Module: restoration¶

References¶

denoise_bilateral¶

skimage.restoration.denoise_bilateral(image, win_size=5, sigma_range=None, sigma_spatial=1, bins=10000, mode='constant', cval=0)¶

Denoise image using bilateral filter.

This is an edge-preserving and noise reducing denoising filter. It averages pixels based on their spatial closeness and radiometric similarity.

Spatial closeness is measured by the gaussian function of the euclidian distance between two pixels and a certain standard deviation (sigma_spatial).

Radiometric similarity is measured by the gaussian function of the euclidian distance between two color values and a certain standard deviation (sigma_range).

Parameters:

image : ndarray Input image. win_size : int Window size for filtering. sigma_range : 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]. 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 : string How to handle values outside the image borders. See scipy.ndimage.map_coordinates for detail. cval : string Used in conjunction with mode ‘constant’, the value outside the image boundaries.

Returns:

denoised : ndarray Denoised image.

References

[R281]

http://users.soe.ucsc.edu/~manduchi/Papers/ICCV98.pdf

denoise_tv_bregman¶

skimage.restoration.denoise_tv_bregman(image, weight, max_iter=100, eps=0.001, 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, optional 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

[R282]

http://en.wikipedia.org/wiki/Total_variation_denoising

[R283]

Tom Goldstein and Stanley Osher, “The Split Bregman Method For L1 Regularized Problems”, ftp://ftp.math.ucla.edu/pub/camreport/cam08-29.pdf

[R284]

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

[R285]

http://www.math.ucsb.edu/~cgarcia/UGProjects/BregmanAlgorithms_JacquelineBush.pdf

denoise_tv_chambolle¶

skimage.restoration.denoise_tv_chambolle(im, weight=50, eps=0.0002, n_iter_max=200, multichannel=False)¶

Perform total-variation denoising on 2D and 3D images.

Parameters:

im : ndarray (2d or 3d) 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 3rd 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 [R286].

References

[R286]

(1, 2) 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 Lena image:

>>> from skimage import color, data
>>> lena = color.rgb2gray(data.lena())[:50, :50]
>>> lena += 0.5 * lena.std() * np.random.randn(*lena.shape)
>>> denoised_lena = denoise_tv_chambolle(lena, 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)

richardson_lucy¶

skimage.restoration.richardson_lucy(image, psf, iterations=50, clip=True)¶

Richardson-Lucy deconvolution.

Parameters:	image : ndarray Input degraded image psf : ndarray The point spread function iterations : int Number of iterations. This parameter play to 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

References

[R288]

http://en.wikipedia.org/wiki/Richardson%E2%80%93Lucy_deconvolution

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)

unsupervised_wiener¶

skimage.restoration.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 recognize 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 dictionary of gibbs parameters. See below. 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:	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. 100 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. 150 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.

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 high probable image since they have the biggest contribution to the mean. At the opposite, the lowest probable image are draw less often since their contribution are 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

[R288]

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) http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-7-1593 http://research.orieux.fr/files/papers/OGR-JOSA10.pdf

Examples

>>> from skimage import color, data, restoration
>>> lena = color.rgb2gray(data.lena())
>>> from scipy.signal import convolve2d
>>> psf = np.ones((5, 5)) / 25
>>> lena = convolve2d(lena, psf, 'same')
>>> lena += 0.1 * lena.std() * np.random.standard_normal(lena.shape)
>>> deconvolved_lena = restoration.unsupervised_wiener(lena, psf)

unwrap_phase¶

skimage.restoration.unwrap_phase(image, wrap_around=False)¶

From image, wrapped to lie in the interval [-pi, pi), recover the original, unwrapped image.

Parameters:	image : 1D, 2D or 3D ndarray of floats, optionally a masked array The values should be in the range [-pi, pi). If a masked array is provided, the masked entries will not be changed, and their values will not be used to guide the unwrapping of neighboring, unmasked values. Masked 1D arrays are not allowed, and will raise a ValueError. wrap_around : bool or sequence of bool When an element of the sequence is True, the unwrapping process will regard the edges along the corresponding axis of the image to be connected and use this connectivity to guide the phase unwrapping process. If only a single boolean is given, it will apply to all axes. Wrap around is not supported for 1D arrays.
Returns:	image_unwrapped : array_like, float32 Unwrapped image of the same shape as the input. If the input image was a masked array, the mask will be preserved.
Raises:	ValueError : If called with a masked 1D array or called with a 1D array and wrap_around=True.

References

[R289]

Miguel Arevallilo Herraez, David R. Burton, Michael J. Lalor, and Munther A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path”, Journal Applied Optics, Vol. 41, No. 35 (2002) 7437,

[R290]

Abdul-Rahman, H., Gdeisat, M., Burton, D., & Lalor, M., “Fast three-dimensional phase-unwrapping algorithm based on sorting by reliability following a non-continuous path. In W. Osten, C. Gorecki, & E. L. Novak (Eds.), Optical Metrology (2005) 32–40, International Society for Optics and Photonics.

Examples

>>> c0, c1 = np.ogrid[-1:1:128j, -1:1:128j]
>>> image = 12 * np.pi * np.exp(-(c0**2 + c1**2))
>>> image_wrapped = np.angle(np.exp(1j * image))
>>> image_unwrapped = unwrap_phase(image_wrapped)
>>> np.std(image_unwrapped - image) < 1e-6   # A constant offset is normal
True

wiener¶

skimage.restoration.wiener(image, psf, balance, reg=None, is_real=True, clip=True)¶

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 artifact). 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 than 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 value of the result above 1 or under -1 are thresholded for skimage pipeline compatibility.

Returns:

im_deconv : (M, N) ndarray The deconvolved image

Notes

This function applies the Wiener filter to a noisy and degraded image by an impulse response (or PSF). If the data model is

y = Hx + n

where n is noise, H the PSF and x the unknown original image, the Wiener filter is

\hat x = F^\dag (|\Lambda_H|^2 + \lambda |\Lambda_D|^2) \Lambda_H^\dag F y

where F and F^\dag are the Fourier and inverse Fourier transfroms respectively, \Lambda_H the transfer function (or the Fourier transfrom of the PSF, see [Hunt] below) and \Lambda_D the filter to penalize the restored image frequencies (Laplacian by default, that is penalization of high frequency). The parameter \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 wrt. data (sometimes called noise amplification or “explosive” solution).

Finally, the use of Fourier space implies a circulant property of H, see [Hunt].

References

[R291]

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) http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-7-1593 http://research.orieux.fr/files/papers/OGR-JOSA10.pdf

[R292]

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

Examples

>>> from skimage import color, data, restoration
>>> lena = color.rgb2gray(data.lena())
>>> from scipy.signal import convolve2d
>>> psf = np.ones((5, 5)) / 25
>>> lena = convolve2d(lena, psf, 'same')
>>> lena += 0.1 * lena.std() * np.random.standard_normal(lena.shape)
>>> deconvolved_lena = restoration.wiener(lena, psf, 1100)