386 lines
15 KiB
Python
386 lines
15 KiB
Python
"""Implementations restoration functions"""
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import numpy as np
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import numpy.random as npr
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from scipy.signal import convolve
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from . import uft
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__keywords__ = "restoration, image, deconvolution"
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def wiener(image, psf, balance, reg=None, is_real=True, clip=True):
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r"""Wiener-Hunt deconvolution
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Return the deconvolution with a Wiener-Hunt approach (i.e. with
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Fourier diagonalisation).
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Parameters
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----------
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image : (M, N) ndarray
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Input degraded image
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psf : ndarray
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Point Spread Function. This is assumed to be the impulse
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response (input image space) if the data-type is real, or the
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transfer function (Fourier space) if the data-type is
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complex. There is no constraints on the shape of the impulse
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response. The transfer function must be of shape `(M, N)` if
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`is_real is True`, `(M, N // 2 + 1)` otherwise (see
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`np.fft.rfftn`).
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balance : float
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The regularisation parameter value that tunes the balance
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between the data adequacy that improve frequency restoration
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and the prior adequacy that reduce frequency restoration (to
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avoid noise artifacts).
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reg : ndarray, optional
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The regularisation operator. The Laplacian by default. It can
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be an impulse response or a transfer function, as for the
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psf. Shape constraint is the same as for the `psf` parameter.
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is_real : boolean, optional
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True by default. Specify if ``psf`` and ``reg`` are provided
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with hermitian hypothesis, that is only half of the frequency
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plane is provided (due to the redundancy of Fourier transform
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of real signal). It's apply only if ``psf`` and/or ``reg`` are
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provided as transfer function. For the hermitian property see
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``uft`` module or ``np.fft.rfftn``.
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clip : boolean, optional
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True by default. If True, pixel values of the result above 1 or
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under -1 are thresholded for skimage pipeline compatibility.
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Returns
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-------
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im_deconv : (M, N) ndarray
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The deconvolved image.
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Examples
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--------
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>>> from skimage import color, data, restoration
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>>> img = color.rgb2gray(data.astronaut())
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>>> from scipy.signal import convolve2d
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>>> psf = np.ones((5, 5)) / 25
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>>> img = convolve2d(img, psf, 'same')
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>>> img += 0.1 * img.std() * np.random.standard_normal(img.shape)
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>>> deconvolved_img = restoration.wiener(img, psf, 1100)
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Notes
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-----
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This function applies the Wiener filter to a noisy and degraded
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image by an impulse response (or PSF). If the data model is
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.. math:: y = Hx + n
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where :math:`n` is noise, :math:`H` the PSF and :math:`x` the
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unknown original image, the Wiener filter is
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.. math::
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\hat x = F^\dagger (|\Lambda_H|^2 + \lambda |\Lambda_D|^2)
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\Lambda_H^\dagger F y
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where :math:`F` and :math:`F^\dagger` are the Fourier and inverse
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Fourier transforms respectively, :math:`\Lambda_H` the transfer
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function (or the Fourier transform of the PSF, see [Hunt] below)
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and :math:`\Lambda_D` the filter to penalize the restored image
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frequencies (Laplacian by default, that is penalization of high
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frequency). The parameter :math:`\lambda` tunes the balance
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between the data (that tends to increase high frequency, even
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those coming from noise), and the regularization.
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These methods are then specific to a prior model. Consequently,
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the application or the true image nature must corresponds to the
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prior model. By default, the prior model (Laplacian) introduce
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image smoothness or pixel correlation. It can also be interpreted
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as high-frequency penalization to compensate the instability of
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the solution with respect to the data (sometimes called noise
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amplification or "explosive" solution).
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Finally, the use of Fourier space implies a circulant property of
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:math:`H`, see [Hunt].
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References
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----------
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.. [1] François Orieux, Jean-François Giovannelli, and Thomas
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Rodet, "Bayesian estimation of regularization and point
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spread function parameters for Wiener-Hunt deconvolution",
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J. Opt. Soc. Am. A 27, 1593-1607 (2010)
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https://www.osapublishing.org/josaa/abstract.cfm?URI=josaa-27-7-1593
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http://research.orieux.fr/files/papers/OGR-JOSA10.pdf
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.. [2] B. R. Hunt "A matrix theory proof of the discrete
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convolution theorem", IEEE Trans. on Audio and
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Electroacoustics, vol. au-19, no. 4, pp. 285-288, dec. 1971
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"""
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if reg is None:
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reg, _ = uft.laplacian(image.ndim, image.shape, is_real=is_real)
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if not np.iscomplexobj(reg):
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reg = uft.ir2tf(reg, image.shape, is_real=is_real)
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if psf.shape != reg.shape:
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trans_func = uft.ir2tf(psf, image.shape, is_real=is_real)
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else:
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trans_func = psf
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wiener_filter = np.conj(trans_func) / (np.abs(trans_func) ** 2 +
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balance * np.abs(reg) ** 2)
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if is_real:
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deconv = uft.uirfft2(wiener_filter * uft.urfft2(image),
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shape=image.shape)
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else:
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deconv = uft.uifft2(wiener_filter * uft.ufft2(image))
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if clip:
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deconv[deconv > 1] = 1
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deconv[deconv < -1] = -1
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return deconv
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def unsupervised_wiener(image, psf, reg=None, user_params=None, is_real=True,
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clip=True):
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"""Unsupervised Wiener-Hunt deconvolution.
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Return the deconvolution with a Wiener-Hunt approach, where the
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hyperparameters are automatically estimated. The algorithm is a
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stochastic iterative process (Gibbs sampler) described in the
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reference below. See also ``wiener`` function.
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Parameters
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----------
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image : (M, N) ndarray
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The input degraded image.
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psf : ndarray
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The impulse response (input image's space) or the transfer
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function (Fourier space). Both are accepted. The transfer
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function is automatically recognized as being complex
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(``np.iscomplexobj(psf)``).
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reg : ndarray, optional
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The regularisation operator. The Laplacian by default. It can
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be an impulse response or a transfer function, as for the psf.
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user_params : dict, optional
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Dictionary of parameters for the Gibbs sampler. See below.
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clip : boolean, optional
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True by default. If true, pixel values of the result above 1 or
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under -1 are thresholded for skimage pipeline compatibility.
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Returns
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-------
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x_postmean : (M, N) ndarray
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The deconvolved image (the posterior mean).
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chains : dict
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The keys ``noise`` and ``prior`` contain the chain list of
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noise and prior precision respectively.
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Other parameters
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----------------
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The keys of ``user_params`` are:
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threshold : float
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The stopping criterion: the norm of the difference between to
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successive approximated solution (empirical mean of object
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samples, see Notes section). 1e-4 by default.
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burnin : int
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The number of sample to ignore to start computation of the
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mean. 15 by default.
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min_iter : int
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The minimum number of iterations. 30 by default.
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max_iter : int
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The maximum number of iterations if ``threshold`` is not
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satisfied. 200 by default.
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callback : callable (None by default)
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A user provided callable to which is passed, if the function
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exists, the current image sample for whatever purpose. The user
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can store the sample, or compute other moments than the
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mean. It has no influence on the algorithm execution and is
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only for inspection.
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Examples
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--------
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>>> from skimage import color, data, restoration
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>>> img = color.rgb2gray(data.astronaut())
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>>> from scipy.signal import convolve2d
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>>> psf = np.ones((5, 5)) / 25
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>>> img = convolve2d(img, psf, 'same')
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>>> img += 0.1 * img.std() * np.random.standard_normal(img.shape)
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>>> deconvolved_img = restoration.unsupervised_wiener(img, psf)
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Notes
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-----
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The estimated image is design as the posterior mean of a
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probability law (from a Bayesian analysis). The mean is defined as
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a sum over all the possible images weighted by their respective
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probability. Given the size of the problem, the exact sum is not
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tractable. This algorithm use of MCMC to draw image under the
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posterior law. The practical idea is to only draw highly probable
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images since they have the biggest contribution to the mean. At the
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opposite, the less probable images are drawn less often since
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their contribution is low. Finally the empirical mean of these
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samples give us an estimation of the mean, and an exact
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computation with an infinite sample set.
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References
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----------
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.. [1] François Orieux, Jean-François Giovannelli, and Thomas
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Rodet, "Bayesian estimation of regularization and point
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spread function parameters for Wiener-Hunt deconvolution",
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J. Opt. Soc. Am. A 27, 1593-1607 (2010)
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https://www.osapublishing.org/josaa/abstract.cfm?URI=josaa-27-7-1593
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http://research.orieux.fr/files/papers/OGR-JOSA10.pdf
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"""
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params = {'threshold': 1e-4, 'max_iter': 200,
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'min_iter': 30, 'burnin': 15, 'callback': None}
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params.update(user_params or {})
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if reg is None:
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reg, _ = uft.laplacian(image.ndim, image.shape, is_real=is_real)
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if not np.iscomplexobj(reg):
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reg = uft.ir2tf(reg, image.shape, is_real=is_real)
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if psf.shape != reg.shape:
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trans_fct = uft.ir2tf(psf, image.shape, is_real=is_real)
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else:
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trans_fct = psf
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# The mean of the object
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x_postmean = np.zeros(trans_fct.shape)
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# The previous computed mean in the iterative loop
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prev_x_postmean = np.zeros(trans_fct.shape)
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# Difference between two successive mean
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delta = np.NAN
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# Initial state of the chain
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gn_chain, gx_chain = [1], [1]
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# The correlation of the object in Fourier space (if size is big,
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# this can reduce computation time in the loop)
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areg2 = np.abs(reg) ** 2
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atf2 = np.abs(trans_fct) ** 2
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# The Fourier transform may change the image.size attribute, so we
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# store it.
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if is_real:
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data_spectrum = uft.urfft2(image.astype(float))
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else:
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data_spectrum = uft.ufft2(image.astype(float))
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# Gibbs sampling
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for iteration in range(params['max_iter']):
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# Sample of Eq. 27 p(circX^k | gn^k-1, gx^k-1, y).
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# weighting (correlation in direct space)
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precision = gn_chain[-1] * atf2 + gx_chain[-1] * areg2 # Eq. 29
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excursion = np.sqrt(0.5) / np.sqrt(precision) * (
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np.random.standard_normal(data_spectrum.shape) +
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1j * np.random.standard_normal(data_spectrum.shape))
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# mean Eq. 30 (RLS for fixed gn, gamma0 and gamma1 ...)
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wiener_filter = gn_chain[-1] * np.conj(trans_fct) / precision
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# sample of X in Fourier space
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x_sample = wiener_filter * data_spectrum + excursion
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if params['callback']:
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params['callback'](x_sample)
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# sample of Eq. 31 p(gn | x^k, gx^k, y)
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gn_chain.append(npr.gamma(image.size / 2,
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2 / uft.image_quad_norm(data_spectrum -
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x_sample *
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trans_fct)))
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# sample of Eq. 31 p(gx | x^k, gn^k-1, y)
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gx_chain.append(npr.gamma((image.size - 1) / 2,
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2 / uft.image_quad_norm(x_sample * reg)))
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# current empirical average
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if iteration > params['burnin']:
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x_postmean = prev_x_postmean + x_sample
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if iteration > (params['burnin'] + 1):
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current = x_postmean / (iteration - params['burnin'])
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previous = prev_x_postmean / (iteration - params['burnin'] - 1)
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delta = np.sum(np.abs(current - previous)) / \
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np.sum(np.abs(x_postmean)) / (iteration - params['burnin'])
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prev_x_postmean = x_postmean
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# stop of the algorithm
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if (iteration > params['min_iter']) and (delta < params['threshold']):
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break
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# Empirical average \approx POSTMEAN Eq. 44
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x_postmean = x_postmean / (iteration - params['burnin'])
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if is_real:
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x_postmean = uft.uirfft2(x_postmean, shape=image.shape)
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else:
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x_postmean = uft.uifft2(x_postmean)
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if clip:
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x_postmean[x_postmean > 1] = 1
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x_postmean[x_postmean < -1] = -1
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return (x_postmean, {'noise': gn_chain, 'prior': gx_chain})
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def richardson_lucy(image, psf, iterations=50, clip=True, filter_epsilon=None):
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"""Richardson-Lucy deconvolution.
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Parameters
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----------
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image : ndarray
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Input degraded image (can be N dimensional).
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psf : ndarray
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The point spread function.
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iterations : int, optional
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Number of iterations. This parameter plays the role of
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regularisation.
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clip : boolean, optional
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True by default. If true, pixel value of the result above 1 or
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under -1 are thresholded for skimage pipeline compatibility.
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filter_epsilon: float, optional
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Value below which intermediate results become 0 to avoid division
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by small numbers.
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Returns
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-------
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im_deconv : ndarray
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The deconvolved image.
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Examples
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--------
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>>> from skimage import img_as_float, data, restoration
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>>> camera = img_as_float(data.camera())
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>>> from scipy.signal import convolve2d
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>>> psf = np.ones((5, 5)) / 25
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>>> camera = convolve2d(camera, psf, 'same')
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>>> camera += 0.1 * camera.std() * np.random.standard_normal(camera.shape)
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>>> deconvolved = restoration.richardson_lucy(camera, psf, 5)
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References
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----------
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.. [1] https://en.wikipedia.org/wiki/Richardson%E2%80%93Lucy_deconvolution
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"""
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float_type = np.promote_types(image.dtype, np.float32)
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image = image.astype(float_type, copy=False)
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psf = psf.astype(float_type, copy=False)
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im_deconv = np.full(image.shape, 0.5, dtype=float_type)
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psf_mirror = np.flip(psf)
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for _ in range(iterations):
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conv = convolve(im_deconv, psf, mode='same')
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if filter_epsilon:
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relative_blur = np.where(conv < filter_epsilon, 0, image / conv)
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else:
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relative_blur = image / conv
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im_deconv *= convolve(relative_blur, psf_mirror, mode='same')
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if clip:
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im_deconv[im_deconv > 1] = 1
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im_deconv[im_deconv < -1] = -1
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return im_deconv
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