:, 0:i]
else:
sino2a[:, i:] = compensateimage[:, i:]
fft_out = fft.fft2(np.vstack((sino, sino2a)))
temp = np.sum(np.abs(fft.fftshift(fft_out)) * mask)
list_metric[count] = temp
count += 1
minpos = np.argmin(list_metric)
rot_centre = centre_fliplr + list_shift[minpos] / 2.0
return rot_centre, list_metric
def _fine_search(self, sino, start_cor, search_radius,
search_step, ratio, drop):
"""
Fine search for finding the rotation center.
"""
# Denoising
(nrow, ncol) = sino.shape
flip_sino = np.fliplr(sino)
search_radius = np.clip(np.abs(search_radius), 1, ncol//10 - 1)
search_step = np.clip(np.abs(search_step), 0.1, 1.1)
start_cor = np.clip(start_cor, search_radius, ncol - search_radius - 1)
cen_fliplr = (ncol - 1.0) / 2.0
list_cor = start_cor + np.arange(
-search_radius, search_radius + search_step, search_step)
comp_sino = np.flipud(sino) # Used to avoid local minima
list_metric = np.zeros(len(list_cor), dtype = np.float32)
mask = self._create_mask(2 * nrow, ncol, 0.5 * ratio * ncol, drop)
for i, cor in enumerate(list_cor):
shift = 2.0*(cor - cen_fliplr)
sino_shift = ndi.interpolation.shift(
flip_sino, (0, shift), order = 3, prefilter = True)
if shift>=0:
shift_int = np.int16(np.ceil(shift))
sino_shift[:,:shift_int] = comp_sino[:,:shift_int]
else:
shift_int = np.int16(np.floor(shift))
sino_shift[:,shift_int:] = comp_sino[:,shift_int:]
mat1 = np.vstack((sino, sino_shift))
list_metric[i] = np.mean(
np.abs(np.fft.fftshift(fft.fft2(mat1)))*mask)
min_pos = np.argmin(list_metric)
cor = list_cor[min_pos]
return cor
def _hilbert(self, data):
pci1 = fft.fft2(fft.fftshift(np.float32(data)))
pci2 = fft.ifftshift(pci1) * self.filter1
fpci0 = fft.ifftshift(fft.ifft2(fft.fftshift(pci2)))
fpci = np.imag(fpci0)
result = fpci
return result
def remove_stripe_based_fft(sinogram, u, n, v, pad=150):
"""
Remove stripes using the method in Ref. [1].
Angular direction is along the axis 0.
Parameters
----------
sinogram : array_like
2D array.
u,n : int
To define the shape of 1D Butterworth low-pass filter.
v : int
Number of rows (* 2) to be applied the filter.
pad : int
Padding for FFT
Returns
-------
ndarray
2D array. Stripe-removed sinogram.
References
----------
.. [1] https://doi.org/10.1063/1.1149043
"""
if pad > 0:
sinogram = np.pad(sinogram, ((pad, pad), (0, 0)), mode='mean')
sinogram = np.pad(sinogram, ((0, 0), (pad, pad)), mode='edge')
(nrow, ncol) = sinogram.shape
window2d = create_2d_window(ncol, nrow, u, v, n)
sinogram = fft.ifft2(
np.fft.ifftshift(np.fft.fftshift(fft.fft2(sinogram)) * window2d))
return np.real(sinogram[pad:nrow - pad, pad:ncol - pad])
def _coarse_search(self, sino, list_shift):
# search minsearch to maxsearch in 1 pixel steps
list_metric = np.zeros(len(list_shift), dtype=np.float32)
(Nrow, Ncol) = sino.shape
# check angles to determine if a sinogram should be chopped off.
# Copy the sinogram and flip left right, to make a full [0:2Pi] sino
sino2 = np.fliplr(sino[1:])
# This image is used for compensating the shift of sino2
compensateimage = np.zeros((Nrow - 1, Ncol), dtype=np.float32)
# Start coarse search in which the shift step is 1
compensateimage[:] = np.flipud(sino)[1:]
mask = self._create_mask(2 * Nrow - 1, Ncol,
0.5 * self.parameters['ratio'] * Ncol)
count = 0
for i in list_shift:
sino2a = np.roll(sino2, i, axis=1)
if i >= 0:
sino2a[:, 0:i] = compensateimage[:, 0:i]
else:
sino2a[:, i:] = compensateimage[:, i:]
list_metric[count] = np.sum(
np.abs(fft.fftshift(fft.fft2(np.vstack(
(sino, sino2a))))) * mask)
count += 1
return list_metric
def _fine_search(self, sino, raw_cor):
(Nrow, Ncol) = sino.shape
centerfliplr = (Ncol + 1.0) / 2.0 - 1.0
# Use to shift the sino2 to the raw CoR
shiftsino = np.int16(2 * (raw_cor - centerfliplr))
sino2 = np.roll(np.fliplr(sino[1:]), shiftsino, axis=1)
lefttake = 0
righttake = Ncol - 1
search_rad = self.parameters['search_radius']
if raw_cor <= centerfliplr:
lefttake = np.ceil(search_rad + 1)
righttake = np.floor(2 * raw_cor - search_rad - 1)
else:
lefttake = np.ceil(raw_cor - (Ncol - 1 - raw_cor) + search_rad + 1)
righttake = np.floor(Ncol - 1 - search_rad - 1)
Ncol1 = righttake - lefttake + 1
mask = self._create_mask(2 * Nrow - 1, Ncol1,
0.5 * self.parameters['ratio'] * Ncol)
numshift = np.int16((2 * search_rad + 1.0) / self.parameters['step'])
listshift = np.linspace(-search_rad, search_rad, num=numshift)
listmetric = np.zeros(len(listshift), dtype=np.float32)
num1 = 0
for i in listshift:
logging.debug("list shift %d", i)
sino2a = ndi.interpolation.shift(sino2, (0, i), prefilter=False)
sinojoin = np.vstack((sino, sino2a))
listmetric[num1] = np.sum(
np.abs(
fft.fftshift(fft.fft2(
sinojoin[:, lefttake:righttake + 1]))) * mask)
num1 = num1 + 1
minpos = np.argmin(listmetric)
rotcenter = raw_cor + listshift[minpos] / 2.0
return rotcenter, listmetric
def _hilbert(self, data):
pci1 = fft.fft2(fft.fftshift(np.float32(data)))
pci2 = fft.ifftshift(pci1)*self.filter1
fpci0 = fft.ifftshift(fft.ifft2(fft.fftshift(pci2)))
fpci = np.imag(fpci0)
result = fpci
return result
def coarse_search_based_integer_shift(sino_0_180,
start_cor=None,
stop_cor=None,
ratio=0.5):
"""
Find center of rotation (CoR) using integer shifts of a 180-360 sinogram.
The 180-360 sinogram is made by flipping horizontally the 0-180 sinogram.
Angular direction is along the axis 0.
Note:
1-pixel shift of the 180-360 sinogram is equivalent to 0.5-pixel shift \
of CoR. Auto-search is limited to the range of [width/4; width - width/4].
Parameters
----------
sino_0_180 : float
Sinogram in the angle range of [0;180].
start_cor : int
Starting point for searching CoR.
stop_cor : int
Ending point for searching CoR.
ratio : float
Ratio between a sample and the width of the sinogram. Default value
of 0.5 works in most cases (even when the sample is larger than the
field_of_view).
Returns
-------
float
Center of rotation with haft-pixel accuracy.
"""
# Denoising. Should not be used for a downsampled sinogram
# sino_0_180 = ndi.gaussian_filter(sino_0_180, (3,1), mode='reflect')
(nrow, ncol) = sino_0_180.shape
if start_cor is None:
start_cor = ncol // 4
if stop_cor is None:
stop_cor = ncol - ncol // 4 - 1
start_cor, stop_cor = np.sort((start_cor, stop_cor))
start_cor = np.int16(np.clip(start_cor, 0, ncol - 1))
stop_cor = np.int16(np.clip(stop_cor, 0, ncol - 1))
cen_fliplr = (ncol - 1.0) / 2.0
sino_180_360 = np.fliplr(sino_0_180)
comp_sino = np.flipud(sino_0_180) # Used to avoid local minima
list_cor = np.arange(start_cor, stop_cor + 0.5, 0.5)
list_metric = np.zeros(len(list_cor), dtype=np.float32)
mask = make_mask(2 * nrow, ncol, 0.5 * ratio * ncol)
for i, cor in enumerate(list_cor):
shift = np.int16(2.0 * (cor - cen_fliplr))
sino_shift = np.roll(sino_180_360, shift, axis=1)
if shift >= 0:
sino_shift[:, :shift] = comp_sino[:, :shift]
else:
sino_shift[:, shift:] = comp_sino[:, shift:]
mat1 = np.vstack((sino_0_180, sino_shift))
list_metric[i] = np.mean(
np.abs(np.fft.fftshift(fft.fft2(mat1))) * mask)
min_pos = np.argmin(list_metric)
cor = list_cor[min_pos]
return cor
def fine_search_based_subpixel_shift(
sino_0_180, start_cor, search_radius=4, search_step=0.25, ratio=0.5):
"""
Find center of rotation (CoR) with sub-pixel accuracy by shifting a\
180-360 sinogram around the coarse CoR. The 180-360 sinogram is made\
by flipping horizontally a 0-180 sinogram.
Angular direction is along the axis 0.
Parameters
----------
sino_0_180 : float
Sinogram in the angle range of [0;180].
start_cor : float
Starting point for searching CoR.
search_radius : float
Searching range = (start_cor +/- search_radius)
search_step : float
Searching step.
ratio : float
Ratio between the sample and the width of the sinogram. Default value
of 0.5 works in most cases (even when a sample is larger than the
field_of_view).
Returns
-------
float
Center of rotation.
"""
# Denoising
sino_0_180 = ndi.gaussian_filter(sino_0_180, (2, 2), mode='reflect')
(nrow, ncol) = sino_0_180.shape
sino_180_360 = np.fliplr(sino_0_180)
search_radius = np.clip(np.abs(search_radius), 1, ncol // 10 - 1)
search_step = np.clip(np.abs(search_step), 0.1, 1.1)
start_cor = np.clip(start_cor, search_radius, ncol - search_radius - 1)
cen_fliplr = (ncol - 1.0) / 2.0
list_cor = start_cor + np.arange(
-search_radius, search_radius + search_step, search_step)
comp_sino = np.flipud(sino_0_180) # Used to avoid local minima
list_metric = np.zeros(len(list_cor), dtype=np.float32)
mask = make_mask(2 * nrow, ncol, 0.5 * ratio * ncol)
for i, cor in enumerate(list_cor):
shift = 2.0 * (cor - cen_fliplr)
sino_shift = ndi.interpolation.shift(
sino_180_360, (0, shift), order=3, prefilter=True)
if shift >= 0:
shift_int = np.int16(np.ceil(shift))
sino_shift[:, :shift_int] = comp_sino[:, :shift_int]
else:
shift_int = np.int16(np.floor(shift))
sino_shift[:, shift_int:] = comp_sino[:, shift_int:]
mat1 = np.vstack((sino_0_180, sino_shift))
list_metric[i] = np.mean(
np.abs(np.fft.fftshift(fft.fft2(mat1))) * mask)
min_pos = np.argmin(list_metric)
cor = list_cor[min_pos]
return cor
def fftFromLiteMap(liteMap,applySlepianTaper = False,nresForSlepian=3.0):
"""
@brief Creates an fft2D object out of a liteMap
@param liteMap The map whose fft is being taken
@param applySlepianTaper If True applies the lowest order taper (to minimize edge-leakage)
@param nresForSlepian If above is True, specifies the resolution of the taeper to use.
"""
ft = fft2D()
ft.Nx = liteMap.Nx
ft.Ny = liteMap.Ny
flTrace.issue("flipper.fftTools",1, "Taking FFT of map with (Ny,Nx)= (%f,%f)"%(ft.Ny,ft.Nx))
ft.pixScaleX = liteMap.pixScaleX
ft.pixScaleY = liteMap.pixScaleY
lx = 2*numpy.pi * fftfreq( ft.Nx, d = ft.pixScaleX )
ly = 2*numpy.pi * fftfreq( ft.Ny, d = ft.pixScaleY )
ix = numpy.mod(numpy.arange(ft.Nx*ft.Ny),ft.Nx)
iy = numpy.arange(ft.Nx*ft.Ny)/ft.Nx
modLMap = numpy.zeros([ft.Ny,ft.Nx])
modLMap[iy,ix] = numpy.sqrt(lx[ix]**2 + ly[iy]**2)
ft.modLMap = modLMap
ft.lx = lx
ft.ly = ly
ft.ix = ix
ft.iy = iy
ft.thetaMap = numpy.zeros([ft.Ny,ft.Nx])
ft.thetaMap[iy[:],ix[:]] = numpy.arctan2(ly[iy[:]],lx[ix[:]])
ft.thetaMap *=180./numpy.pi
map = liteMap.data.copy()
#map = map0.copy()
#map[:,:] =map0[::-1,:]
taper = map.copy()*0.0 + 1.0
if (applySlepianTaper) :
try:
f = open(taperDir + os.path.sep + 'taper_Ny%d_Nx%d_Nres%3.1f'%(ft.Ny,ft.Nx,nresForSlepian))
taper = pickle.load(f)
f.close()
except:
taper = slepianTaper00(ft.Nx,ft.Ny,nresForSlepian)
f = open(taperDir + os.path.sep + 'taper_Ny%d_Nx%d_Nres%3.1f'%(ft.Ny,ft.Nx,nresForSlepian),mode="w")
pickle.dump(taper,f)
f.close()
ft.kMap = fft2(map*taper)
del map, modLMap, lx, ly
return ft
def process_frames(self, data):
sino = data[0]
sino2 = np.fliplr(sino[1:])
(Nrow, Ncol) = sino.shape
mask = self._create_mask(
2*Nrow-1, Ncol, 0.5*self.parameters['ratio']*Ncol)
FT1 = fft.fftshift(fft.fft2(np.vstack((sino, sino2))))
sino = fft.ifft2(fft.ifftshift(FT1 - FT1*mask))
return sino[0:Nrow].real
def sampling_op_forward(image, mask_as_image):
""" Sampling operator
:param image: Assumed to be an array with shape [N,N]
:param mask_as_image:
:param mask_as_image: Mask of shape [N,N] where middle of the image
corresponds to zero frequency. Values should be 0,1 or
True/False.
:return: An array with shape [N,N] where some of the data have been zeroed out.
"""
N = max(image.shape)
fourier_coeff = fftw.fftshift(fftw.fft2(image)) / N
fourier_coeff_zero = np.multiply(mask_as_image, fourier_coeff)
return fourier_coeff_zero
def incoherent():
# example of sparse image in time domain, diffuse in Fourier domain
s = np.random.random((50, 50))
mask = s < .98
s[mask] = 0
fs = fftpack.fft2(s)
fs = np.abs(fs)
plt.subplot(121)
plt.imshow(1 - s, cmap=plt.cm.Greys_r, interpolation='nearest')
plt.xticks([])
plt.yticks([])
plt.subplot(122)
plt.imshow(fs, cmap=plt.cm.Greys_r, interpolation='nearest')
plt.xticks([])
plt.yticks([])
plt.savefig('incoherent.pdf')
plt.clf()
def incoherent():
# example of sparse image in time domain, diffuse in Fourier domain
s = np.random.random((50,50))
mask = s < .98
s[mask] = 0
fs = fftpack.fft2(s)
fs = np.abs(fs)
plt.subplot(121)
plt.imshow(1-s, cmap=plt.cm.Greys_r, interpolation='nearest')
plt.xticks([])
plt.yticks([])
plt.subplot(122)
plt.imshow(fs, cmap=plt.cm.Greys_r, interpolation='nearest')
plt.xticks([])
plt.yticks([])
plt.savefig('incoherent.pdf')
plt.clf()
def process_frames(self, data):
sinogram = data[0]
(height, _) = sinogram.shape
if height % 2 == 0:
height = height - 1
sinofit = np.abs(
savgol_filter(sinogram, height, self.order, axis=0, mode='mirror'))
sinofit2 = np.pad(sinofit, ((0, 0), (self.pad, self.pad)), mode='edge')
sinofit2 = np.pad(sinofit2, ((self.pad, self.pad), (0, 0)),
mode='mean')
sinofitsmooth = np.real(
fft.ifft2(fft.fft2(sinofit2 * self.matsign) * self.window2d) *
self.matsign)
sinofitsmooth = sinofitsmooth[self.pad:self.height1 - self.pad,
self.pad:self.width1 - self.pad]
num1 = np.mean(sinofit)
num2 = np.mean(sinofitsmooth)
sinofitsmooth = num1 * sinofitsmooth / num2
return sinogram / sinofit * sinofitsmooth
def process_frames(self, data):
sinogram = data[0]
(height, _) = sinogram.shape
if height%2==0:
height = height - 1
sinofit = np.abs(savgol_filter(
sinogram, height, self.order, axis=0, mode = 'mirror'))
sinofit2 = np.pad(
sinofit,((0, 0), (self.pad, self.pad)), mode = 'edge')
sinofit2 = np.pad(
sinofit2,((self.pad, self.pad), (0, 0)), mode = 'mean')
sinofitsmooth = np.real(fft.ifft2(fft.fft2(
sinofit2*self.matsign)*self.window2d)*self.matsign)
sinofitsmooth = sinofitsmooth[self.pad:self.height1-self.pad,
self.pad:self.width1-self.pad]
num1 = np.mean(sinofit)
num2 = np.mean(sinofitsmooth)
sinofitsmooth = num1*sinofitsmooth/num2
return sinogram/sinofit*sinofitsmooth
def _coarse_search(self, sino, start_cor, stop_cor, ratio, drop):
"""
Coarse search for finding the rotation center.
"""
(nrow, ncol) = sino.shape
start_cor, stop_cor = np.sort((start_cor,stop_cor))
start_cor = np.int16(np.clip(start_cor, 0, ncol-1))
stop_cor = np.int16(np.clip(stop_cor, 0, ncol-1))
cen_fliplr = (ncol - 1.0) / 2.0
# Flip left-right the [0:Pi ] sinogram to make a full [0;2Pi] sinogram
flip_sino = np.fliplr(sino)
# Below image is used for compensating the shift of the [Pi;2Pi] sinogram
# It helps to avoid local minima.
comp_sino = np.flipud(sino)
list_cor = np.arange(start_cor, stop_cor + 1.0)
list_metric = np.zeros(len(list_cor), dtype=np.float32)
mask = self._create_mask(2 * nrow, ncol, 0.5 * ratio * ncol, drop)
sino_sino = np.vstack((sino, flip_sino))
for i, cor in enumerate(list_cor):
shift = np.int16(2.0*(cor - cen_fliplr))
_sino = sino_sino[nrow:]
_sino[...] = np.roll(flip_sino, shift, axis=1)
if shift >= 0:
_sino[:, :shift] = comp_sino[:, :shift]
else:
_sino[:, shift:] = comp_sino[:, shift:]
list_metric[i] = np.mean(
np.abs(np.fft.fftshift(fft.fft2(sino_sino)))*mask)
minpos = np.argmin(list_metric)
if minpos==0:
logging.warn('!!! WARNING !!! Global minimum is out of'
' the searching range. Please extend smin')
self.error_msg_1 = "!!! WARNING !!! Global minimum is out of "\
"the searching range. Please extend smin"
if minpos==len(list_metric)-1:
logging.warn('!!! WARNING !!! Global minimum is out of'
' the searching range. Please extend smax')
self.error_msg_2 = "!!! WARNING !!! Global minimum is out of "\
"the searching range. Please extend smax"
rot_centre = list_cor[minpos]
return rot_centre
def _coarse_search(self, sino, start_cor, stop_cor, ratio, drop):
"""
Coarse search for finding the rotation center.
"""
(nrow, ncol) = sino.shape
start_cor, stop_cor = np.sort((start_cor, stop_cor))
start_cor = np.int16(np.clip(start_cor, 0, ncol - 1))
stop_cor = np.int16(np.clip(stop_cor, 0, ncol - 1))
cen_fliplr = (ncol - 1.0) / 2.0
# Flip left-right the [0:Pi ] sinogram to make a full [0;2Pi] sinogram
flip_sino = np.fliplr(sino)
# Below image is used for compensating the shift of the [Pi;2Pi] sinogram
# It helps to avoid local minima.
comp_sino = np.flipud(sino)
list_cor = np.arange(start_cor, stop_cor + 1.0)
list_metric = np.zeros(len(list_cor), dtype=np.float32)
mask = self._create_mask(2 * nrow, ncol, 0.5 * ratio * ncol, drop)
sino_sino = np.vstack((sino, flip_sino))
for i, cor in enumerate(list_cor):
shift = np.int16(2.0 * (cor - cen_fliplr))
_sino = sino_sino[nrow:]
_sino[...] = np.roll(flip_sino, shift, axis=1)
if shift >= 0:
_sino[:, :shift] = comp_sino[:, :shift]
else:
_sino[:, shift:] = comp_sino[:, shift:]
list_metric[i] = np.mean(
np.abs(np.fft.fftshift(fft.fft2(sino_sino))) * mask)
minpos = np.argmin(list_metric)
if minpos == 0:
self.error_msg_1 = "!!! WARNING !!! Global minimum is out of "\
"the searching range. Please extend smin"
logging.warn(self.error_msg_1)
cu.user_message(self.error_msg_1)
if minpos == len(list_metric) - 1:
self.error_msg_2 = "!!! WARNING !!! Global minimum is out of "\
"the searching range. Please extend smax"
logging.warn(self.error_msg_2)
cu.user_message(self.error_msg_2)
rot_centre = list_cor[minpos]
return rot_centre
def apply_filter(self, mat, window, pattern, pad_width):
(nrow, ncol) = mat.shape
if pattern == "PROJECTION":
top_drop = 10 # To remove the time stamp at some data
mat_pad = np.pad(mat[top_drop:],
((pad_width + top_drop, pad_width),
(pad_width, pad_width)),
mode="edge")
win_pad = np.pad(window, pad_width, mode="edge")
mat_dec = fft.ifft2(
fft.fft2(-np.log(mat_pad)) / fft.ifftshift(win_pad))
mat_dec = np.abs(mat_dec[pad_width:pad_width + nrow,
pad_width:pad_width + ncol])
else:
mat_pad = np.pad(-np.log(mat), ((0, 0), (pad_width, pad_width)),
mode='edge')
win_pad = np.pad(window, ((0, 0), (pad_width, pad_width)),
mode="edge")
mat_fft = np.fft.fftshift(fft.fft(mat_pad), axes=1) / win_pad
mat_dec = fft.ifft(np.fft.ifftshift(mat_fft, axes=1))
mat_dec = np.abs(mat_dec[:, pad_width:pad_width + ncol])
return np.float32(np.exp(-mat_dec))
def fft_image(image):
"""
Perform an FFT on the supplied image array.
Parameters
----------
image : ndarray
An array of the image
Returns
-------
namedtuple
A namedtuple containing the the fast-fourier transform, the
amplitude and phase.
"""
image_fft = fft2(image)
image_amplitude = np.abs(image_fft)
image_phase = np.angle(image_fft)
result = namedtuple('Image', 'FFT Amplitude Phase')
return result(image_fft, image_amplitude, image_phase)
def _2d_filter(mat, sigmax, sigmay, pad):
"""
Filtering an image using 2D Gaussian window.
---------
Parameters: - mat: 2D array.
- sigmax, sigmay: sigmas of the window.
- pad: padding for FFT
---------
Return: - filtered image.
"""
matpad = np.pad(mat, ((0, 0), (pad, pad)), mode='edge')
matpad = np.pad(matpad, ((pad, pad), (0, 0)), mode='mean')
(nrow, ncol) = matpad.shape
win2d = _2d_window_ellipse(nrow, ncol, sigmax, sigmay)
listx = np.arange(0, ncol)
listy = np.arange(0, nrow)
x, y = np.meshgrid(listx, listy)
matsign = np.power(-1.0, x + y)
# matfilter = np.real(ifft2(fft2(matpad*matsign)*win2d)*matsign)
matfilter = np.real(
fft_vo.ifft2(fft_vo.fft2(matpad * matsign) * win2d) * matsign)
return matfilter[pad:nrow - pad, pad:ncol - pad]
def fft_image(image):
"""
Perform an FFT on the supplied image array.
Parameters
----------
image : ndarray
An array of the image
Returns
-------
namedtuple
A namedtuple containing the the fast-fourier transform, the
amplitude and phase.
"""
image_fft = fft2(image)
image_amplitude = np.abs(image_fft)
image_phase = np.angle(image_fft)
result = namedtuple('Image', 'FFT Amplitude Phase')
return result(image_fft, image_amplitude, image_phase)
def sample_image(im, k_mask_idx1, k_mask_idx2):
"""
Creates the fourier samples the AUTOMAP network is trained to recover.
The parameters k_mask_idx1 and k_mask_idx2 cointains the row and column
indices, respectively, of the samples the network is trained to recover.
It is assumed that these indices have the same ordering of the coefficents,
as the network is used to recover.
:param im: Image, assumed of size [batch_size, height, width]. The intensity
values of the image should lie in the range [0, 1].
:param k_maks_idx1: Row indices of the Fourier samples
:param k_maks_idx2: Column indices of the Fourier samples
:return: Fourier samples in the format the AUTOMAP network expect
"""
# Scale the image to the right range
im1 = 4096 * im
batch_size = im1.shape[0]
nbr_samples = k_mask_idx1.shape[0]
samp_batch = np.zeros([batch_size, 2 * nbr_samples], dtype=np.float32)
for i in range(batch_size):
single_im = np.squeeze(im1[i, :, :])
fft_im = fftw.fft2(single_im)
samples = fft_im[k_mask_idx1, k_mask_idx2]
samples_real = np.real(samples)
samples_imag = np.imag(np.conj(samples))
samples_concat = np.squeeze(
np.concatenate((samples_real, samples_imag)))
samples_concat = (0.0075 / (2 * 4096)) * samples_concat
samp_batch[i] = samples_concat
return samp_batch
def _coarse_search(self, sino, list_shift):
# search minsearch to maxsearch in 1 pixel steps
list_metric = np.zeros(len(list_shift), dtype=np.float32)
(Nrow, Ncol) = sino.shape
# check angles to determine if a sinogram should be chopped off.
# Copy the sinogram and flip left right, to make a full [0:2Pi] sino
sino2 = np.fliplr(sino[1:])
# This image is used for compensating the shift of sino2
compensateimage = np.zeros((Nrow-1, Ncol), dtype=np.float32)
# Start coarse search in which the shift step is 1
compensateimage[:] = np.flipud(sino)[1:]
mask = self._create_mask(2*Nrow-1, Ncol,
0.5*self.parameters['ratio']*Ncol)
count = 0
for i in list_shift:
sino2a = np.roll(sino2, i, axis=1)
if i >= 0:
sino2a[:, 0:i] = compensateimage[:, 0:i]
else:
sino2a[:, i:] = compensateimage[:, i:]
list_metric[count] = np.sum(
np.abs(fft.fftshift(fft.fft2(np.vstack((sino, sino2a)))))*mask)
count += 1
return list_metric
def _reconstruct(self, propagation_distance, fourier_mask=None):
"""
Reconstruct the wave at a single ``propagation_distance`` for a single ``wavelength``.
Parameters
----------
propagation_distance : float
Propagation distance [m]
spectral_peak : integer pair [x,y]
Centroid of spectral peak for wavelength in power spectrum of hologram FT
fourier_mask : array_like or None, optional
Fourier-domain mask. If None (default), a mask is determined from the position of the
main spectral peak.
Returns
-------
reconstructed_wave : `~numpy.ndarray` ndim 3
The reconstructed wave as an array of dimensions (X, Y, wavelengths)
"""
x_peak, y_peak = self.spectral_peak
# Calculate mask radius. TODO: Update 250 to an automated guess based on input values.
if self.rebin_factor != 1:
mask_radius = 150. / self.rebin_factor
elif self.crop_fraction is not None and self.crop_fraction != 0:
mask_radius = 150. * self.crop_fraction
else:
mask_radius = 150.
# Either use a Fourier-domain mask based on coords of spectral peak,
# or a user-specified mask
if fourier_mask is None:
mask = self.real_image_mask(x_peak, y_peak, mask_radius)
else:
mask = np.asarray(fourier_mask, dtype=np.bool)
mask = np.atleast_3d(mask)
# Calculate Fourier transform of impulse response function
G = self.fourier_trans_of_impulse_resp_func(
np.atleast_1d([propagation_distance] *
self.wavelength.size).reshape((1, 1, -1)) -
self.chromatic_shift)
# Now calculate digital phase mask. First center the spectral peak for each channel
x_peak, y_peak = x_peak.reshape(-1), y_peak.reshape(-1)
shifted_ft_hologram = np.empty_like(np.atleast_3d(mask),
dtype=np.complex128)
for channel in range(self.wavelength.size):
shifted_ft_hologram[:, :, channel] = arrshift(
self.ft_hologram * mask[:, :, channel],
[-x_peak[channel], -y_peak[channel]],
axes=(0, 1))
# Apodize the result
psi = self.apodize(shifted_ft_hologram * G)
digital_phase_mask = self.get_digital_phase_mask(psi)
# Reconstruct the image
# fftshift is independent of channel
psi = np.empty_like(np.atleast_3d(shifted_ft_hologram))
for channel in range(psi.shape[2]):
psi[:, :, channel] = arrshift(fftshift(
fft2(self.apodize(self.hologram) *
digital_phase_mask[:, :, channel],
axes=(0, 1))) * mask[:, :, channel],
[-x_peak[channel], -y_peak[channel]],
axes=(0, 1))
psi *= G
return fftshift(ifft2(psi, axes=(0, 1)), axes=(0, 1))
def perfft2(im, compute_P=True, compute_spatial=False):
"""
Moisan's Periodic plus Smooth Image Decomposition. The image is
decomposed into two parts:
im = s + p
where 's' is the 'smooth' component with mean 0, and 'p' is the 'periodic'
component which has no sharp discontinuities when one moves cyclically
across the image boundaries.
useage: S, [P, s, p] = perfft2(im)
where: im is the image
S is the FFT of the smooth component
P is the FFT of the periodic component, returned if
compute_P (default)
s & p are the smooth and periodic components in the spatial
domain, returned if compute_spatial
By default this function returns `P` and `S`, the FFTs of the periodic and
smooth components respectively. If `compute_spatial=True`, the spatial
domain components 'p' and 's' are also computed.
This code is adapted from Lionel Moisan's Scilab function 'perdecomp.sci'
"Periodic plus Smooth Image Decomposition" 07/2012 available at:
<http://www.mi.parisdescartes.fr/~moisan/p+s>
"""
if im.dtype not in ['float32', 'float64']:
im = np.float64(im)
rows, cols = im.shape
# Compute the boundary image which is equal to the image discontinuity
# values across the boundaries at the edges and is 0 elsewhere
s = np.zeros_like(im)
s[0, :] = im[0, :] - im[-1, :]
s[-1, :] = -s[0, :]
s[:, 0] = s[:, 0] + im[:, 0] - im[:, -1]
s[:, -1] = s[:, -1] - im[:, 0] + im[:, -1]
# Generate grid upon which to compute the filter for the boundary image
# in the frequency domain. Note that cos is cyclic hence the grid
# values can range from 0 .. 2*pi rather than 0 .. pi and then pi .. 0
x, y = (2 * np.pi * np.arange(0, v) / float(v) for v in (cols, rows))
cx, cy = np.meshgrid(x, y)
denom = (2. * (2. - np.cos(cx) - np.cos(cy)))
denom[0, 0] = 1. # avoid / 0
S = fft2(s) / denom
S[0, 0] = 0 # enforce zero mean
if compute_P or compute_spatial:
P = fft2(im) - S
if compute_spatial:
s = ifft2(S).real
p = im - s
return S, P, s, p
else:
return S, P
else:
return S
def perfft2(im, compute_P=True, compute_spatial=False):
"""
Moisan's Periodic plus Smooth Image Decomposition. The image is
decomposed into two parts:
im = s + p
where 's' is the 'smooth' component with mean 0, and 'p' is the 'periodic'
component which has no sharp discontinuities when one moves cyclically
across the image boundaries.
useage: S, [P, s, p] = perfft2(im)
where: im is the image
S is the FFT of the smooth component
P is the FFT of the periodic component, returned if
compute_P (default)
s & p are the smooth and periodic components in the spatial
domain, returned if compute_spatial
By default this function returns `P` and `S`, the FFTs of the periodic and
smooth components respectively. If `compute_spatial=True`, the spatial
domain components 'p' and 's' are also computed.
This code is adapted from Lionel Moisan's Scilab function 'perdecomp.sci'
"Periodic plus Smooth Image Decomposition" 07/2012 available at:
<http://www.mi.parisdescartes.fr/~moisan/p+s>
"""
if im.dtype not in ['float32', 'float64']:
im = np.float64(im)
rows, cols = im.shape
# Compute the boundary image which is equal to the image discontinuity
# values across the boundaries at the edges and is 0 elsewhere
s = np.zeros_like(im)
s[0, :] = im[0, :] - im[-1, :]
s[-1, :] = -s[0, :]
s[:, 0] = s[:, 0] + im[:, 0] - im[:, -1]
s[:, -1] = s[:, -1] - im[:, 0] + im[:, -1]
# Generate grid upon which to compute the filter for the boundary image
# in the frequency domain. Note that cos is cyclic hence the grid
# values can range from 0 .. 2*pi rather than 0 .. pi and then pi .. 0
x, y = (2 * np.pi * np.arange(0, v) / float(v) for v in (cols, rows))
cx, cy = np.meshgrid(x, y)
denom = (2. * (2. - np.cos(cx) - np.cos(cy)))
denom[0, 0] = 1. # avoid / 0
S = fft2(s) / denom
S[0, 0] = 0 # enforce zero mean
if compute_P or compute_spatial:
P = fft2(im) - S
if compute_spatial:
s = ifft2(S).real
p = im - s
return S, P, s, p
else:
return S, P
else:
return S
def cross_correlation_2d(pixels1, pixels2):
'''Align the second image with the first using max cross-correlation
returns the x,y offsets to add to image1's indexes to align it with
image2
Many of the ideas here are based on the paper, "Fast Normalized
Cross-Correlation" by J.P. Lewis
(http://www.idiom.com/~zilla/Papers/nvisionInterface/nip.html)
which is frequently cited when addressing this problem.
'''
#
# We double the size of the image to get a field of zeros
# for the parts of one image that don't overlap the displaced
# second image.
#
# Since we're going into the frequency domain, if the images are of
# different sizes, we can make the FFT shape large enough to capture
# the period of the largest image - the smaller just will have zero
# amplitude at that frequency.
#
s = np.maximum(pixels1.shape, pixels2.shape)
fshape = s*2
#
# Calculate the # of pixels at a particular point
#
i,j = np.mgrid[-s[0]:s[0],
-s[1]:s[1]]
unit = np.abs(i*j).astype(float)
unit[unit<1]=1 # keeps from dividing by zero in some places
#
# Normalize the pixel values around zero which does not affect the
# correlation, keeps some of the sums of multiplications from
# losing precision and precomputes t(x-u,y-v) - t_mean
#
pixels1 = np.nan_to_num(pixels1-np.nanmean(pixels1))
pixels2 = np.nan_to_num(pixels2-np.nanmean(pixels2))
#
# Lewis uses an image, f and a template t. He derives a normalized
# cross correlation, ncc(u,v) =
# sum((f(x,y)-f_mean(u,v))*(t(x-u,y-v)-t_mean),x,y) /
# sqrt(sum((f(x,y)-f_mean(u,v))**2,x,y) * (sum((t(x-u,y-v)-t_mean)**2,x,y)
#
# From here, he finds that the numerator term, f_mean(u,v)*(t...) is zero
# leaving f(x,y)*(t(x-u,y-v)-t_mean) which is a convolution of f
# by t-t_mean.
#
fp1 = fft2(pixels1.astype('float32'),fshape)
fp2 = fft2(pixels2.astype('float32'),fshape)
corr12 = ifft2(fp1 * fp2.conj()).real
#
# Use the trick of Lewis here - compute the cumulative sums
# in a fashion that accounts for the parts that are off the
# edge of the template.
#
# We do this in quadrants:
# q0 q1
# q2 q3
# For the first,
# q0 is the sum over pixels1[i:,j:] - sum i,j backwards
# q1 is the sum over pixels1[i:,:j] - sum i backwards, j forwards
# q2 is the sum over pixels1[:i,j:] - sum i forwards, j backwards
# q3 is the sum over pixels1[:i,:j] - sum i,j forwards
#
# The second is done as above but reflected lr and ud
#
p1_si = pixels1.shape[0]
p1_sj = pixels1.shape[1]
p1_sum = np.zeros(fshape)
p1_sum[:p1_si,:p1_sj] = cumsum_quadrant(pixels1, False, False)
p1_sum[:p1_si,-p1_sj:] = cumsum_quadrant(pixels1, False, True)
p1_sum[-p1_si:,:p1_sj] = cumsum_quadrant(pixels1, True, False)
p1_sum[-p1_si:,-p1_sj:] = cumsum_quadrant(pixels1, True, True)
#
# Divide the sum over the # of elements summed-over
#
p1_mean = p1_sum / unit
p2_si = pixels2.shape[0]
p2_sj = pixels2.shape[1]
p2_sum = np.zeros(fshape)
p2_sum[:p2_si,:p2_sj] = cumsum_quadrant(pixels2, False, False)
p2_sum[:p2_si,-p2_sj:] = cumsum_quadrant(pixels2, False, True)
p2_sum[-p2_si:,:p2_sj] = cumsum_quadrant(pixels2, True, False)
p2_sum[-p2_si:,-p2_sj:] = cumsum_quadrant(pixels2, True, True)
p2_sum = np.fliplr(np.flipud(p2_sum))
p2_mean = p2_sum / unit
#
# Once we have the means for u,v, we can caluclate the
# variance-like parts of the equation. We have to multiply
# the mean^2 by the # of elements being summed-over
# to account for the mean being summed that many times.
#
p1sd = np.sum(pixels1**2) - p1_mean**2 * np.product(s)
p2sd = np.sum(pixels2**2) - p2_mean**2 * np.product(s)
#
# There's always chance of roundoff error for a zero value
# resulting in a negative sd, so limit the sds here
#
sd = np.sqrt(np.maximum(p1sd * p2sd, 0))
with warnings.catch_warnings():
warnings.simplefilter("ignore")
corrnorm = corr12 / sd
#
# There's not much information for points where the standard
# deviation is less than 1/100 of the maximum. We exclude these
# from consideration.
#
corrnorm[(unit < np.product(s) / 2) &
(sd < np.mean(sd) / 100)] = 0
# Also exclude possibilites with few observed pixels.
corrnorm[unit < np.product(s) / 4] = 0
return corrnorm
def FFT2(I):
[Ny, Nx] = np.shape(I)
FT = fftshift(fft2(ifftshift(I))) / np.sqrt(Ny) / np.sqrt(Nx)
return FT
def cross_correlation_2d(pixels1, pixels2):
'''Align the second image with the first using max cross-correlation
returns the x,y offsets to add to image1's indexes to align it with
image2
Many of the ideas here are based on the paper, "Fast Normalized
Cross-Correlation" by J.P. Lewis
(http://www.idiom.com/~zilla/Papers/nvisionInterface/nip.html)
which is frequently cited when addressing this problem.
'''
#
# We double the size of the image to get a field of zeros
# for the parts of one image that don't overlap the displaced
# second image.
#
# Since we're going into the frequency domain, if the images are of
# different sizes, we can make the FFT shape large enough to capture
# the period of the largest image - the smaller just will have zero
# amplitude at that frequency.
#
s = np.maximum(pixels1.shape, pixels2.shape)
fshape = s*2
#
# Calculate the # of pixels at a particular point
#
i, j = np.mgrid[-s[0]:s[0],
-s[1]:s[1]]
unit = np.abs(i*j).astype(float)
unit[unit < 1] = 1 # keeps from dividing by zero in some places
#
# Normalize the pixel values around zero which does not affect the
# correlation, keeps some of the sums of multiplications from
# losing precision and precomputes t(x-u,y-v) - t_mean
#
pixels1 = np.nan_to_num(pixels1-nanmean(pixels1))
pixels2 = np.nan_to_num(pixels2-nanmean(pixels2))
#
# Lewis uses an image, f and a template t. He derives a normalized
# cross correlation, ncc(u,v) =
# sum((f(x,y)-f_mean(u,v))*(t(x-u,y-v)-t_mean),x,y) /
# sqrt(sum((f(x,y)-f_mean(u,v))**2,x,y) * (sum((t(x-u,y-v)-t_mean)**2,x,y)
#
# From here, he finds that the numerator term, f_mean(u,v)*(t...) is zero
# leaving f(x,y)*(t(x-u,y-v)-t_mean) which is a convolution of f
# by t-t_mean.
#
fp1 = fft2(pixels1.astype('float32'), fshape)
fp2 = fft2(pixels2.astype('float32'), fshape)
corr12 = ifft2(fp1 * fp2.conj()).real
#
# Use the trick of Lewis here - compute the cumulative sums
# in a fashion that accounts for the parts that are off the
# edge of the template.
#
# We do this in quadrants:
# q0 q1
# q2 q3
# For the first,
# q0 is the sum over pixels1[i:,j:] - sum i,j backwards
# q1 is the sum over pixels1[i:,:j] - sum i backwards, j forwards
# q2 is the sum over pixels1[:i,j:] - sum i forwards, j backwards
# q3 is the sum over pixels1[:i,:j] - sum i,j forwards
#
# The second is done as above but reflected lr and ud
#
p1_si = pixels1.shape[0]
p1_sj = pixels1.shape[1]
p1_sum = np.zeros(fshape)
p1_sum[:p1_si, :p1_sj] = cumsum_quadrant(pixels1, False, False)
p1_sum[:p1_si, -p1_sj:] = cumsum_quadrant(pixels1, False, True)
p1_sum[-p1_si:, :p1_sj] = cumsum_quadrant(pixels1, True, False)
p1_sum[-p1_si:, -p1_sj:] = cumsum_quadrant(pixels1, True, True)
#
# Divide the sum over the # of elements summed-over
#
p1_mean = old_div(p1_sum, unit)
p2_si = pixels2.shape[0]
p2_sj = pixels2.shape[1]
p2_sum = np.zeros(fshape)
p2_sum[:p2_si, :p2_sj] = cumsum_quadrant(pixels2, False, False)
p2_sum[:p2_si, -p2_sj:] = cumsum_quadrant(pixels2, False, True)
p2_sum[-p2_si:, :p2_sj] = cumsum_quadrant(pixels2, True, False)
p2_sum[-p2_si:, -p2_sj:] = cumsum_quadrant(pixels2, True, True)
p2_sum = np.fliplr(np.flipud(p2_sum))
p2_mean = old_div(p2_sum, unit)
#
# Once we have the means for u,v, we can caluclate the
# variance-like parts of the equation. We have to multiply
# the mean^2 by the # of elements being summed-over
# to account for the mean being summed that many times.
#
p1sd = np.sum(pixels1**2) - p1_mean**2 * np.product(s)
p2sd = np.sum(pixels2**2) - p2_mean**2 * np.product(s)
#
# There's always chance of roundoff error for a zero value
# resulting in a negative sd, so limit the sds here
#
sd = np.sqrt(np.maximum(p1sd * p2sd, 0))
with warnings.catch_warnings():
warnings.simplefilter("ignore")
corrnorm = old_div(corr12, sd)
#
# There's not much information for points where the standard
# deviation is less than 1/100 of the maximum. We exclude these
# from consideration.
#
corrnorm[(unit < old_div(np.product(s), 2)) &
(sd < old_div(np.mean(sd), 100))] = 0
# Also exclude possibilites with few observed pixels.
corrnorm[unit < old_div(np.product(s), 4)] = 0
return corrnorm