# coding: utf-8 import scipy.stats import numpy as np from math import ceil from .. import img_as_float from ..restoration._denoise_cy import _denoise_bilateral, _denoise_tv_bregman from .._shared.utils import skimage_deprecation, warn import pywt import skimage.color as color import numbers def denoise_bilateral(image, win_size=None, sigma_color=None, sigma_spatial=1, bins=10000, mode='constant', cval=0, multichannel=None): """Denoise image using bilateral filter. This is an edge-preserving, denoising filter. It averages pixels based on their spatial closeness and radiometric similarity [1]_. Spatial closeness is measured by the Gaussian function of the Euclidean distance between two pixels and a certain standard deviation (`sigma_spatial`). Radiometric similarity is measured by the Gaussian function of the Euclidean distance between two color values and a certain standard deviation (`sigma_color`). Parameters ---------- image : ndarray, shape (M, N[, 3]) Input image, 2D grayscale or RGB. win_size : int Window size for filtering. If win_size is not specified, it is calculated as ``max(5, 2 * ceil(3 * sigma_spatial) + 1)``. sigma_color : float Standard deviation for grayvalue/color distance (radiometric similarity). A larger value results in averaging of pixels with larger radiometric differences. Note, that the image will be converted using the `img_as_float` function and thus the standard deviation is in respect to the range ``[0, 1]``. If the value is ``None`` the standard deviation of the ``image`` will be used. sigma_spatial : float Standard deviation for range distance. A larger value results in averaging of pixels with larger spatial differences. bins : int Number of discrete values for Gaussian weights of color filtering. A larger value results in improved accuracy. mode : {'constant', 'edge', 'symmetric', 'reflect', 'wrap'} How to handle values outside the image borders. See `numpy.pad` for detail. cval : string Used in conjunction with mode 'constant', the value outside the image boundaries. multichannel : bool Whether the last axis of the image is to be interpreted as multiple channels or another spatial dimension. Returns ------- denoised : ndarray Denoised image. References ---------- .. [1] http://users.soe.ucsc.edu/~manduchi/Papers/ICCV98.pdf Examples -------- >>> from skimage import data, img_as_float >>> astro = img_as_float(data.astronaut()) >>> astro = astro[220:300, 220:320] >>> noisy = astro + 0.6 * astro.std() * np.random.random(astro.shape) >>> noisy = np.clip(noisy, 0, 1) >>> denoised = denoise_bilateral(noisy, sigma_color=0.05, sigma_spatial=15) """ if multichannel is None: warn('denoise_bilateral will default to multichannel=False in v0.15') multichannel = True if multichannel: if image.ndim != 3: if image.ndim == 2: raise ValueError("Use ``multichannel=False`` for 2D grayscale " "images. The last axis of the input image " "must be multiple color channels not another " "spatial dimension.") else: raise ValueError("Bilateral filter is only implemented for " "2D grayscale images (image.ndim == 2) and " "2D multichannel (image.ndim == 3) images, " "but the input image has {0} dimensions. " "".format(image.ndim)) elif image.shape[2] not in (3, 4): if image.shape[2] > 4: msg = ("The last axis of the input image is interpreted as " "channels. Input image with shape {0} has {1} channels " "in last axis. ``denoise_bilateral`` is implemented " "for 2D grayscale and color images only") warn(msg.format(image.shape, image.shape[2])) else: msg = "Input image must be grayscale, RGB, or RGBA; " \ "but has shape {0}." warn(msg.format(image.shape)) else: if image.ndim > 2: raise ValueError("Bilateral filter is not implemented for " "grayscale images of 3 or more dimensions, " "but input image has {0} dimension. Use " "``multichannel=True`` for 2-D RGB " "images.".format(image.shape)) if win_size is None: win_size = max(5, 2 * int(ceil(3 * sigma_spatial)) + 1) return _denoise_bilateral(image, win_size, sigma_color, sigma_spatial, bins, mode, cval) def denoise_tv_bregman(image, weight, max_iter=100, eps=1e-3, isotropic=True): """Perform total-variation denoising using split-Bregman optimization. Total-variation denoising (also know as total-variation regularization) tries to find an image with less total-variation under the constraint of being similar to the input image, which is controlled by the regularization parameter ([1]_, [2]_, [3]_, [4]_). Parameters ---------- image : ndarray Input data to be denoised (converted using img_as_float`). weight : float Denoising weight. The smaller the `weight`, the more denoising (at the expense of less similarity to the `input`). The regularization parameter `lambda` is chosen as `2 * weight`. eps : float, optional Relative difference of the value of the cost function that determines the stop criterion. The algorithm stops when:: SUM((u(n) - u(n-1))**2) < eps max_iter : int, optional Maximal number of iterations used for the optimization. isotropic : boolean, optional Switch between isotropic and anisotropic TV denoising. Returns ------- u : ndarray Denoised image. References ---------- .. [1] http://en.wikipedia.org/wiki/Total_variation_denoising .. [2] Tom Goldstein and Stanley Osher, "The Split Bregman Method For L1 Regularized Problems", ftp://ftp.math.ucla.edu/pub/camreport/cam08-29.pdf .. [3] Pascal Getreuer, "Rudin–Osher–Fatemi Total Variation Denoising using Split Bregman" in Image Processing On Line on 2012–05–19, http://www.ipol.im/pub/art/2012/g-tvd/article_lr.pdf .. [4] http://www.math.ucsb.edu/~cgarcia/UGProjects/BregmanAlgorithms_JacquelineBush.pdf """ return _denoise_tv_bregman(image, weight, max_iter, eps, isotropic) def _denoise_tv_chambolle_nd(image, weight=0.1, eps=2.e-4, n_iter_max=200): """Perform total-variation denoising on n-dimensional images. Parameters ---------- image : ndarray n-D input data to be denoised. weight : float, optional Denoising weight. The greater `weight`, the more denoising (at the expense of fidelity to `input`). eps : float, optional Relative difference of the value of the cost function that determines the stop criterion. The algorithm stops when: (E_(n-1) - E_n) < eps * E_0 n_iter_max : int, optional Maximal number of iterations used for the optimization. Returns ------- out : ndarray Denoised array of floats. Notes ----- Rudin, Osher and Fatemi algorithm. """ ndim = image.ndim p = np.zeros((image.ndim, ) + image.shape, dtype=image.dtype) g = np.zeros_like(p) d = np.zeros_like(image) i = 0 while i < n_iter_max: if i > 0: # d will be the (negative) divergence of p d = -p.sum(0) slices_d = [slice(None), ] * ndim slices_p = [slice(None), ] * (ndim + 1) for ax in range(ndim): slices_d[ax] = slice(1, None) slices_p[ax+1] = slice(0, -1) slices_p[0] = ax d[tuple(slices_d)] += p[tuple(slices_p)] slices_d[ax] = slice(None) slices_p[ax+1] = slice(None) out = image + d else: out = image E = (d ** 2).sum() # g stores the gradients of out along each axis # e.g. g[0] is the first order finite difference along axis 0 slices_g = [slice(None), ] * (ndim + 1) for ax in range(ndim): slices_g[ax+1] = slice(0, -1) slices_g[0] = ax g[tuple(slices_g)] = np.diff(out, axis=ax) slices_g[ax+1] = slice(None) norm = np.sqrt((g ** 2).sum(axis=0))[np.newaxis, ...] E += weight * norm.sum() tau = 1. / (2.*ndim) norm *= tau / weight norm += 1. p -= tau * g p /= norm E /= float(image.size) if i == 0: E_init = E E_previous = E else: if np.abs(E_previous - E) < eps * E_init: break else: E_previous = E i += 1 return out def denoise_tv_chambolle(image, weight=0.1, eps=2.e-4, n_iter_max=200, multichannel=False): """Perform total-variation denoising on n-dimensional images. Parameters ---------- image : ndarray of ints, uints or floats Input data to be denoised. `image` can be of any numeric type, but it is cast into an ndarray of floats for the computation of the denoised image. weight : float, optional Denoising weight. The greater `weight`, the more denoising (at the expense of fidelity to `input`). eps : float, optional Relative difference of the value of the cost function that determines the stop criterion. The algorithm stops when: (E_(n-1) - E_n) < eps * E_0 n_iter_max : int, optional Maximal number of iterations used for the optimization. multichannel : bool, optional Apply total-variation denoising separately for each channel. This option should be true for color images, otherwise the denoising is also applied in the channels dimension. Returns ------- out : ndarray Denoised image. Notes ----- Make sure to set the multichannel parameter appropriately for color images. The principle of total variation denoising is explained in http://en.wikipedia.org/wiki/Total_variation_denoising The principle of total variation denoising is to minimize the total variation of the image, which can be roughly described as the integral of the norm of the image gradient. Total variation denoising tends to produce "cartoon-like" images, that is, piecewise-constant images. This code is an implementation of the algorithm of Rudin, Fatemi and Osher that was proposed by Chambolle in [1]_. References ---------- .. [1] A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, Springer, 2004, 20, 89-97. Examples -------- 2D example on astronaut image: >>> from skimage import color, data >>> img = color.rgb2gray(data.astronaut())[:50, :50] >>> img += 0.5 * img.std() * np.random.randn(*img.shape) >>> denoised_img = denoise_tv_chambolle(img, weight=60) 3D example on synthetic data: >>> x, y, z = np.ogrid[0:20, 0:20, 0:20] >>> mask = (x - 22)**2 + (y - 20)**2 + (z - 17)**2 < 8**2 >>> mask = mask.astype(np.float) >>> mask += 0.2*np.random.randn(*mask.shape) >>> res = denoise_tv_chambolle(mask, weight=100) """ im_type = image.dtype if not im_type.kind == 'f': image = img_as_float(image) if multichannel: out = np.zeros_like(image) for c in range(image.shape[-1]): out[..., c] = _denoise_tv_chambolle_nd(image[..., c], weight, eps, n_iter_max) else: out = _denoise_tv_chambolle_nd(image, weight, eps, n_iter_max) return out def _bayes_thresh(details, var): """BayesShrink threshold for a zero-mean details coeff array.""" # Equivalent to: dvar = np.var(details) for 0-mean details array dvar = np.mean(details*details) eps = np.finfo(details.dtype).eps thresh = var / np.sqrt(max(dvar - var, eps)) return thresh def _universal_thresh(img, sigma): """ Universal threshold used by the VisuShrink method """ return sigma*np.sqrt(2*np.log(img.size)) def _sigma_est_dwt(detail_coeffs, distribution='Gaussian'): """Calculate the robust median estimator of the noise standard deviation. Parameters ---------- detail_coeffs : ndarray The detail coefficients corresponding to the discrete wavelet transform of an image. distribution : str The underlying noise distribution. Returns ------- sigma : float The estimated noise standard deviation (see section 4.2 of [1]_). References ---------- .. [1] D. L. Donoho and I. M. Johnstone. "Ideal spatial adaptation by wavelet shrinkage." Biometrika 81.3 (1994): 425-455. DOI:10.1093/biomet/81.3.425 """ # Consider regions with detail coefficients exactly zero to be masked out detail_coeffs = detail_coeffs[np.nonzero(detail_coeffs)] if distribution.lower() == 'gaussian': # 75th quantile of the underlying, symmetric noise distribution denom = scipy.stats.norm.ppf(0.75) sigma = np.median(np.abs(detail_coeffs)) / denom else: raise ValueError("Only Gaussian noise estimation is currently " "supported") return sigma def _wavelet_threshold(image, wavelet, method=None, threshold=None, sigma=None, mode='soft', wavelet_levels=None): """Perform wavelet thresholding. Parameters ---------- image : ndarray (2d or 3d) of ints, uints or floats Input data to be denoised. `image` can be of any numeric type, but it is cast into an ndarray of floats for the computation of the denoised image. wavelet : string The type of wavelet to perform. Can be any of the options pywt.wavelist outputs. For example, this may be any of ``{db1, db2, db3, db4, haar}``. method : {'BayesShrink', 'VisuShrink'}, optional Thresholding method to be used. The currently supported methods are "BayesShrink" [1]_ and "VisuShrink" [2]_. If it is set to None, a user-specified ``threshold`` must be supplied instead. threshold : float, optional The thresholding value to apply during wavelet coefficient thresholding. The default value (None) uses the selected ``method`` to estimate appropriate threshold(s) for noise removal. sigma : float, optional The standard deviation of the noise. The noise is estimated when sigma is None (the default) by the method in [2]_. mode : {'soft', 'hard'}, optional An optional argument to choose the type of denoising performed. It noted that choosing soft thresholding given additive noise finds the best approximation of the original image. wavelet_levels : int or None, optional The number of wavelet decomposition levels to use. The default is three less than the maximum number of possible decomposition levels (see Notes below). Returns ------- out : ndarray Denoised image. References ---------- .. [1] Chang, S. Grace, Bin Yu, and Martin Vetterli. "Adaptive wavelet thresholding for image denoising and compression." Image Processing, IEEE Transactions on 9.9 (2000): 1532-1546. DOI: 10.1109/83.862633 .. [2] D. L. Donoho and I. M. Johnstone. "Ideal spatial adaptation by wavelet shrinkage." Biometrika 81.3 (1994): 425-455. DOI: 10.1093/biomet/81.3.425 """ wavelet = pywt.Wavelet(wavelet) # original_extent is used to workaround PyWavelets issue #80 # odd-sized input results in an image with 1 extra sample after waverecn original_extent = tuple(slice(s) for s in image.shape) # Determine the number of wavelet decomposition levels if wavelet_levels is None: # Determine the maximum number of possible levels for image dlen = wavelet.dec_len wavelet_levels = np.min( [pywt.dwt_max_level(s, dlen) for s in image.shape]) # Skip coarsest wavelet scales (see Notes in docstring). wavelet_levels = max(wavelet_levels - 3, 1) coeffs = pywt.wavedecn(image, wavelet=wavelet, level=wavelet_levels) # Detail coefficients at each decomposition level dcoeffs = coeffs[1:] if sigma is None: # Estimate the noise via the method in [2]_ detail_coeffs = dcoeffs[-1]['d' * image.ndim] sigma = _sigma_est_dwt(detail_coeffs, distribution='Gaussian') if method is not None and threshold is not None: warn(("Thresholding method {} selected. The user-specified threshold " "will be ignored.").format(method)) if threshold is None: var = sigma**2 if method is None: raise ValueError( "If method is None, a threshold must be provided.") elif method == "BayesShrink": # The BayesShrink thresholds from [1]_ in docstring threshold = [{key: _bayes_thresh(level[key], var) for key in level} for level in dcoeffs] elif method == "VisuShrink": # The VisuShrink thresholds from [2]_ in docstring threshold = _universal_thresh(image, sigma) else: raise ValueError("Unrecognized method: {}".format(method)) if np.isscalar(threshold): # A single threshold for all coefficient arrays denoised_detail = [{key: pywt.threshold(level[key], value=threshold, mode=mode) for key in level} for level in dcoeffs] else: # Dict of unique threshold coefficients for each detail coeff. array denoised_detail = [{key: pywt.threshold(level[key], value=thresh[key], mode=mode) for key in level} for thresh, level in zip(threshold, dcoeffs)] denoised_coeffs = [coeffs[0]] + denoised_detail return pywt.waverecn(denoised_coeffs, wavelet)[original_extent] def denoise_wavelet(image, sigma=None, wavelet='db1', mode='soft', wavelet_levels=None, multichannel=False, convert2ycbcr=False, method='BayesShrink'): """Perform wavelet denoising on an image. Parameters ---------- image : ndarray ([M[, N[, ...P]][, C]) of ints, uints or floats Input data to be denoised. `image` can be of any numeric type, but it is cast into an ndarray of floats for the computation of the denoised image. sigma : float or list, optional The noise standard deviation used when computing the wavelet detail coefficient threshold(s). When None (default), the noise standard deviation is estimated via the method in [2]_. wavelet : string, optional The type of wavelet to perform and can be any of the options ``pywt.wavelist`` outputs. The default is `'db1'`. For example, ``wavelet`` can be any of ``{'db2', 'haar', 'sym9'}`` and many more. mode : {'soft', 'hard'}, optional An optional argument to choose the type of denoising performed. It noted that choosing soft thresholding given additive noise finds the best approximation of the original image. wavelet_levels : int or None, optional The number of wavelet decomposition levels to use. The default is three less than the maximum number of possible decomposition levels. multichannel : bool, optional Apply wavelet denoising separately for each channel (where channels correspond to the final axis of the array). convert2ycbcr : bool, optional If True and multichannel True, do the wavelet denoising in the YCbCr colorspace instead of the RGB color space. This typically results in better performance for RGB images. method : {'BayesShrink', 'VisuShrink'}, optional Thresholding method to be used. The currently supported methods are "BayesShrink" [1]_ and "VisuShrink" [2]_. Defaults to "BayesShrink". Returns ------- out : ndarray Denoised image. Notes ----- The wavelet domain is a sparse representation of the image, and can be thought of similarly to the frequency domain of the Fourier transform. Sparse representations have most values zero or near-zero and truly random noise is (usually) represented by many small values in the wavelet domain. Setting all values below some threshold to 0 reduces the noise in the image, but larger thresholds also decrease the detail present in the image. If the input is 3D, this function performs wavelet denoising on each color plane separately. The output image is clipped between either [-1, 1] and [0, 1] depending on the input image range. When YCbCr conversion is done, every color channel is scaled between 0 and 1, and `sigma` values are applied to these scaled color channels. Many wavelet coefficient thresholding approaches have been proposed. By default, ``denoise_wavelet`` applies BayesShrink, which is an adaptive thresholding method that computes separate thresholds for each wavelet sub-band as described in [1]_. If ``method == "VisuShrink"``, a single "universal threshold" is applied to all wavelet detail coefficients as described in [2]_. This threshold is designed to remove all Gaussian noise at a given ``sigma`` with high probability, but tends to produce images that appear overly smooth. References ---------- .. [1] Chang, S. Grace, Bin Yu, and Martin Vetterli. "Adaptive wavelet thresholding for image denoising and compression." Image Processing, IEEE Transactions on 9.9 (2000): 1532-1546. DOI: 10.1109/83.862633 .. [2] D. L. Donoho and I. M. Johnstone. "Ideal spatial adaptation by wavelet shrinkage." Biometrika 81.3 (1994): 425-455. DOI: 10.1093/biomet/81.3.425 Examples -------- >>> from skimage import color, data >>> img = img_as_float(data.astronaut()) >>> img = color.rgb2gray(img) >>> img += 0.1 * np.random.randn(*img.shape) >>> img = np.clip(img, 0, 1) >>> denoised_img = denoise_wavelet(img, sigma=0.1) """ if method not in ["BayesShrink", "VisuShrink"]: raise ValueError( ('Invalid method: {}. The currently supported methods are ' '"BayesShrink" and "VisuShrink"').format(method)) image = img_as_float(image) if multichannel: if isinstance(sigma, numbers.Number) or sigma is None: sigma = [sigma] * image.shape[-1] if multichannel: if convert2ycbcr: out = color.rgb2ycbcr(image) for i in range(3): # renormalizing this color channel to live in [0, 1] min, max = out[..., i].min(), out[..., i].max() channel = out[..., i] - min channel /= max - min out[..., i] = denoise_wavelet(channel, wavelet=wavelet, method=method, sigma=sigma[i], mode=mode, wavelet_levels=wavelet_levels) out[..., i] = out[..., i] * (max - min) out[..., i] += min out = color.ycbcr2rgb(out) else: out = np.empty_like(image) for c in range(image.shape[-1]): out[..., c] = _wavelet_threshold(image[..., c], wavelet=wavelet, method=method, sigma=sigma[c], mode=mode, wavelet_levels=wavelet_levels) else: out = _wavelet_threshold(image, wavelet=wavelet, method=method, sigma=sigma, mode=mode, wavelet_levels=wavelet_levels) clip_range = (-1, 1) if image.min() < 0 else (0, 1) return np.clip(out, *clip_range) def estimate_sigma(image, average_sigmas=False, multichannel=False): """ Robust wavelet-based estimator of the (Gaussian) noise standard deviation. Parameters ---------- image : ndarray Image for which to estimate the noise standard deviation. average_sigmas : bool, optional If true, average the channel estimates of `sigma`. Otherwise return a list of sigmas corresponding to each channel. multichannel : bool Estimate sigma separately for each channel. Returns ------- sigma : float or list Estimated noise standard deviation(s). If `multichannel` is True and `average_sigmas` is False, a separate noise estimate for each channel is returned. Otherwise, the average of the individual channel estimates is returned. Notes ----- This function assumes the noise follows a Gaussian distribution. The estimation algorithm is based on the median absolute deviation of the wavelet detail coefficients as described in section 4.2 of [1]_. References ---------- .. [1] D. L. Donoho and I. M. Johnstone. "Ideal spatial adaptation by wavelet shrinkage." Biometrika 81.3 (1994): 425-455. DOI:10.1093/biomet/81.3.425 Examples -------- >>> import skimage.data >>> from skimage import img_as_float >>> img = img_as_float(skimage.data.camera()) >>> sigma = 0.1 >>> img = img + sigma * np.random.standard_normal(img.shape) >>> sigma_hat = estimate_sigma(img, multichannel=False) """ if multichannel: nchannels = image.shape[-1] sigmas = [estimate_sigma( image[..., c], multichannel=False) for c in range(nchannels)] if average_sigmas: sigmas = np.mean(sigmas) return sigmas elif image.shape[-1] <= 4: msg = ("image is size {0} on the last axis, but multichannel is " "False. If this is a color image, please set multichannel " "to True for proper noise estimation.") warn(msg.format(image.shape[-1])) coeffs = pywt.dwtn(image, wavelet='db2') detail_coeffs = coeffs['d' * image.ndim] return _sigma_est_dwt(detail_coeffs, distribution='Gaussian')