def grad(self, scalar):
self.scalar_input_test(scalar)
scalar_hat = self.fft(scalar)
return fft.irfftn(1.0j * self.K * (2 * np.pi / self.L) * scalar_hat,
axes=(1, 2),
threads=self.num_threads)
def uirdftn(inarray, ndim=None, *args, **kwargs):
"""N-dim real unitary discrete Fourier transform
This transform consider the Hermitian property of the transform
from complex to real real input.
Parameters
----------
inarray : ndarray
The array to transform.
ndim : int, optional
The `ndim` last axis along wich to compute the transform. All
axes by default.
Returns
-------
outarray : array-like (the last ndim as (N - 1) * 2 lenght)
"""
if not ndim:
ndim = inarray.ndim
return irfftn(inarray, axes=range(-ndim, 0), *args, **kwargs) * np.sqrt(
np.prod(inarray.shape[-ndim:-1]) * (inarray.shape[-1] - 1) * 2)
def grad_invlap(self, scalar):
self.scalar_input_test(scalar)
scalar_hat = self.fft(scalar)
return fft.irfftn(-1.0j * self.KoverK2 * (2 * np.pi / self.L)**(-1.0) *
scalar_hat,
axes=(1, 2),
threads=self.num_threads)
def fft_gaussian_filter(img, sigma):
"""FFT gaussian convolution
Parameters
----------
img : ndarray
Image to convolve with a gaussian kernel
sigma : int or sequence
The sigma(s) of the gaussian kernel in _real space_
Returns
-------
filt_img : ndarray
The filtered image
"""
# This doesn't help agreement but it will make things faster
# pull the shape
s1 = np.array(img.shape)
# s2 = np.array([int(s * 4) for s in _normalize_sequence(sigma, img.ndim)])
shape = s1 # + s2 - 1
# calculate a nice shape
fshape = [sig.fftpack.helper.next_fast_len(int(d)) for d in shape]
# pad out with reflection
pad_img = fft_pad(img, fshape, "reflect")
# calculate the padding
padding = tuple(_calc_pad(o, n) for o, n in zip(img.shape, pad_img.shape))
# so that we can calculate the cropping, maybe this should be integrated
# into `fft_pad` ...
fslice = tuple(
slice(s, -e) if e != 0 else slice(s, None) for s, e in padding)
# fourier transfrom and apply the filter
kimg = rfftn(pad_img, fshape)
filt_kimg = fourier_gaussian(kimg, sigma, pad_img.shape[-1])
# inverse FFT and return.
return irfftn(filt_kimg, fshape)[fslice]
def div(self, vector):
""" Take divergence of vector """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
return fft.irfftn(np.sum(
1j * self.K * (2 * np.pi / self.L) * vector_hat, 0),
threads=self.num_threads)
def h1norm(self, vector):
""" L2 norm of a vector field """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
grad_vx = fft.irfftn(1.0j * self.K * (2 * np.pi / self.L) *
vector_hat[0],
axes=(1, 2),
threads=self.num_threads)
grad_vy = fft.irfftn(1.0j * self.K * (2 * np.pi / self.L) *
vector_hat[1],
axes=(1, 2),
threads=self.num_threads)
integrand = (grad_vx[0]**2.0 + grad_vx[1]**2.0 + grad_vy[0]**2.0 +
grad_vy[1]**2.0)
return np.sum(np.ravel(integrand) * self.h**2)**0.5
def fftconvolve(in1, in2, mode="same", threads=1):
"""Same as above but with pyfftw added in"""
in1 = np.asarray(in1)
in2 = np.asarray(in2)
if in1.ndim == in2.ndim == 0: # scalar inputs
return in1 * in2
elif not in1.ndim == in2.ndim:
raise ValueError("in1 and in2 should have the same dimensionality")
elif in1.size == 0 or in2.size == 0: # empty arrays
return np.array([])
s1 = np.array(in1.shape)
s2 = np.array(in2.shape)
complex_result = (np.issubdtype(in1.dtype, complex)
or np.issubdtype(in2.dtype, complex))
shape = s1 + s2 - 1
# Check that input sizes are compatible with 'valid' mode
if sig._inputs_swap_needed(mode, s1, s2):
# Convolution is commutative; order doesn't have any effect on output
in1, s1, in2, s2 = in2, s2, in1, s1
# Speed up FFT by padding to optimal size for FFTPACK
fshape = [sig.fftpack.helper.next_fast_len(int(d)) for d in shape]
fslice = tuple([slice(0, int(sz)) for sz in shape])
# Pre-1.9 NumPy FFT routines are not threadsafe. For older NumPys, make
# sure we only call rfftn/irfftn from one thread at a time.
if not complex_result and (sig._rfft_mt_safe
or sig._rfft_lock.acquire(False)):
try:
sp1 = rfftn(in1, fshape, threads=threads)
sp2 = rfftn(in2, fshape, threads=threads)
ret = (irfftn(sp1 * sp2, fshape,
threads=threads)[fslice].copy())
finally:
if not sig._rfft_mt_safe:
sig._rfft_lock.release()
else:
# If we're here, it's either because we need a complex result, or we
# failed to acquire _rfft_lock (meaning rfftn isn't threadsafe and
# is already in use by another thread). In either case, use the
# (threadsafe but slower) SciPy complex-FFT routines instead.
sp1 = fftn(in1, fshape, threads=threads)
sp2 = fftn(in2, fshape, threads=threads)
ret = ifftn(sp1 * sp2, threads=threads)[fslice].copy()
if not complex_result:
ret = ret.real
if mode == "full":
return ret
elif mode == "same":
return sig._centered(ret, s1)
elif mode == "valid":
return sig._centered(ret, s1 - s2 + 1)
else:
raise ValueError("Acceptable mode flags are 'valid',"
" 'same', or 'full'.")
def curl(self, vector):
""" Perform curl of vector """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
w = fft.irfftn(1j * self.K[0] *
(2.0 * np.pi / self.L) * vector_hat[1] -
1j * self.K[1] * (2.0 * np.pi / self.L) * vector_hat[0],
threads=self.num_threads)
return w
def fftconvolve(in1, in2, mode="same", threads=1):
"""Same as above but with pyfftw added in"""
in1 = np.asarray(in1)
in2 = np.asarray(in2)
if in1.ndim == in2.ndim == 0: # scalar inputs
return in1 * in2
elif not in1.ndim == in2.ndim:
raise ValueError("in1 and in2 should have the same dimensionality")
elif in1.size == 0 or in2.size == 0: # empty arrays
return np.array([])
s1 = np.array(in1.shape)
s2 = np.array(in2.shape)
complex_result = (np.issubdtype(in1.dtype, complex) or
np.issubdtype(in2.dtype, complex))
shape = s1 + s2 - 1
# Check that input sizes are compatible with 'valid' mode
if sig._inputs_swap_needed(mode, s1, s2):
# Convolution is commutative; order doesn't have any effect on output
in1, s1, in2, s2 = in2, s2, in1, s1
# Speed up FFT by padding to optimal size for FFTPACK
fshape = [sig.fftpack.helper.next_fast_len(int(d)) for d in shape]
fslice = tuple([slice(0, int(sz)) for sz in shape])
# Pre-1.9 NumPy FFT routines are not threadsafe. For older NumPys, make
# sure we only call rfftn/irfftn from one thread at a time.
if not complex_result and (sig._rfft_mt_safe or sig._rfft_lock.acquire(False)):
try:
sp1 = rfftn(in1, fshape, threads=threads)
sp2 = rfftn(in2, fshape, threads=threads)
ret = (irfftn(sp1 * sp2, fshape, threads=threads)[fslice].copy())
finally:
if not sig._rfft_mt_safe:
sig._rfft_lock.release()
else:
# If we're here, it's either because we need a complex result, or we
# failed to acquire _rfft_lock (meaning rfftn isn't threadsafe and
# is already in use by another thread). In either case, use the
# (threadsafe but slower) SciPy complex-FFT routines instead.
sp1 = fftn(in1, fshape, threads=threads)
sp2 = fftn(in2, fshape, threads=threads)
ret = ifftn(sp1 * sp2, threads=threads)[fslice].copy()
if not complex_result:
ret = ret.real
if mode == "full":
return ret
elif mode == "same":
return sig._centered(ret, s1)
elif mode == "valid":
return sig._centered(ret, s1 - s2 + 1)
else:
raise ValueError("Acceptable mode flags are 'valid',"
" 'same', or 'full'.")
def get_smoothed(self, filter):
if isinstance(filter, Number):
filter = filters.GaussianFilter(radius=filter)
if filter in self._smoothed:
return self._smoothed[filter]
knorm = self.knorm
fft_field = self.white_noise_fft * filter.W(knorm)
field = fft.irfftn(fft_field)
self._smoothed[filter] = field
return field
def fftconvolvefast(in1, in2):
s1 = np.array(in1.shape)
s2 = np.array(in2.shape)
shape = s1 + s2 - 1
fshape = [fftpack.helper.next_fast_len(int(d)) for d in shape]
fslice = tuple([slice(0, int(sz)) for sz in shape])
sp1 = rfftn(in1, fshape)
sp2 = rfftn(in2, fshape)
ret = (irfftn(sp1 * sp2, fshape)[fslice].copy())
return _centered(ret, s1 - s2 + 1)
def generate_white_noise(self):
"""Generate a white noise with the relevant power spectrum.
Returns
-------
white_noise : np.ndarray
"""
# Compute the k grid
d = self.Lbox / self.dimensions / (2 * np.pi)
all_k = [fft.fftfreq(self.dimensions, d=d)] * (self.Ndim - 1) + [
fft.rfftfreq(self.dimensions, d=d)
]
self.kgrid = kgrid = np.array(np.meshgrid(*all_k, indexing="ij"))
self.knorm = knorm = np.sqrt(np.sum(kgrid**2, axis=0))
# Compute Pk
Pk = np.zeros_like(knorm)
mask = knorm > 0
Pk[mask] = self.Pk(knorm[mask] * 2)
# Compute white noise (in Fourier space)
mu = np.random.standard_normal([self.dimensions] * self.Ndim)
muk = fft.rfftn(mu)
deltak = muk * np.sqrt(Pk)
# Compute field in real space
white_noise = fft.irfftn(deltak)
# Normalize variance
deltak_smoothed = deltak * self.filter.W(knorm)
field = fft.irfftn(deltak_smoothed)
std = field.std()
self.white_noise_fft = deltak * self.sigma8 / std
self.white_noise = white_noise * self.sigma8 / std
return self.white_noise
def smooth(self, R):
"""Smooth the data at scale R (in pixel unit)."""
if isinstance(R, unyt_quantity):
R = float(R.to('pixel'))
if R in self.data_smooth:
return self.data_smooth[R]
logger.debug('Smoothing at scale %.3f', R)
if R > 0:
data_f = self.data_raw_f * np.exp(-self.k2 * R**2 / 2)
else:
data_f = self.data_raw_f.copy()
self.data_smooth_f[R] = data_f
self.data_smooth[R] = fft.irfftn(data_f, **self.FFT_args)
return self.data_smooth[R]
def ftgr(fg, grid, real=False):
"""Fourier transform function fg from G space to R space.
Args:
fg (np.ndarray): G space function. shape == (grid.n1, grid.n2, grid.n3).
grid (FFTGrid): FFT grid on which fg is defined.
real (bool): if True, fg contains only iG1 >= 0 and thus has the shape
of (grid.n1 // 2 + 1, grid.n2, grid.n3), after FT R space function is real.
Returns:
R space function.
"""
if real:
assert fg.shape == (grid.n1h, grid.n2, grid.n3)
fgzyx = fg.T
return grid.N * irfftn(fgzyx, s=(grid.n1, grid.n2, grid.n3)).T
else:
assert fg.shape == (grid.n1, grid.n2, grid.n3)
return grid.N * ifftn(fg)
def uirfftn(inarray, dim=None, shape=None):
"""N-dimensional inverse real unitary Fourier transform.
This transform considers the Hermitian property of the transform
from complex to real input.
Parameters
----------
inarray : ndarray
The array to transform.
dim : int, optional
The last axis along which to compute the transform. All
axes by default.
shape : tuple of int, optional
The shape of the output. The shape of ``rfft`` is ambiguous in
case of odd-valued input shape. In this case, this parameter
should be provided. See ``irfftn``.
Returns
-------
outarray : ndarray
The unitary N-D inverse real Fourier transform of ``inarray``.
Notes
-----
The ``uirfft`` function assumes that the output array is
real-valued. Consequently, the input is assumed to have a Hermitian
property and redundant values are implicit.
Examples
--------
>>> input = np.ones((5, 5, 5))
>>> output = uirfftn(urfftn(input), shape=input.shape)
>>> np.allclose(input, output)
True
>>> output.shape
(5, 5, 5)
"""
if dim is None:
dim = inarray.ndim
outarray = irfftn(inarray, shape, axes=range(-dim, 0))
return outarray * np.sqrt(np.prod(outarray.shape[-dim:]))
def _ifft(arry):
"""Find inverse FFT of arry.
Parameters
----------
spectrum: array_like
The spectrum of a real-valued signal
Returns
-------
array_like
The spectrum transformed back to physical space
"""
slicer = (base_slices +
tuple(slice(None) for dim in arry.shape[ndims:]))
big_result = irfftn(arry,
axes=axes,
s=computational_shape,
threads=NUM_THREADS,
planner_effort=PLANNER_EFFORT)
return big_result[slicer]
def fft_gaussian_filter(img, sigma):
"""FFT gaussian convolution
Parameters
----------
img : ndarray
Image to convolve with a gaussian kernel
sigma : int or sequence
The sigma(s) of the gaussian kernel in _real space_
Returns
-------
filt_img : ndarray
The filtered image
"""
# This doesn't help agreement but it will make things faster
# pull the shape
s1 = np.array(img.shape)
# if any of the sizes is 32 or smaller, revert to proper filter
if any(s1 < 33):
warnings.warn(("Input is small along a dimension,"
" will revert to `gaussian_filter`"))
return gaussian_filter(img, sigma)
# s2 = np.array([int(s * 4) for s in _normalize_sequence(sigma, img.ndim)])
shape = s1 # + s2 - 1
# calculate a nice shape
fshape = [sig.fftpack.helper.next_fast_len(int(d)) for d in shape]
# pad out with reflection
pad_img = fft_pad(img, fshape, "reflect")
# calculate the padding
padding = tuple(_calc_pad(o, n) for o, n in zip(img.shape, pad_img.shape))
# so that we can calculate the cropping, maybe this should be integrated
# into `fft_pad` ...
fslice = tuple(slice(s, -e) if e != 0 else slice(s, None)
for s, e in padding)
# fourier transfrom and apply the filter
kimg = rfftn(pad_img, fshape)
filt_kimg = fourier_gaussian(kimg, sigma, pad_img.shape[-1])
# inverse FFT and return.
return irfftn(filt_kimg, fshape)[fslice]
def fftconvolve_fast(data, kernel, **kwargs):
"""A faster version of fft convolution
In this case the kernel ifftshifted before FFT but the data is not.
This can be done because the effect of fourier convolution is to
"wrap" around the data edges so whether we ifftshift before FFT
and then fftshift after it makes no difference so we can skip the
step entirely.
"""
# TODO: add error checking like in the above and add functionality
# for complex inputs. Also could add options for different types of
# padding.
dshape = np.array(data.shape)
kshape = np.array(kernel.shape)
# find maximum dimensions
maxshape = np.max((dshape, kshape), 0)
# calculate a nice shape
fshape = [sig.fftpack.helper.next_fast_len(int(d)) for d in maxshape]
# pad out with reflection
pad_data = fft_pad(data, fshape, "reflect")
# calculate padding
padding = tuple(
_calc_pad(o, n) for o, n in zip(data.shape, pad_data.shape))
# so that we can calculate the cropping, maybe this should be integrated
# into `fft_pad` ...
fslice = tuple(
slice(s, -e) if e != 0 else slice(s, None) for s, e in padding)
if kernel.shape != pad_data.shape:
# its been assumed that the background of the kernel has already been
# removed and that the kernel has already been centered
kernel = fft_pad(kernel, pad_data.shape, mode='constant')
k_kernel = rfftn(ifftshift(kernel), pad_data.shape, **kwargs)
k_data = rfftn(pad_data, pad_data.shape, **kwargs)
convolve_data = irfftn(k_kernel * k_data, pad_data.shape, **kwargs)
# return data with same shape as original data
return convolve_data[fslice]
def cross_correlation_3d(pixels1, pixels2):
'''Align the second image with the first using max cross-correlation
returns the z,y,x 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.
'''
s = np.maximum(pixels1.shape, pixels2.shape)
fshape = s*2
#
# Calculate the # of pixels at a particular point
#
i, j, k = np.mgrid[-s[0]:s[0], -s[1]:s[1], -s[2]:s[2] ]
unit = np.abs(i*j*k).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 = rfftn(pixels1.astype('float32'), fshape, axes=(0, 1, 2))
fp2 = rfftn(pixels2.astype('float32'), fshape, axes=(0, 1, 2))
corr12 = irfftn(fp1 * fp2.conj(), axes=(0, 1, 2)).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
#
def get_cumsums(im, fshape):
im_si = im.shape[0]
im_sj = im.shape[1]
im_sk = im.shape[2]
im_sum = np.zeros(fshape)
im_sum[:im_si, :im_sj, :im_sk] = cumsum_quadrant(im, False, False, False)
im_sum[:im_si, :im_sj, -im_sk:] = cumsum_quadrant(im, False, False, True)
im_sum[:im_si, -im_sj:, :im_sk] = cumsum_quadrant(im, False, True, True)
im_sum[:im_si, -im_sj:, -im_sk:] = cumsum_quadrant(im, False, True, False)
im_sum[-im_si:, :im_sj, :im_sk] = cumsum_quadrant(im, True, False, True)
im_sum[-im_si:, :im_sj, -im_sk:] = cumsum_quadrant(im, True, False, False)
im_sum[-im_si:, -im_sj:, :im_sk] = cumsum_quadrant(im, True, True, True)
im_sum[-im_si:, -im_sj:, -im_sk:] = cumsum_quadrant(im, True, True, False)
#
# Divide the sum over the # of elements summed-over
#
return old_div(im_sum, unit)
p1_mean = get_cumsums(pixels1, fshape)
p2_mean = get_cumsums(pixels2, fshape)
#
# 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
def compute_derivatives(self, R):
"""Compute the 1st and 2nd derivatives of the (Fourier) input.
Argument
--------
R: float
The smoothing scale.
Return
------
indices: (ndim, ndim, M)
The M indices of the hessian. M = ndim*(ndim+1)/2.
"""
if R not in self.data_smooth_f:
self.smooth(R)
data_f = self.data_smooth_f[R]
ndim = self.ndim
kgrid = self.kgrid
ihess = 0
indices = np.zeros((ndim, ndim), dtype=int)
logger.debug('Computing hessian and gradient in Fourier space.')
# Compute hessian and gradient in Fourier space
for idim in range(ndim):
k1 = kgrid[idim]
self.grad_f[idim, ...] = ne.evaluate('data_f * 1j * k1')
for idim2 in range(idim, ndim):
k2 = kgrid[idim2]
grad_f = self.grad_f[idim, ...]
self.hess_f[ihess, ...] = ne.evaluate('grad_f * 1j * k2')
indices[idim, idim2] = indices[idim2, idim] = ihess
ihess += 1
logger.debug('Inverse FFT of gradient')
# Get them back in real space
self.grad[...] = fft.irfftn(self.grad_f,
axes=range(1, ndim + 1),
**self.FFT_args)
logger.debug('Inverse FFT of hessian')
self.hess[...] = fft.irfftn(self.hess_f,
axes=range(1, ndim + 1),
**self.FFT_args)
logger.debug('Curvature')
if ndim == 3:
a = self.hess[indices[0, 0]]
b = self.hess[indices[1, 1]]
c = self.hess[indices[2, 2]]
d = self.hess[indices[0, 1]]
e = self.hess[indices[0, 2]]
f = self.hess[indices[1, 2]]
self.curvature[...] = ne.evaluate(
'a*b*c - c*d**2 - b*e**2 + 2*d*e*f - a*f**2',
local_dict=dict(a=a, b=b, c=c, d=d, e=e, f=f))
elif ndim == 2:
a = self.hess[indices[0, 0]]
b = self.hess[indices[1, 1]]
c = self.hess[indices[0, 1]]
self.curvature[...] = ne.evaluate('a*b - c**2',
local_dict=dict(a=a, b=b, c=c))
else:
self.curvature[...] = np.linalg.det(self.hess[indices,
...].T.copy()).T
return indices
def ifft(*args, **kwargs):
return fftw.irfftn(*args, **kwargs, threads=2)
def cross_correlation_3d(pixels1, pixels2):
'''Align the second image with the first using max cross-correlation
returns the z,y,x 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.
'''
s = np.maximum(pixels1.shape, pixels2.shape)
fshape = s*2
#
# Calculate the # of pixels at a particular point
#
i,j,k = np.mgrid[-s[0]:s[0], -s[1]:s[1], -s[2]:s[2] ]
unit = np.abs(i*j*k).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 = rfftn(pixels1.astype('float32'), fshape, axes=(0, 1, 2))
fp2 = rfftn(pixels2.astype('float32'), fshape, axes=(0, 1, 2))
corr12 = irfftn(fp1 * fp2.conj(), axes=(0, 1, 2)).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
#
def get_cumsums(im, fshape):
im_si = im.shape[0]
im_sj = im.shape[1]
im_sk = im.shape[2]
im_sum = np.zeros(fshape)
im_sum[:im_si,:im_sj,:im_sk] = cumsum_quadrant(im, False, False, False)
im_sum[:im_si,:im_sj,-im_sk:] = cumsum_quadrant(im, False, False, True)
im_sum[:im_si,-im_sj:,:im_sk] = cumsum_quadrant(im, False, True, True)
im_sum[:im_si,-im_sj:,-im_sk:] = cumsum_quadrant(im, False, True, False)
im_sum[-im_si:,:im_sj,:im_sk] = cumsum_quadrant(im, True, False, True)
im_sum[-im_si:,:im_sj,-im_sk:] = cumsum_quadrant(im, True, False, False)
im_sum[-im_si:,-im_sj:,:im_sk] = cumsum_quadrant(im, True, True, True)
im_sum[-im_si:,-im_sj:,-im_sk:] = cumsum_quadrant(im, True, True, False)
#
# Divide the sum over the # of elements summed-over
#
return im_sum / unit
p1_mean = get_cumsums(pixels1, fshape)
p2_mean = get_cumsums(pixels2, fshape)
#
# 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 ifft(self, scalar_hat):
""" Performs inverse fft of scalar field """
self.scalar_hat_input_test(scalar_hat)
return fft.irfftn(scalar_hat, threads=self.num_threads)
def ifft(self, vector_hat):
""" Performs inverse fft of vector hat field """
self.vector_hat_input_test(vector_hat)
return fft.irfftn(vector_hat, axes=(1, 2), threads=self.num_threads)