"""
Port of Manuel Guizar's code from:
http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation
"""
import itertools
import warnings
import numpy as np
from scipy.fft import fftn, ifftn, fftfreq
from scipy import ndimage as ndi
from ._masked_phase_cross_correlation import _masked_phase_cross_correlation
def _upsampled_dft(data, upsampled_region_size, upsample_factor=1, axis_offsets=None):
"""
Upsampled DFT by matrix multiplication.
This code is intended to provide the same result as if the following
operations were performed:
- Embed the array "data" in an array that is ``upsample_factor`` times
larger in each dimension. ifftshift to bring the center of the
image to (1,1).
- Take the FFT of the larger array.
- Extract an ``[upsampled_region_size]`` region of the result, starting
with the ``[axis_offsets+1]`` element.
It achieves this result by computing the DFT in the output array without
the need to zeropad. Much faster and memory efficient than the zero-padded
FFT approach if ``upsampled_region_size`` is much smaller than
``data.size * upsample_factor``.
Parameters
----------
data : array
The input data array (DFT of original data) to upsample.
upsampled_region_size : integer or tuple of integers, optional
The size of the region to be sampled. If one integer is provided, it
is duplicated up to the dimensionality of ``data``.
upsample_factor : integer, optional
The upsampling factor. Defaults to 1.
axis_offsets : tuple of integers, optional
The offsets of the region to be sampled. Defaults to None (uses
image center)
Returns
-------
output : ndarray
The upsampled DFT of the specified region.
"""
# if people pass in an integer, expand it to a list of equal-sized sections
if not hasattr(upsampled_region_size, "__iter__"):
upsampled_region_size = [
upsampled_region_size,
] * data.ndim
else:
if len(upsampled_region_size) != data.ndim:
raise ValueError(
"shape of upsampled region sizes must be equal "
"to input data's number of dimensions."
)
if axis_offsets is None:
axis_offsets = [
0,
] * data.ndim
else:
if len(axis_offsets) != data.ndim:
raise ValueError(
"number of axis offsets must be equal to input "
"data's number of dimensions."
)
im2pi = 1j * 2 * np.pi
dim_properties = list(zip(data.shape, upsampled_region_size, axis_offsets))
for n_items, ups_size, ax_offset in dim_properties[::-1]:
kernel = (np.arange(ups_size) - ax_offset)[:, None] * fftfreq(
n_items, upsample_factor
)
kernel = np.exp(-im2pi * kernel)
# use kernel with same precision as the data
kernel = kernel.astype(data.dtype, copy=False)
# Equivalent to:
# data[i, j, k] = kernel[i, :] @ data[j, k].T
data = np.tensordot(kernel, data, axes=(1, -1))
return data
def _compute_phasediff(cross_correlation_max):
"""
Compute global phase difference between the two images (should be
zero if images are non-negative).
Parameters
----------
cross_correlation_max : complex
The complex value of the cross correlation at its maximum point.
"""
return np.arctan2(cross_correlation_max.imag, cross_correlation_max.real)
def _compute_error(cross_correlation_max, src_amp, target_amp):
"""
Compute RMS error metric between ``src_image`` and ``target_image``.
Parameters
----------
cross_correlation_max : complex
The complex value of the cross correlation at its maximum point.
src_amp : float
The normalized average image intensity of the source image
target_amp : float
The normalized average image intensity of the target image
"""
amp = src_amp * target_amp
if amp == 0:
warnings.warn(
"Could not determine RMS error between images with the normalized "
f"average intensities {src_amp!r} and {target_amp!r}. Either the "
"reference or moving image may be empty.",
UserWarning,
stacklevel=3,
)
with np.errstate(invalid="ignore"):
error = 1.0 - cross_correlation_max * cross_correlation_max.conj() / amp
return np.sqrt(np.abs(error))
def _disambiguate_shift(reference_image, moving_image, shift):
"""Determine the correct real-space shift based on periodic shift.
When determining a translation shift from phase cross-correlation in
Fourier space, the shift is only correct to within a period of the image
size along each axis, resulting in $2^n$ possible shifts, where $n$ is the
number of dimensions of the image. This function checks the
cross-correlation in real space for each of those shifts, and returns the
one with the highest cross-correlation.
The strategy we use is to perform the shift on the moving image *using the
'grid-wrap' mode* in `scipy.ndimage`. The moving image's original borders
then define $2^n$ quadrants, which we cross-correlate with the reference
image in turn using slicing. The entire operation is thus $O(2^n + m)$,
where $m$ is the number of pixels in the image (and typically dominates).
Parameters
----------
reference_image : numpy array
The reference (non-moving) image.
moving_image : numpy array
The moving image: applying the shift to this image overlays it on the
reference image. Must be the same shape as the reference image.
shift : ndarray
The shift to apply to each axis of the moving image, *modulo* image
size. The length of ``shift`` must be equal to ``moving_image.ndim``.
Returns
-------
real_shift : ndarray
The shift disambiguated in real space.
"""
shape = reference_image.shape
positive_shift = [shift_i % s for shift_i, s in zip(shift, shape)]
negative_shift = [shift_i - s for shift_i, s in zip(positive_shift, shape)]
subpixel = np.any(np.array(shift) % 1 != 0)
interp_order = 3 if subpixel else 0
shifted = ndi.shift(moving_image, shift, mode='grid-wrap', order=interp_order)
indices = np.round(positive_shift).astype(int)
splits_per_dim = [(slice(0, i), slice(i, None)) for i in indices]
max_corr = -1.0
max_slice = None
for test_slice in itertools.product(*splits_per_dim):
reference_tile = np.reshape(reference_image[test_slice], -1)
moving_tile = np.reshape(shifted[test_slice], -1)
corr = -1.0
if reference_tile.size > 2:
# In the case of zero std, np.corrcoef returns NaN and warns
# about division by zero. This is expected and handled below.
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=RuntimeWarning)
corr = np.corrcoef(reference_tile, moving_tile)[0, 1]
if corr > max_corr:
max_corr = corr
max_slice = test_slice
if max_slice is None:
warnings.warn(
f"Could not determine real-space shift for periodic shift {shift!r} "
f"as requested by `disambiguate=True` (disambiguation is degenerate).",
stacklevel=3,
)
return shift
real_shift_acc = []
for sl, pos_shift, neg_shift in zip(max_slice, positive_shift, negative_shift):
real_shift_acc.append(pos_shift if sl.stop is None else neg_shift)
return np.array(real_shift_acc)
[文档]
def phase_cross_correlation(
reference_image,
moving_image,
*,
upsample_factor=1,
space="real",
disambiguate=False,
reference_mask=None,
moving_mask=None,
overlap_ratio=0.3,
normalization="phase",
):
"""Efficient subpixel image translation registration by cross-correlation.
This code gives the same precision as the FFT upsampled cross-correlation
in a fraction of the computation time and with reduced memory requirements.
It obtains an initial estimate of the cross-correlation peak by an FFT and
then refines the shift estimation by upsampling the DFT only in a small
neighborhood of that estimate by means of a matrix-multiply DFT [1]_.
Parameters
----------
reference_image : array
Reference image.
moving_image : array
Image to register. Must be same dimensionality as
``reference_image``.
upsample_factor : int, optional
Upsampling factor. Images will be registered to within
``1 / upsample_factor`` of a pixel. For example
``upsample_factor == 20`` means the images will be registered
within 1/20th of a pixel. Default is 1 (no upsampling).
Not used if any of ``reference_mask`` or ``moving_mask`` is not None.
space : string, one of "real" or "fourier", optional
Defines how the algorithm interprets input data. "real" means
data will be FFT'd to compute the correlation, while "fourier"
data will bypass FFT of input data. Case insensitive. Not
used if any of ``reference_mask`` or ``moving_mask`` is not
None.
disambiguate : bool
The shift returned by this function is only accurate *modulo* the
image shape, due to the periodic nature of the Fourier transform. If
this parameter is set to ``True``, the *real* space cross-correlation
is computed for each possible shift, and the shift with the highest
cross-correlation within the overlapping area is returned.
reference_mask : ndarray
Boolean mask for ``reference_image``. The mask should evaluate
to ``True`` (or 1) on valid pixels. ``reference_mask`` should
have the same shape as ``reference_image``.
moving_mask : ndarray or None, optional
Boolean mask for ``moving_image``. The mask should evaluate to ``True``
(or 1) on valid pixels. ``moving_mask`` should have the same shape
as ``moving_image``. If ``None``, ``reference_mask`` will be used.
overlap_ratio : float, optional
Minimum allowed overlap ratio between images. The correlation for
translations corresponding with an overlap ratio lower than this
threshold will be ignored. A lower `overlap_ratio` leads to smaller
maximum translation, while a higher `overlap_ratio` leads to greater
robustness against spurious matches due to small overlap between
masked images. Used only if one of ``reference_mask`` or
``moving_mask`` is not None.
normalization : {"phase", None}
The type of normalization to apply to the cross-correlation. This
parameter is unused when masks (`reference_mask` and `moving_mask`) are
supplied.
Returns
-------
shift : ndarray
Shift vector (in pixels) required to register ``moving_image``
with ``reference_image``. Axis ordering is consistent with
the axis order of the input array.
error : float
Translation invariant normalized RMS error between
``reference_image`` and ``moving_image``. For masked cross-correlation
this error is not available and NaN is returned.
phasediff : float
Global phase difference between the two images (should be
zero if images are non-negative). For masked cross-correlation
this phase difference is not available and NaN is returned.
Notes
-----
The use of cross-correlation to estimate image translation has a long
history dating back to at least [2]_. The "phase correlation"
method (selected by ``normalization="phase"``) was first proposed in [3]_.
Publications [1]_ and [2]_ use an unnormalized cross-correlation
(``normalization=None``). Which form of normalization is better is
application-dependent. For example, the phase correlation method works
well in registering images under different illumination, but is not very
robust to noise. In a high noise scenario, the unnormalized method may be
preferable.
When masks are provided, a masked normalized cross-correlation algorithm is
used [5]_, [6]_.
References
----------
.. [1] Manuel Guizar-Sicairos, Samuel T. Thurman, and James R. Fienup,
"Efficient subpixel image registration algorithms,"
Optics Letters 33, 156-158 (2008). :DOI:`10.1364/OL.33.000156`
.. [2] P. Anuta, Spatial registration of multispectral and multitemporal
digital imagery using fast Fourier transform techniques, IEEE Trans.
Geosci. Electron., vol. 8, no. 4, pp. 353–368, Oct. 1970.
:DOI:`10.1109/TGE.1970.271435`.
.. [3] C. D. Kuglin D. C. Hines. The phase correlation image alignment
method, Proceeding of IEEE International Conference on Cybernetics
and Society, pp. 163-165, New York, NY, USA, 1975, pp. 163–165.
.. [4] James R. Fienup, "Invariant error metrics for image reconstruction"
Optics Letters 36, 8352-8357 (1997). :DOI:`10.1364/AO.36.008352`
.. [5] Dirk Padfield. Masked Object Registration in the Fourier Domain.
IEEE Transactions on Image Processing, vol. 21(5),
pp. 2706-2718 (2012). :DOI:`10.1109/TIP.2011.2181402`
.. [6] D. Padfield. "Masked FFT registration". In Proc. Computer Vision and
Pattern Recognition, pp. 2918-2925 (2010).
:DOI:`10.1109/CVPR.2010.5540032`
"""
if (reference_mask is not None) or (moving_mask is not None):
shift = _masked_phase_cross_correlation(
reference_image, moving_image, reference_mask, moving_mask, overlap_ratio
)
return shift, np.nan, np.nan
# images must be the same shape
if reference_image.shape != moving_image.shape:
raise ValueError("images must be same shape")
# assume complex data is already in Fourier space
if space.lower() == 'fourier':
src_freq = reference_image
target_freq = moving_image
# real data needs to be fft'd.
elif space.lower() == 'real':
src_freq = fftn(reference_image)
target_freq = fftn(moving_image)
else:
raise ValueError('space argument must be "real" of "fourier"')
# Whole-pixel shift - Compute cross-correlation by an IFFT
shape = src_freq.shape
image_product = src_freq * target_freq.conj()
if normalization == "phase":
eps = np.finfo(image_product.real.dtype).eps
image_product /= np.maximum(np.abs(image_product), 100 * eps)
elif normalization is not None:
raise ValueError("normalization must be either phase or None")
cross_correlation = ifftn(image_product)
# Locate maximum
maxima = np.unravel_index(
np.argmax(np.abs(cross_correlation)), cross_correlation.shape
)
midpoint = np.array([np.fix(axis_size / 2) for axis_size in shape])
float_dtype = image_product.real.dtype
shift = np.stack(maxima).astype(float_dtype, copy=False)
shift[shift > midpoint] -= np.array(shape)[shift > midpoint]
if upsample_factor == 1:
src_amp = np.sum(np.real(src_freq * src_freq.conj()))
src_amp /= src_freq.size
target_amp = np.sum(np.real(target_freq * target_freq.conj()))
target_amp /= target_freq.size
CCmax = cross_correlation[maxima]
# If upsampling > 1, then refine estimate with matrix multiply DFT
else:
# Initial shift estimate in upsampled grid
upsample_factor = np.array(upsample_factor, dtype=float_dtype)
shift = np.round(shift * upsample_factor) / upsample_factor
upsampled_region_size = np.ceil(upsample_factor * 1.5)
# Center of output array at dftshift + 1
dftshift = np.fix(upsampled_region_size / 2.0)
# Matrix multiply DFT around the current shift estimate
sample_region_offset = dftshift - shift * upsample_factor
cross_correlation = _upsampled_dft(
image_product.conj(),
upsampled_region_size,
upsample_factor,
sample_region_offset,
).conj()
# Locate maximum and map back to original pixel grid
maxima = np.unravel_index(
np.argmax(np.abs(cross_correlation)), cross_correlation.shape
)
CCmax = cross_correlation[maxima]
maxima = np.stack(maxima).astype(float_dtype, copy=False)
maxima -= dftshift
shift += maxima / upsample_factor
src_amp = np.sum(np.real(src_freq * src_freq.conj()))
target_amp = np.sum(np.real(target_freq * target_freq.conj()))
# If its only one row or column the shift along that dimension has no
# effect. We set to zero.
for dim in range(src_freq.ndim):
if shape[dim] == 1:
shift[dim] = 0
if disambiguate:
if space.lower() != 'real':
reference_image = ifftn(reference_image)
moving_image = ifftn(moving_image)
shift = _disambiguate_shift(reference_image, moving_image, shift)
# Redirect user to masked_phase_cross_correlation if NaNs are observed
if np.isnan(CCmax) or np.isnan(src_amp) or np.isnan(target_amp):
raise ValueError(
"NaN values found, please remove NaNs from your "
"input data or use the `reference_mask`/`moving_mask` "
"keywords, eg: "
"phase_cross_correlation(reference_image, moving_image, "
"reference_mask=~np.isnan(reference_image), "
"moving_mask=~np.isnan(moving_image))"
)
return shift, _compute_error(CCmax, src_amp, target_amp), _compute_phasediff(CCmax)
