"""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(np.float)) else: data_spectrum = uft.ufft2(image.astype(np.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): """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. Returns ------- im_deconv : ndarray The deconvolved image. Examples -------- >>> from skimage import color, data, restoration >>> camera = color.rgb2gray(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 """ image = image.astype(np.float) psf = psf.astype(np.float) im_deconv = np.full(image.shape, 0.5) psf_mirror = psf[::-1, ::-1] for _ in range(iterations): relative_blur = image / convolve(im_deconv, psf, mode='same') 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