277 lines
12 KiB
Python
277 lines
12 KiB
Python
"""
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Port of Manuel Guizar's code from:
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http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation
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"""
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import numpy as np
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from .._shared.fft import fftmodule as fft
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from ._masked_phase_cross_correlation import _masked_phase_cross_correlation
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def _upsampled_dft(data, upsampled_region_size,
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upsample_factor=1, axis_offsets=None):
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"""
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Upsampled DFT by matrix multiplication.
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This code is intended to provide the same result as if the following
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operations were performed:
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- Embed the array "data" in an array that is ``upsample_factor`` times
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larger in each dimension. ifftshift to bring the center of the
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image to (1,1).
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- Take the FFT of the larger array.
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- Extract an ``[upsampled_region_size]`` region of the result, starting
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with the ``[axis_offsets+1]`` element.
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It achieves this result by computing the DFT in the output array without
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the need to zeropad. Much faster and memory efficient than the zero-padded
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FFT approach if ``upsampled_region_size`` is much smaller than
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``data.size * upsample_factor``.
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Parameters
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----------
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data : array
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The input data array (DFT of original data) to upsample.
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upsampled_region_size : integer or tuple of integers, optional
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The size of the region to be sampled. If one integer is provided, it
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is duplicated up to the dimensionality of ``data``.
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upsample_factor : integer, optional
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The upsampling factor. Defaults to 1.
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axis_offsets : tuple of integers, optional
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The offsets of the region to be sampled. Defaults to None (uses
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image center)
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Returns
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-------
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output : ndarray
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The upsampled DFT of the specified region.
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"""
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# if people pass in an integer, expand it to a list of equal-sized sections
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if not hasattr(upsampled_region_size, "__iter__"):
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upsampled_region_size = [upsampled_region_size, ] * data.ndim
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else:
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if len(upsampled_region_size) != data.ndim:
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raise ValueError("shape of upsampled region sizes must be equal "
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"to input data's number of dimensions.")
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if axis_offsets is None:
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axis_offsets = [0, ] * data.ndim
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else:
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if len(axis_offsets) != data.ndim:
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raise ValueError("number of axis offsets must be equal to input "
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"data's number of dimensions.")
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im2pi = 1j * 2 * np.pi
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dim_properties = list(zip(data.shape, upsampled_region_size, axis_offsets))
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for (n_items, ups_size, ax_offset) in dim_properties[::-1]:
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kernel = ((np.arange(ups_size) - ax_offset)[:, None]
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* fft.fftfreq(n_items, upsample_factor))
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kernel = np.exp(-im2pi * kernel)
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# Equivalent to:
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# data[i, j, k] = kernel[i, :] @ data[j, k].T
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data = np.tensordot(kernel, data, axes=(1, -1))
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return data
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def _compute_phasediff(cross_correlation_max):
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"""
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Compute global phase difference between the two images (should be
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zero if images are non-negative).
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Parameters
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----------
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cross_correlation_max : complex
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The complex value of the cross correlation at its maximum point.
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"""
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return np.arctan2(cross_correlation_max.imag, cross_correlation_max.real)
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def _compute_error(cross_correlation_max, src_amp, target_amp):
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"""
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Compute RMS error metric between ``src_image`` and ``target_image``.
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Parameters
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----------
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cross_correlation_max : complex
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The complex value of the cross correlation at its maximum point.
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src_amp : float
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The normalized average image intensity of the source image
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target_amp : float
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The normalized average image intensity of the target image
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"""
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error = 1.0 - cross_correlation_max * cross_correlation_max.conj() /\
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(src_amp * target_amp)
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return np.sqrt(np.abs(error))
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def phase_cross_correlation(reference_image, moving_image, *,
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upsample_factor=1, space="real",
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return_error=True, reference_mask=None,
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moving_mask=None, overlap_ratio=0.3):
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"""Efficient subpixel image translation registration by cross-correlation.
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This code gives the same precision as the FFT upsampled cross-correlation
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in a fraction of the computation time and with reduced memory requirements.
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It obtains an initial estimate of the cross-correlation peak by an FFT and
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then refines the shift estimation by upsampling the DFT only in a small
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neighborhood of that estimate by means of a matrix-multiply DFT.
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Parameters
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----------
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reference_image : array
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Reference image.
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moving_image : array
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Image to register. Must be same dimensionality as
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``reference_image``.
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upsample_factor : int, optional
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Upsampling factor. Images will be registered to within
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``1 / upsample_factor`` of a pixel. For example
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``upsample_factor == 20`` means the images will be registered
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within 1/20th of a pixel. Default is 1 (no upsampling).
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Not used if any of ``reference_mask`` or ``moving_mask`` is not None.
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space : string, one of "real" or "fourier", optional
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Defines how the algorithm interprets input data. "real" means
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data will be FFT'd to compute the correlation, while "fourier"
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data will bypass FFT of input data. Case insensitive. Not
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used if any of ``reference_mask`` or ``moving_mask`` is not
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None.
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return_error : bool, optional
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Returns error and phase difference if on, otherwise only
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shifts are returned. Has noeffect if any of ``reference_mask`` or
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``moving_mask`` is not None. In this case only shifts is returned.
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reference_mask : ndarray
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Boolean mask for ``reference_image``. The mask should evaluate
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to ``True`` (or 1) on valid pixels. ``reference_mask`` should
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have the same shape as ``reference_image``.
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moving_mask : ndarray or None, optional
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Boolean mask for ``moving_image``. The mask should evaluate to ``True``
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(or 1) on valid pixels. ``moving_mask`` should have the same shape
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as ``moving_image``. If ``None``, ``reference_mask`` will be used.
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overlap_ratio : float, optional
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Minimum allowed overlap ratio between images. The correlation for
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translations corresponding with an overlap ratio lower than this
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threshold will be ignored. A lower `overlap_ratio` leads to smaller
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maximum translation, while a higher `overlap_ratio` leads to greater
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robustness against spurious matches due to small overlap between
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masked images. Used only if one of ``reference_mask`` or
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``moving_mask`` is None.
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Returns
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-------
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shifts : ndarray
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Shift vector (in pixels) required to register ``moving_image``
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with ``reference_image``. Axis ordering is consistent with
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numpy (e.g. Z, Y, X)
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error : float
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Translation invariant normalized RMS error between
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``reference_image`` and ``moving_image``.
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phasediff : float
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Global phase difference between the two images (should be
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zero if images are non-negative).
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References
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----------
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.. [1] Manuel Guizar-Sicairos, Samuel T. Thurman, and James R. Fienup,
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"Efficient subpixel image registration algorithms,"
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Optics Letters 33, 156-158 (2008). :DOI:`10.1364/OL.33.000156`
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.. [2] James R. Fienup, "Invariant error metrics for image reconstruction"
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Optics Letters 36, 8352-8357 (1997). :DOI:`10.1364/AO.36.008352`
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.. [3] Dirk Padfield. Masked Object Registration in the Fourier Domain.
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IEEE Transactions on Image Processing, vol. 21(5),
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pp. 2706-2718 (2012). :DOI:`10.1109/TIP.2011.2181402`
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.. [4] D. Padfield. "Masked FFT registration". In Proc. Computer Vision and
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Pattern Recognition, pp. 2918-2925 (2010).
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:DOI:`10.1109/CVPR.2010.5540032`
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"""
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if (reference_mask is not None) or (moving_mask is not None):
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return _masked_phase_cross_correlation(reference_image, moving_image,
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reference_mask, moving_mask,
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overlap_ratio)
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# images must be the same shape
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if reference_image.shape != moving_image.shape:
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raise ValueError("images must be same shape")
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# assume complex data is already in Fourier space
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if space.lower() == 'fourier':
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src_freq = reference_image
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target_freq = moving_image
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# real data needs to be fft'd.
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elif space.lower() == 'real':
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src_freq = fft.fftn(reference_image)
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target_freq = fft.fftn(moving_image)
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else:
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raise ValueError('space argument must be "real" of "fourier"')
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# Whole-pixel shift - Compute cross-correlation by an IFFT
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shape = src_freq.shape
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image_product = src_freq * target_freq.conj()
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cross_correlation = fft.ifftn(image_product)
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# Locate maximum
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maxima = np.unravel_index(np.argmax(np.abs(cross_correlation)),
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cross_correlation.shape)
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midpoints = np.array([np.fix(axis_size / 2) for axis_size in shape])
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shifts = np.stack(maxima).astype(np.float64)
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shifts[shifts > midpoints] -= np.array(shape)[shifts > midpoints]
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if upsample_factor == 1:
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if return_error:
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src_amp = np.sum(np.real(src_freq * src_freq.conj()))
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src_amp /= src_freq.size
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target_amp = np.sum(np.real(target_freq * target_freq.conj()))
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target_amp /= target_freq.size
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CCmax = cross_correlation[maxima]
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# If upsampling > 1, then refine estimate with matrix multiply DFT
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else:
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# Initial shift estimate in upsampled grid
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shifts = np.round(shifts * upsample_factor) / upsample_factor
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upsampled_region_size = np.ceil(upsample_factor * 1.5)
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# Center of output array at dftshift + 1
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dftshift = np.fix(upsampled_region_size / 2.0)
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upsample_factor = np.array(upsample_factor, dtype=np.float64)
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# Matrix multiply DFT around the current shift estimate
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sample_region_offset = dftshift - shifts*upsample_factor
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cross_correlation = _upsampled_dft(image_product.conj(),
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upsampled_region_size,
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upsample_factor,
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sample_region_offset).conj()
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# Locate maximum and map back to original pixel grid
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maxima = np.unravel_index(np.argmax(np.abs(cross_correlation)),
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cross_correlation.shape)
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CCmax = cross_correlation[maxima]
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maxima = np.stack(maxima).astype(np.float64) - dftshift
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shifts = shifts + maxima / upsample_factor
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if return_error:
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src_amp = np.sum(np.real(src_freq * src_freq.conj()))
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target_amp = np.sum(np.real(target_freq * target_freq.conj()))
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# If its only one row or column the shift along that dimension has no
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# effect. We set to zero.
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for dim in range(src_freq.ndim):
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if shape[dim] == 1:
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shifts[dim] = 0
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if return_error:
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# Redirect user to masked_phase_cross_correlation if NaNs are observed
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if np.isnan(CCmax) or np.isnan(src_amp) or np.isnan(target_amp):
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raise ValueError(
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"NaN values found, please remove NaNs from your "
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"input data or use the `reference_mask`/`moving_mask` "
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"keywords, eg: "
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"phase_cross_correlation(reference_image, moving_image, "
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"reference_mask=~np.isnan(reference_image), "
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"moving_mask=~np.isnan(moving_image))")
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return shifts, _compute_error(CCmax, src_amp, target_amp),\
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_compute_phasediff(CCmax)
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else:
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return shifts
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