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Thibault Barnouin
2024-05-24 12:15:06 +02:00
parent 4e17f40534
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package/lib/cross_correlation.py Executable file
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"""
Library functions for phase cross-correlation computation.
"""
# Prefer FFTs via the new scipy.fft module when available (SciPy 1.4+)
# Otherwise fall back to numpy.fft.
# Like numpy 1.15+ scipy 1.3+ is also using pocketfft, but a newer
# C++/pybind11 version called pypocketfft
try:
import scipy.fft as fft
except ImportError:
import numpy.fft as fft
import numpy as np
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``.
----------
Inputs:
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]
* fft.fftfreq(n_items, upsample_factor))
kernel = np.exp(-im2pi * kernel)
# 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).
----------
Inputs:
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``.
----------
Inputs:
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
"""
error = 1.0 - cross_correlation_max * cross_correlation_max.conj() /\
(src_amp * target_amp)
return np.sqrt(np.abs(error))
def phase_cross_correlation(reference_image, moving_image, *,
upsample_factor=1, space="real",
return_error=True, overlap_ratio=0.3):
"""
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.
----------
Inputs:
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.
return_error : bool, optional
Returns error and phase difference if on, otherwise only
shifts are returned.
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.
----------
Returns:
shifts : ndarray
Shift vector (in pixels) required to register ``moving_image``
with ``reference_image``. Axis ordering is consistent with
numpy (e.g. Z, Y, X)
error : float
Translation invariant normalized RMS error between
``reference_image`` and ``moving_image``.
phasediff : float
Global phase difference between the two images (should be
zero if images are non-negative).
----------
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] James R. Fienup, "Invariant error metrics for image reconstruction"
Optics Letters 36, 8352-8357 (1997). :DOI:`10.1364/AO.36.008352`
[3] 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`
[4] D. Padfield. "Masked FFT registration". In Proc. Computer Vision and
Pattern Recognition, pp. 2918-2925 (2010).
:DOI:`10.1109/CVPR.2010.5540032`
"""
# 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 = fft.fftn(reference_image)
target_freq = fft.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()
cross_correlation = fft.ifftn(image_product)
# Locate maximum
maxima = np.unravel_index(np.argmax(np.abs(cross_correlation)),
cross_correlation.shape)
midpoints = np.array([np.fix(axis_size / 2) for axis_size in shape])
shifts = np.stack(maxima).astype(np.float64)
shifts[shifts > midpoints] -= np.array(shape)[shifts > midpoints]
if upsample_factor == 1:
if return_error:
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
shifts = np.round(shifts * 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)
upsample_factor = np.array(upsample_factor, dtype=np.float64)
# Matrix multiply DFT around the current shift estimate
sample_region_offset = dftshift - shifts*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(np.float64) - dftshift
shifts = shifts + maxima / upsample_factor
if return_error:
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:
shifts[dim] = 0
if return_error:
# 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")
return shifts, _compute_error(CCmax, src_amp, target_amp), \
_compute_phasediff(CCmax)
else:
return shifts