forked from 170010011/fr
386 lines
14 KiB
Python
386 lines
14 KiB
Python
|
"""Implementations restoration functions"""
|
||
|
|
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
import numpy.random as npr
|
||
|
|
from scipy.signal import convolve
|
||
|
|
|
||
|
|
from . import uft
|
||
|
|
|
||
|
|
__keywords__ = "restoration, image, deconvolution"
|
||
|
|
|
||
|
|
|
||
|
|
def wiener(image, psf, balance, reg=None, is_real=True, clip=True):
|
||
|
|
r"""Wiener-Hunt deconvolution
|
||
|
|
|
||
|
|
Return the deconvolution with a Wiener-Hunt approach (i.e. with
|
||
|
|
Fourier diagonalisation).
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
image : (M, N) ndarray
|
||
|
|
Input degraded image
|
||
|
|
psf : ndarray
|
||
|
|
Point Spread Function. This is assumed to be the impulse
|
||
|
|
response (input image space) if the data-type is real, or the
|
||
|
|
transfer function (Fourier space) if the data-type is
|
||
|
|
complex. There is no constraints on the shape of the impulse
|
||
|
|
response. The transfer function must be of shape `(M, N)` if
|
||
|
|
`is_real is True`, `(M, N // 2 + 1)` otherwise (see
|
||
|
|
`np.fft.rfftn`).
|
||
|
|
balance : float
|
||
|
|
The regularisation parameter value that tunes the balance
|
||
|
|
between the data adequacy that improve frequency restoration
|
||
|
|
and the prior adequacy that reduce frequency restoration (to
|
||
|
|
avoid noise artifacts).
|
||
|
|
reg : ndarray, optional
|
||
|
|
The regularisation operator. The Laplacian by default. It can
|
||
|
|
be an impulse response or a transfer function, as for the
|
||
|
|
psf. Shape constraint is the same as for the `psf` parameter.
|
||
|
|
is_real : boolean, optional
|
||
|
|
True by default. Specify if ``psf`` and ``reg`` are provided
|
||
|
|
with hermitian hypothesis, that is only half of the frequency
|
||
|
|
plane is provided (due to the redundancy of Fourier transform
|
||
|
|
of real signal). It's apply only if ``psf`` and/or ``reg`` are
|
||
|
|
provided as transfer function. For the hermitian property see
|
||
|
|
``uft`` module or ``np.fft.rfftn``.
|
||
|
|
clip : boolean, optional
|
||
|
|
True by default. If True, pixel values of the result above 1 or
|
||
|
|
under -1 are thresholded for skimage pipeline compatibility.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
im_deconv : (M, N) ndarray
|
||
|
|
The deconvolved image.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from skimage import color, data, restoration
|
||
|
|
>>> img = color.rgb2gray(data.astronaut())
|
||
|
|
>>> from scipy.signal import convolve2d
|
||
|
|
>>> psf = np.ones((5, 5)) / 25
|
||
|
|
>>> img = convolve2d(img, psf, 'same')
|
||
|
|
>>> img += 0.1 * img.std() * np.random.standard_normal(img.shape)
|
||
|
|
>>> deconvolved_img = restoration.wiener(img, psf, 1100)
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This function applies the Wiener filter to a noisy and degraded
|
||
|
|
image by an impulse response (or PSF). If the data model is
|
||
|
|
|
||
|
|
.. math:: y = Hx + n
|
||
|
|
|
||
|
|
where :math:`n` is noise, :math:`H` the PSF and :math:`x` the
|
||
|
|
unknown original image, the Wiener filter is
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
\hat x = F^\dagger (|\Lambda_H|^2 + \lambda |\Lambda_D|^2)
|
||
|
|
\Lambda_H^\dagger F y
|
||
|
|
|
||
|
|
where :math:`F` and :math:`F^\dagger` are the Fourier and inverse
|
||
|
|
Fourier transforms respectively, :math:`\Lambda_H` the transfer
|
||
|
|
function (or the Fourier transform of the PSF, see [Hunt] below)
|
||
|
|
and :math:`\Lambda_D` the filter to penalize the restored image
|
||
|
|
frequencies (Laplacian by default, that is penalization of high
|
||
|
|
frequency). The parameter :math:`\lambda` tunes the balance
|
||
|
|
between the data (that tends to increase high frequency, even
|
||
|
|
those coming from noise), and the regularization.
|
||
|
|
|
||
|
|
These methods are then specific to a prior model. Consequently,
|
||
|
|
the application or the true image nature must corresponds to the
|
||
|
|
prior model. By default, the prior model (Laplacian) introduce
|
||
|
|
image smoothness or pixel correlation. It can also be interpreted
|
||
|
|
as high-frequency penalization to compensate the instability of
|
||
|
|
the solution with respect to the data (sometimes called noise
|
||
|
|
amplification or "explosive" solution).
|
||
|
|
|
||
|
|
Finally, the use of Fourier space implies a circulant property of
|
||
|
|
:math:`H`, see [Hunt].
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] François Orieux, Jean-François Giovannelli, and Thomas
|
||
|
|
Rodet, "Bayesian estimation of regularization and point
|
||
|
|
spread function parameters for Wiener-Hunt deconvolution",
|
||
|
|
J. Opt. Soc. Am. A 27, 1593-1607 (2010)
|
||
|
|
|
||
|
|
https://www.osapublishing.org/josaa/abstract.cfm?URI=josaa-27-7-1593
|
||
|
|
|
||
|
|
http://research.orieux.fr/files/papers/OGR-JOSA10.pdf
|
||
|
|
|
||
|
|
.. [2] B. R. Hunt "A matrix theory proof of the discrete
|
||
|
|
convolution theorem", IEEE Trans. on Audio and
|
||
|
|
Electroacoustics, vol. au-19, no. 4, pp. 285-288, dec. 1971
|
||
|
|
"""
|
||
|
|
if reg is None:
|
||
|
|
reg, _ = uft.laplacian(image.ndim, image.shape, is_real=is_real)
|
||
|
|
if not np.iscomplexobj(reg):
|
||
|
|
reg = uft.ir2tf(reg, image.shape, is_real=is_real)
|
||
|
|
|
||
|
|
if psf.shape != reg.shape:
|
||
|
|
trans_func = uft.ir2tf(psf, image.shape, is_real=is_real)
|
||
|
|
else:
|
||
|
|
trans_func = psf
|
||
|
|
|
||
|
|
wiener_filter = np.conj(trans_func) / (np.abs(trans_func) ** 2 +
|
||
|
|
balance * np.abs(reg) ** 2)
|
||
|
|
if is_real:
|
||
|
|
deconv = uft.uirfft2(wiener_filter * uft.urfft2(image),
|
||
|
|
shape=image.shape)
|
||
|
|
else:
|
||
|
|
deconv = uft.uifft2(wiener_filter * uft.ufft2(image))
|
||
|
|
|
||
|
|
if clip:
|
||
|
|
deconv[deconv > 1] = 1
|
||
|
|
deconv[deconv < -1] = -1
|
||
|
|
|
||
|
|
return deconv
|
||
|
|
|
||
|
|
|
||
|
|
def unsupervised_wiener(image, psf, reg=None, user_params=None, is_real=True,
|
||
|
|
clip=True):
|
||
|
|
"""Unsupervised Wiener-Hunt deconvolution.
|
||
|
|
|
||
|
|
Return the deconvolution with a Wiener-Hunt approach, where the
|
||
|
|
hyperparameters are automatically estimated. The algorithm is a
|
||
|
|
stochastic iterative process (Gibbs sampler) described in the
|
||
|
|
reference below. See also ``wiener`` function.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
image : (M, N) ndarray
|
||
|
|
The input degraded image.
|
||
|
|
psf : ndarray
|
||
|
|
The impulse response (input image's space) or the transfer
|
||
|
|
function (Fourier space). Both are accepted. The transfer
|
||
|
|
function is automatically recognized as being complex
|
||
|
|
(``np.iscomplexobj(psf)``).
|
||
|
|
reg : ndarray, optional
|
||
|
|
The regularisation operator. The Laplacian by default. It can
|
||
|
|
be an impulse response or a transfer function, as for the psf.
|
||
|
|
user_params : dict, optional
|
||
|
|
Dictionary of parameters for the Gibbs sampler. See below.
|
||
|
|
clip : boolean, optional
|
||
|
|
True by default. If true, pixel values of the result above 1 or
|
||
|
|
under -1 are thresholded for skimage pipeline compatibility.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
x_postmean : (M, N) ndarray
|
||
|
|
The deconvolved image (the posterior mean).
|
||
|
|
chains : dict
|
||
|
|
The keys ``noise`` and ``prior`` contain the chain list of
|
||
|
|
noise and prior precision respectively.
|
||
|
|
|
||
|
|
Other parameters
|
||
|
|
----------------
|
||
|
|
The keys of ``user_params`` are:
|
||
|
|
|
||
|
|
threshold : float
|
||
|
|
The stopping criterion: the norm of the difference between to
|
||
|
|
successive approximated solution (empirical mean of object
|
||
|
|
samples, see Notes section). 1e-4 by default.
|
||
|
|
burnin : int
|
||
|
|
The number of sample to ignore to start computation of the
|
||
|
|
mean. 15 by default.
|
||
|
|
min_iter : int
|
||
|
|
The minimum number of iterations. 30 by default.
|
||
|
|
max_iter : int
|
||
|
|
The maximum number of iterations if ``threshold`` is not
|
||
|
|
satisfied. 200 by default.
|
||
|
|
callback : callable (None by default)
|
||
|
|
A user provided callable to which is passed, if the function
|
||
|
|
exists, the current image sample for whatever purpose. The user
|
||
|
|
can store the sample, or compute other moments than the
|
||
|
|
mean. It has no influence on the algorithm execution and is
|
||
|
|
only for inspection.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from skimage import color, data, restoration
|
||
|
|
>>> img = color.rgb2gray(data.astronaut())
|
||
|
|
>>> from scipy.signal import convolve2d
|
||
|
|
>>> psf = np.ones((5, 5)) / 25
|
||
|
|
>>> img = convolve2d(img, psf, 'same')
|
||
|
|
>>> img += 0.1 * img.std() * np.random.standard_normal(img.shape)
|
||
|
|
>>> deconvolved_img = restoration.unsupervised_wiener(img, psf)
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The estimated image is design as the posterior mean of a
|
||
|
|
probability law (from a Bayesian analysis). The mean is defined as
|
||
|
|
a sum over all the possible images weighted by their respective
|
||
|
|
probability. Given the size of the problem, the exact sum is not
|
||
|
|
tractable. This algorithm use of MCMC to draw image under the
|
||
|
|
posterior law. The practical idea is to only draw highly probable
|
||
|
|
images since they have the biggest contribution to the mean. At the
|
||
|
|
opposite, the less probable images are drawn less often since
|
||
|
|
their contribution is low. Finally the empirical mean of these
|
||
|
|
samples give us an estimation of the mean, and an exact
|
||
|
|
computation with an infinite sample set.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] François Orieux, Jean-François Giovannelli, and Thomas
|
||
|
|
Rodet, "Bayesian estimation of regularization and point
|
||
|
|
spread function parameters for Wiener-Hunt deconvolution",
|
||
|
|
J. Opt. Soc. Am. A 27, 1593-1607 (2010)
|
||
|
|
|
||
|
|
https://www.osapublishing.org/josaa/abstract.cfm?URI=josaa-27-7-1593
|
||
|
|
|
||
|
|
http://research.orieux.fr/files/papers/OGR-JOSA10.pdf
|
||
|
|
"""
|
||
|
|
params = {'threshold': 1e-4, 'max_iter': 200,
|
||
|
|
'min_iter': 30, 'burnin': 15, 'callback': None}
|
||
|
|
params.update(user_params or {})
|
||
|
|
|
||
|
|
if reg is None:
|
||
|
|
reg, _ = uft.laplacian(image.ndim, image.shape, is_real=is_real)
|
||
|
|
if not np.iscomplexobj(reg):
|
||
|
|
reg = uft.ir2tf(reg, image.shape, is_real=is_real)
|
||
|
|
|
||
|
|
if psf.shape != reg.shape:
|
||
|
|
trans_fct = uft.ir2tf(psf, image.shape, is_real=is_real)
|
||
|
|
else:
|
||
|
|
trans_fct = psf
|
||
|
|
|
||
|
|
# The mean of the object
|
||
|
|
x_postmean = np.zeros(trans_fct.shape)
|
||
|
|
# The previous computed mean in the iterative loop
|
||
|
|
prev_x_postmean = np.zeros(trans_fct.shape)
|
||
|
|
|
||
|
|
# Difference between two successive mean
|
||
|
|
delta = np.NAN
|
||
|
|
|
||
|
|
# Initial state of the chain
|
||
|
|
gn_chain, gx_chain = [1], [1]
|
||
|
|
|
||
|
|
# The correlation of the object in Fourier space (if size is big,
|
||
|
|
# this can reduce computation time in the loop)
|
||
|
|
areg2 = np.abs(reg) ** 2
|
||
|
|
atf2 = np.abs(trans_fct) ** 2
|
||
|
|
|
||
|
|
# The Fourier transform may change the image.size attribute, so we
|
||
|
|
# store it.
|
||
|
|
if is_real:
|
||
|
|
data_spectrum = uft.urfft2(image.astype(float))
|
||
|
|
else:
|
||
|
|
data_spectrum = uft.ufft2(image.astype(float))
|
||
|
|
|
||
|
|
# Gibbs sampling
|
||
|
|
for iteration in range(params['max_iter']):
|
||
|
|
# Sample of Eq. 27 p(circX^k | gn^k-1, gx^k-1, y).
|
||
|
|
|
||
|
|
# weighting (correlation in direct space)
|
||
|
|
precision = gn_chain[-1] * atf2 + gx_chain[-1] * areg2 # Eq. 29
|
||
|
|
excursion = np.sqrt(0.5) / np.sqrt(precision) * (
|
||
|
|
np.random.standard_normal(data_spectrum.shape) +
|
||
|
|
1j * np.random.standard_normal(data_spectrum.shape))
|
||
|
|
|
||
|
|
# mean Eq. 30 (RLS for fixed gn, gamma0 and gamma1 ...)
|
||
|
|
wiener_filter = gn_chain[-1] * np.conj(trans_fct) / precision
|
||
|
|
|
||
|
|
# sample of X in Fourier space
|
||
|
|
x_sample = wiener_filter * data_spectrum + excursion
|
||
|
|
if params['callback']:
|
||
|
|
params['callback'](x_sample)
|
||
|
|
|
||
|
|
# sample of Eq. 31 p(gn | x^k, gx^k, y)
|
||
|
|
gn_chain.append(npr.gamma(image.size / 2,
|
||
|
|
2 / uft.image_quad_norm(data_spectrum -
|
||
|
|
x_sample *
|
||
|
|
trans_fct)))
|
||
|
|
|
||
|
|
# sample of Eq. 31 p(gx | x^k, gn^k-1, y)
|
||
|
|
gx_chain.append(npr.gamma((image.size - 1) / 2,
|
||
|
|
2 / uft.image_quad_norm(x_sample * reg)))
|
||
|
|
|
||
|
|
# current empirical average
|
||
|
|
if iteration > params['burnin']:
|
||
|
|
x_postmean = prev_x_postmean + x_sample
|
||
|
|
|
||
|
|
if iteration > (params['burnin'] + 1):
|
||
|
|
current = x_postmean / (iteration - params['burnin'])
|
||
|
|
previous = prev_x_postmean / (iteration - params['burnin'] - 1)
|
||
|
|
|
||
|
|
delta = np.sum(np.abs(current - previous)) / \
|
||
|
|
np.sum(np.abs(x_postmean)) / (iteration - params['burnin'])
|
||
|
|
|
||
|
|
prev_x_postmean = x_postmean
|
||
|
|
|
||
|
|
# stop of the algorithm
|
||
|
|
if (iteration > params['min_iter']) and (delta < params['threshold']):
|
||
|
|
break
|
||
|
|
|
||
|
|
# Empirical average \approx POSTMEAN Eq. 44
|
||
|
|
x_postmean = x_postmean / (iteration - params['burnin'])
|
||
|
|
if is_real:
|
||
|
|
x_postmean = uft.uirfft2(x_postmean, shape=image.shape)
|
||
|
|
else:
|
||
|
|
x_postmean = uft.uifft2(x_postmean)
|
||
|
|
|
||
|
|
if clip:
|
||
|
|
x_postmean[x_postmean > 1] = 1
|
||
|
|
x_postmean[x_postmean < -1] = -1
|
||
|
|
|
||
|
|
return (x_postmean, {'noise': gn_chain, 'prior': gx_chain})
|
||
|
|
|
||
|
|
|
||
|
|
def richardson_lucy(image, psf, iterations=50, clip=True, filter_epsilon=None):
|
||
|
|
"""Richardson-Lucy deconvolution.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
image : ndarray
|
||
|
|
Input degraded image (can be N dimensional).
|
||
|
|
psf : ndarray
|
||
|
|
The point spread function.
|
||
|
|
iterations : int, optional
|
||
|
|
Number of iterations. This parameter plays the role of
|
||
|
|
regularisation.
|
||
|
|
clip : boolean, optional
|
||
|
|
True by default. If true, pixel value of the result above 1 or
|
||
|
|
under -1 are thresholded for skimage pipeline compatibility.
|
||
|
|
filter_epsilon: float, optional
|
||
|
|
Value below which intermediate results become 0 to avoid division
|
||
|
|
by small numbers.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
im_deconv : ndarray
|
||
|
|
The deconvolved image.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from skimage import img_as_float, data, restoration
|
||
|
|
>>> camera = img_as_float(data.camera())
|
||
|
|
>>> from scipy.signal import convolve2d
|
||
|
|
>>> psf = np.ones((5, 5)) / 25
|
||
|
|
>>> camera = convolve2d(camera, psf, 'same')
|
||
|
|
>>> camera += 0.1 * camera.std() * np.random.standard_normal(camera.shape)
|
||
|
|
>>> deconvolved = restoration.richardson_lucy(camera, psf, 5)
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Richardson%E2%80%93Lucy_deconvolution
|
||
|
|
"""
|
||
|
|
float_type = np.promote_types(image.dtype, np.float32)
|
||
|
|
image = image.astype(float_type, copy=False)
|
||
|
|
psf = psf.astype(float_type, copy=False)
|
||
|
|
im_deconv = np.full(image.shape, 0.5, dtype=float_type)
|
||
|
|
psf_mirror = np.flip(psf)
|
||
|
|
|
||
|
|
for _ in range(iterations):
|
||
|
|
conv = convolve(im_deconv, psf, mode='same')
|
||
|
|
if filter_epsilon:
|
||
|
|
relative_blur = np.where(conv < filter_epsilon, 0, image / conv)
|
||
|
|
else:
|
||
|
|
relative_blur = image / conv
|
||
|
|
im_deconv *= convolve(relative_blur, psf_mirror, mode='same')
|
||
|
|
|
||
|
|
if clip:
|
||
|
|
im_deconv[im_deconv > 1] = 1
|
||
|
|
im_deconv[im_deconv < -1] = -1
|
||
|
|
|
||
|
|
return im_deconv
|