# coding: utf-8 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 warnings import pywt def denoise_bilateral(image, win_size=None, sigma_color=None, sigma_spatial=1, bins=10000, mode='constant', cval=0, multichannel=True, sigma_range=None): """Denoise image using bilateral filter. This is an edge-preserving, denoising filter. It averages pixels based on their spatial closeness and radiometric similarity. 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: 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: warnings.warn("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.".format(image.shape, image.shape[2])) else: msg = "Input image must be grayscale, RGB, or RGBA; " \ "but has shape {0}." warnings.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 sigma_range is not None: warn('`sigma_range` has been deprecated in favor of ' '`sigma_color`. The `sigma_range` keyword argument ' 'will be removed in v0.14', skimage_deprecation) #If sigma_range is provided, assign it to sigma_color sigma_color = sigma_range 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. 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(im, weight=0.1, eps=2.e-4, n_iter_max=200): """Perform total-variation denoising on n-dimensional images. Parameters ---------- im : 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 = im.ndim p = np.zeros((im.ndim, ) + im.shape, dtype=im.dtype) g = np.zeros_like(p) d = np.zeros_like(im) 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[slices_d] += p[slices_p] slices_d[ax] = slice(None) slices_p[ax+1] = slice(None) out = im + d else: out = im 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[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(im.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(im, weight=0.1, eps=2.e-4, n_iter_max=200, multichannel=False): """Perform total-variation denoising on n-dimensional images. Parameters ---------- im : ndarray of ints, uints or floats Input data to be denoised. `im` 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 = im.dtype if not im_type.kind == 'f': im = img_as_float(im) if multichannel: out = np.zeros_like(im) for c in range(im.shape[-1]): out[..., c] = _denoise_tv_chambolle_nd(im[..., c], weight, eps, n_iter_max) else: out = _denoise_tv_chambolle_nd(im, weight, eps, n_iter_max) return out def _wavelet_threshold(img, wavelet, threshold=None, sigma=None, mode='soft'): """Performs wavelet denoising. Parameters ---------- img : ndarray (2d or 3d) of ints, uints or floats Input data to be denoised. `img` 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}``. sigma : float, optional The standard deviation of the noise. The noise is estimated when sigma is None (the default) by the method in [2]_. threshold : float, optional The thresholding value. All wavelet coefficients less than this value are set to 0. The default value (None) uses the SureShrink method found in [1]_ to remove noise. 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. 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 """ coeffs = pywt.wavedecn(img, wavelet=wavelet) detail_coeffs = coeffs[-1]['d' * img.ndim] if sigma is None: # Estimates via the noise via method in [2] sigma = np.median(np.abs(detail_coeffs)) / 0.67448975019608171 if threshold is None: # The BayesShrink threshold from [1]_ in docstring threshold = sigma**2 / np.sqrt(max(img.var() - sigma**2, 0)) denoised_detail = [{key: pywt.threshold(level[key], value=threshold, mode=mode) for key in level} for level in coeffs[1:]] denoised_root = pywt.threshold(coeffs[0], value=threshold, mode=mode) denoised_coeffs = [denoised_root] + [d for d in denoised_detail] return pywt.waverecn(denoised_coeffs, wavelet) def denoise_wavelet(img, sigma=None, wavelet='db1', mode='soft', multichannel=False): """Performs wavelet denoising on an image. Parameters ---------- img : ndarray ([M[, N[, ...P]][, C]) of ints, uints or floats Input data to be denoised. `img` can be of any numeric type, but it is cast into an ndarray of floats for the computation of the denoised image. sigma : float, optional The noise standard deviation used when computing the threshold adaptively as described in [1]_. 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. multichannel : bool, optional Apply wavelet denoising separately for each channel (where channels correspond to the final axis of the array). 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. 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) """ img = img_as_float(img) if multichannel: out = np.empty_like(img) for c in range(img.shape[-1]): out[..., c] = _wavelet_threshold(img[..., c], wavelet=wavelet, mode=mode, sigma=sigma) else: out = _wavelet_threshold(img, wavelet=wavelet, mode=mode, sigma=sigma) clip_range = (-1, 1) if img.min() < 0 else (0, 1) return np.clip(out, *clip_range)