Module: restoration
¶
Image restoration module.

Create a ball kernel for restoration.rolling_ball. 
Calibrate a denoising function and return optimal Jinvariant version. 


Cycle spinning (repeatedly apply func to shifted versions of x). 
Denoise image using bilateral filter. 

Perform nonlocal means denoising on 2D or 3D grayscale images, and 2D RGB images. 

Perform totalvariation denoising using splitBregman optimization. 

Perform totalvariation denoising on ndimensional images. 


Perform wavelet denoising on an image. 

Create an ellipoid kernel for restoration.rolling_ball. 

Robust waveletbased estimator of the (Gaussian) noise standard deviation. 
Inpaint masked points in image with biharmonic equations. 


RichardsonLucy deconvolution. 

Estimate background intensity by rolling/translating a kernel. 
Unsupervised WienerHunt deconvolution. 


Recover the original from a wrapped phase image. 

WienerHunt deconvolution 
ball_kernel¶

skimage.restoration.
ball_kernel
(radius, ndim)[source]¶ Create a ball kernel for restoration.rolling_ball.
 Parameters
 radiusint
Radius of the ball.
 ndimint
Number of dimensions of the ball.
ndim
should match the dimensionality of the image the kernel will be applied to.
 Returns
 kernelndarray
The kernel containing the surface intensity of the top half of the ellipsoid.
See also
calibrate_denoiser¶

skimage.restoration.
calibrate_denoiser
(image, denoise_function, denoise_parameters, *, stride=4, approximate_loss=True, extra_output=False)[source]¶ Calibrate a denoising function and return optimal Jinvariant version.
The returned function is partially evaluated with optimal parameter values set for denoising the input image.
 Parameters
 imagendarray
Input data to be denoised (converted using img_as_float).
 denoise_functionfunction
Denoising function to be calibrated.
 denoise_parametersdict of list
Ranges of parameters for denoise_function to be calibrated over.
 strideint, optional
Stride used in masking procedure that converts denoise_function to Jinvariance.
 approximate_lossbool, optional
Whether to approximate the selfsupervised loss used to evaluate the denoiser by only computing it on one masked version of the image. If False, the runtime will be a factor of stride**image.ndim longer.
 extra_outputbool, optional
If True, return parameters and losses in addition to the calibrated denoising function
 Returns
 best_denoise_functionfunction
The optimal Jinvariant version of denoise_function.
 If extra_output is True, the following tuple is also returned:
 (parameters_tested, losses)tuple (list of dict, list of int)
List of parameters tested for denoise_function, as a dictionary of kwargs Selfsupervised loss for each set of parameters in parameters_tested.
Notes
The calibration procedure uses a selfsupervised meansquareerror loss to evaluate the performance of Jinvariant versions of denoise_function. The minimizer of the selfsupervised loss is also the minimizer of the groundtruth loss (i.e., the true MSE error) [1]. The returned function can be used on the original noisy image, or other images with similar characteristics.
 Increasing the stride increases the performance of best_denoise_function
at the expense of increasing its runtime. It has no effect on the runtime of the calibration.
References
 1
J. Batson & L. Royer. Noise2Self: Blind Denoising by SelfSupervision, International Conference on Machine Learning, p. 524533 (2019).
Examples
>>> from skimage import color, data >>> from skimage.restoration import denoise_wavelet >>> import numpy as np >>> img = color.rgb2gray(data.astronaut()[:50, :50]) >>> noisy = img + 0.5 * img.std() * np.random.randn(*img.shape) >>> parameters = {'sigma': np.arange(0.1, 0.4, 0.02)} >>> denoising_function = calibrate_denoiser(noisy, denoise_wavelet, ... denoise_parameters=parameters) >>> denoised_img = denoising_function(img)
cycle_spin¶

skimage.restoration.
cycle_spin
(x, func, max_shifts, shift_steps=1, num_workers=None, multichannel=False, func_kw={})[source]¶ Cycle spinning (repeatedly apply func to shifted versions of x).
 Parameters
 xarraylike
Data for input to
func
. funcfunction
A function to apply to circularly shifted versions of
x
. Should takex
as its first argument. Any additional arguments can be supplied viafunc_kw
. max_shiftsint or tuple
If an integer, shifts in
range(0, max_shifts+1)
will be used along each axis ofx
. If a tuple,range(0, max_shifts[i]+1)
will be along axis i. shift_stepsint or tuple, optional
The step size for the shifts applied along axis, i, are::
range((0, max_shifts[i]+1, shift_steps[i]))
. If an integer is provided, the same step size is used for all axes. num_workersint or None, optional
The number of parallel threads to use during cycle spinning. If set to
None
, the full set of available cores are used. multichannelbool, optional
Whether to treat the final axis as channels (no cycle shifts are performed over the channels axis).
 func_kwdict, optional
Additional keyword arguments to supply to
func
.
 Returns
 avg_ynp.ndarray
The output of
func(x, **func_kw)
averaged over all combinations of the specified axis shifts.
Notes
Cycle spinning was proposed as a way to approach shiftinvariance via performing several circular shifts of a shiftvariant transform [1].
For a nlevel discrete wavelet transforms, one may wish to perform all shifts up to
max_shifts = 2**n  1
. In practice, much of the benefit can often be realized with only a small number of shifts per axis.For transforms such as the blockwise discrete cosine transform, one may wish to evaluate shifts up to the block size used by the transform.
References
 1
R.R. Coifman and D.L. Donoho. “TranslationInvariant DeNoising”. Wavelets and Statistics, Lecture Notes in Statistics, vol.103. Springer, New York, 1995, pp.125150. DOI:10.1007/9781461225447_9
Examples
>>> import skimage.data >>> from skimage import img_as_float >>> from skimage.restoration import denoise_wavelet, cycle_spin >>> img = img_as_float(skimage.data.camera()) >>> sigma = 0.1 >>> img = img + sigma * np.random.standard_normal(img.shape) >>> denoised = cycle_spin(img, func=denoise_wavelet, ... max_shifts=3)
denoise_bilateral¶

skimage.restoration.
denoise_bilateral
(image, win_size=None, sigma_color=None, sigma_spatial=1, bins=10000, mode='constant', cval=0, multichannel=False)[source]¶ Denoise image using bilateral filter.
 Parameters
 imagendarray, shape (M, N[, 3])
Input image, 2D grayscale or RGB.
 win_sizeint
Window size for filtering. If win_size is not specified, it is calculated as
max(5, 2 * ceil(3 * sigma_spatial) + 1)
. sigma_colorfloat
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 isNone
the standard deviation of theimage
will be used. sigma_spatialfloat
Standard deviation for range distance. A larger value results in averaging of pixels with larger spatial differences.
 binsint
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. cvalstring
Used in conjunction with mode ‘constant’, the value outside the image boundaries.
 multichannelbool
Whether the last axis of the image is to be interpreted as multiple channels or another spatial dimension.
 Returns
 denoisedndarray
Denoised image.
Notes
This is an edgepreserving, 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).
References
 1
C. Tomasi and R. Manduchi. “Bilateral Filtering for Gray and Color Images.” IEEE International Conference on Computer Vision (1998) 839846. DOI:10.1109/ICCV.1998.710815
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, ... multichannel=True)
Examples using skimage.restoration.denoise_bilateral
¶
denoise_nl_means¶

skimage.restoration.
denoise_nl_means
(image, patch_size=7, patch_distance=11, h=0.1, multichannel=False, fast_mode=True, sigma=0.0, *, preserve_range=None)[source]¶ Perform nonlocal means denoising on 2D or 3D grayscale images, and 2D RGB images.
 Parameters
 image2D or 3D ndarray
Input image to be denoised, which can be 2D or 3D, and grayscale or RGB (for 2D images only, see
multichannel
parameter). patch_sizeint, optional
Size of patches used for denoising.
 patch_distanceint, optional
Maximal distance in pixels where to search patches used for denoising.
 hfloat, optional
Cutoff distance (in gray levels). The higher h, the more permissive one is in accepting patches. A higher h results in a smoother image, at the expense of blurring features. For a Gaussian noise of standard deviation sigma, a rule of thumb is to choose the value of h to be sigma of slightly less.
 multichannelbool, optional
Whether the last axis of the image is to be interpreted as multiple channels or another spatial dimension.
 fast_modebool, optional
If True (default value), a fast version of the nonlocal means algorithm is used. If False, the original version of nonlocal means is used. See the Notes section for more details about the algorithms.
 sigmafloat, optional
The standard deviation of the (Gaussian) noise. If provided, a more robust computation of patch weights is computed that takes the expected noise variance into account (see Notes below).
 preserve_rangebool, optional
Whether to keep the original range of values. Otherwise, the input image is converted according to the conventions of img_as_float. Also see https://scikitimage.org/docs/dev/user_guide/data_types.html
 Returns
 resultndarray
Denoised image, of same shape as image.
Notes
The nonlocal means algorithm is well suited for denoising images with specific textures. The principle of the algorithm is to average the value of a given pixel with values of other pixels in a limited neighbourhood, provided that the patches centered on the other pixels are similar enough to the patch centered on the pixel of interest.
In the original version of the algorithm [1], corresponding to
fast=False
, the computational complexity is:image.size * patch_size ** image.ndim * patch_distance ** image.ndim
Hence, changing the size of patches or their maximal distance has a strong effect on computing times, especially for 3D images.
However, the default behavior corresponds to
fast_mode=True
, for which another version of nonlocal means [2] is used, corresponding to a complexity of:image.size * patch_distance ** image.ndim
The computing time depends only weakly on the patch size, thanks to the computation of the integral of patches distances for a given shift, that reduces the number of operations [1]. Therefore, this algorithm executes faster than the classic algorithm (
fast_mode=False
), at the expense of using twice as much memory. This implementation has been proven to be more efficient compared to other alternatives, see e.g. [3].Compared to the classic algorithm, all pixels of a patch contribute to the distance to another patch with the same weight, no matter their distance to the center of the patch. This coarser computation of the distance can result in a slightly poorer denoising performance. Moreover, for small images (images with a linear size that is only a few times the patch size), the classic algorithm can be faster due to boundary effects.
The image is padded using the reflect mode of
skimage.util.pad
before denoising.If the noise standard deviation, sigma, is provided a more robust computation of patch weights is used. Subtracting the known noise variance from the computed patch distances improves the estimates of patch similarity, giving a moderate improvement to denoising performance [4]. It was also mentioned as an option for the fast variant of the algorithm in [3].
When sigma is provided, a smaller h should typically be used to avoid oversmoothing. The optimal value for h depends on the image content and noise level, but a reasonable starting point is
h = 0.8 * sigma
when fast_mode is True, orh = 0.6 * sigma
when fast_mode is False.References
 1(1,2)
A. Buades, B. Coll, & JM. Morel. A nonlocal algorithm for image denoising. In CVPR 2005, Vol. 2, pp. 6065, IEEE. DOI:10.1109/CVPR.2005.38
 2
J. Darbon, A. Cunha, T.F. Chan, S. Osher, and G.J. Jensen, Fast nonlocal filtering applied to electron cryomicroscopy, in 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2008, pp. 13311334. DOI:10.1109/ISBI.2008.4541250
 3(1,2)
Jacques Froment. ParameterFree Fast Pixelwise NonLocal Means Denoising. Image Processing On Line, 2014, vol. 4, pp. 300326. DOI:10.5201/ipol.2014.120
 4
A. Buades, B. Coll, & JM. Morel. NonLocal Means Denoising. Image Processing On Line, 2011, vol. 1, pp. 208212. DOI:10.5201/ipol.2011.bcm_nlm
Examples
>>> a = np.zeros((40, 40)) >>> a[10:10, 10:10] = 1. >>> a += 0.3 * np.random.randn(*a.shape) >>> denoised_a = denoise_nl_means(a, 7, 5, 0.1)
denoise_tv_bregman¶

skimage.restoration.
denoise_tv_bregman
(image, weight, max_iter=100, eps=0.001, isotropic=True, *, multichannel=False)[source]¶ Perform totalvariation denoising using splitBregman optimization.
Totalvariation denoising (also know as totalvariation regularization) tries to find an image with less totalvariation under the constraint of being similar to the input image, which is controlled by the regularization parameter ([1], [2], [3], [4]).
 Parameters
 imagendarray
Input data to be denoised (converted using img_as_float`).
 weightfloat
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.
 epsfloat, optional
Relative difference of the value of the cost function that determines the stop criterion. The algorithm stops when:
SUM((u(n)  u(n1))**2) < eps
 max_iterint, optional
Maximal number of iterations used for the optimization.
 isotropicboolean, optional
Switch between isotropic and anisotropic TV denoising.
 multichannelbool, optional
Apply totalvariation denoising separately for each channel. This option should be true for color images, otherwise the denoising is also applied in the channels dimension.
 Returns
 undarray
Denoised image.
References
 1
 2
Tom Goldstein and Stanley Osher, “The Split Bregman Method For L1 Regularized Problems”, ftp://ftp.math.ucla.edu/pub/camreport/cam0829.pdf
 3
Pascal Getreuer, “Rudin–Osher–Fatemi Total Variation Denoising using Split Bregman” in Image Processing On Line on 2012–05–19, https://www.ipol.im/pub/art/2012/gtvd/article_lr.pdf
 4
https://web.math.ucsb.edu/~cgarcia/UGProjects/BregmanAlgorithms_JacquelineBush.pdf
denoise_tv_chambolle¶

skimage.restoration.
denoise_tv_chambolle
(image, weight=0.1, eps=0.0002, n_iter_max=200, multichannel=False)[source]¶ Perform totalvariation denoising on ndimensional images.
 Parameters
 imagendarray 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.
 weightfloat, optional
Denoising weight. The greater weight, the more denoising (at the expense of fidelity to input).
 epsfloat, optional
Relative difference of the value of the cost function that determines the stop criterion. The algorithm stops when:
(E_(n1)  E_n) < eps * E_0
 n_iter_maxint, optional
Maximal number of iterations used for the optimization.
 multichannelbool, optional
Apply totalvariation denoising separately for each channel. This option should be true for color images, otherwise the denoising is also applied in the channels dimension.
 Returns
 outndarray
Denoised image.
Notes
Make sure to set the multichannel parameter appropriately for color images.
The principle of total variation denoising is explained in https://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 “cartoonlike” images, that is, piecewiseconstant 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, 8997.
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(float) >>> mask += 0.2*np.random.randn(*mask.shape) >>> res = denoise_tv_chambolle(mask, weight=100)
denoise_wavelet¶

skimage.restoration.
denoise_wavelet
(image, sigma=None, wavelet='db1', mode='soft', wavelet_levels=None, multichannel=False, convert2ycbcr=False, method='BayesShrink', rescale_sigma=True)[source]¶ Perform wavelet denoising on an image.
 Parameters
 imagendarray ([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.
 sigmafloat 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].
 waveletstring, 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_levelsint or None, optional
The number of wavelet decomposition levels to use. The default is three less than the maximum number of possible decomposition levels.
 multichannelbool, optional
Apply wavelet denoising separately for each channel (where channels correspond to the final axis of the array).
 convert2ycbcrbool, 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”.
 rescale_sigmabool, optional
If False, no rescaling of the userprovided
sigma
will be performed. The default ofTrue
rescales sigma appropriately if the image is rescaled internally.New in version 0.16:
rescale_sigma
was introduced in 0.16
 Returns
 outndarray
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 nearzero 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.
Changed in version 0.16: For floating point inputs, the original input range is maintained and there is no clipping applied to the output. Other input types will be converted to a floating point value in the range [1, 1] or [0, 1] depending on the input image range. Unless
rescale_sigma = False
, any internal rescaling applied to theimage
will also be applied tosigma
to maintain the same relative amplitude.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 subband 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 givensigma
with high probability, but tends to produce images that appear overly smooth.Although any of the wavelets from
PyWavelets
can be selected, the thresholding methods assume an orthogonal wavelet transform and may not choose the threshold appropriately for biorthogonal wavelets. Orthogonal wavelets are desirable because white noise in the input remains white noise in the subbands. Biorthogonal wavelets lead to colored noise in the subbands. Additionally, the orthogonal wavelets in PyWavelets are orthonormal so that noise variance in the subbands remains identical to the noise variance of the input. Example orthogonal wavelets are the Daubechies (e.g. ‘db2’) or symmlet (e.g. ‘sym2’) families.References
 1(1,2)
Chang, S. Grace, Bin Yu, and Martin Vetterli. “Adaptive wavelet thresholding for image denoising and compression.” Image Processing, IEEE Transactions on 9.9 (2000): 15321546. DOI:10.1109/83.862633
 2(1,2,3)
D. L. Donoho and I. M. Johnstone. “Ideal spatial adaptation by wavelet shrinkage.” Biometrika 81.3 (1994): 425455. 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, rescale_sigma=True)
ellipsoid_kernel¶

skimage.restoration.
ellipsoid_kernel
(shape, intensity)[source]¶ Create an ellipoid kernel for restoration.rolling_ball.
 Parameters
 shapearraylike
Length of the principal axis of the ellipsoid (excluding the intensity axis). The kernel needs to have the same dimensionality as the image it will be applied to.
 intensityint
Length of the intensity axis of the ellipsoid.
 Returns
 kernelndarray
The kernel containing the surface intensity of the top half of the ellipsoid.
See also
Examples using skimage.restoration.ellipsoid_kernel
¶
estimate_sigma¶

skimage.restoration.
estimate_sigma
(image, average_sigmas=False, multichannel=False)[source]¶ Robust waveletbased estimator of the (Gaussian) noise standard deviation.
 Parameters
 imagendarray
Image for which to estimate the noise standard deviation.
 average_sigmasbool, optional
If true, average the channel estimates of sigma. Otherwise return a list of sigmas corresponding to each channel.
 multichannelbool
Estimate sigma separately for each channel.
 Returns
 sigmafloat 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): 425455. 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)
inpaint_biharmonic¶

skimage.restoration.
inpaint_biharmonic
(image, mask, multichannel=False)[source]¶ Inpaint masked points in image with biharmonic equations.
 Parameters
 image(M[, N[, …, P]][, C]) ndarray
Input image.
 mask(M[, N[, …, P]]) ndarray
Array of pixels to be inpainted. Have to be the same shape as one of the ‘image’ channels. Unknown pixels have to be represented with 1, known pixels  with 0.
 multichannelboolean, optional
If True, the last image dimension is considered as a color channel, otherwise as spatial.
 Returns
 out(M[, N[, …, P]][, C]) ndarray
Input image with masked pixels inpainted.
References
 1
N.S.Hoang, S.B.Damelin, “On surface completion and image inpainting by biharmonic functions: numerical aspects”, arXiv:1707.06567
 2
C. K. Chui and H. N. Mhaskar, MRA ContextualRecovery Extension of Smooth Functions on Manifolds, Appl. and Comp. Harmonic Anal., 28 (2010), 104113, DOI:10.1016/j.acha.2009.04.004
Examples
>>> img = np.tile(np.square(np.linspace(0, 1, 5)), (5, 1)) >>> mask = np.zeros_like(img) >>> mask[2, 2:] = 1 >>> mask[1, 3:] = 1 >>> mask[0, 4:] = 1 >>> out = inpaint_biharmonic(img, mask)
richardson_lucy¶

skimage.restoration.
richardson_lucy
(image, psf, iterations=50, clip=True, filter_epsilon=None)[source]¶ RichardsonLucy deconvolution.
 Parameters
 imagendarray
Input degraded image (can be N dimensional).
 psfndarray
The point spread function.
 iterationsint, optional
Number of iterations. This parameter plays the role of regularisation.
 clipboolean, 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_deconvndarray
The deconvolved image.
References
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)
rolling_ball¶

skimage.restoration.
rolling_ball
(image, *, radius=100, kernel=None, nansafe=False, num_threads=None)[source]¶ Estimate background intensity by rolling/translating a kernel.
This rolling ball algorithm estimates background intensity for a ndimage in case of uneven exposure. It is a generalization of the frequently used rolling ball algorithm [1].
 Parameters
 imagendarray
The image to be filtered.
 radiusint, optional
Radius of a ball shaped kernel to be rolled/translated in the image. Used if
kernel = None
. kernelndarray, optional
The kernel to be rolled/translated in the image. It must have the same number of dimensions as
image
. Kernel is filled with the intensity of the kernel at that position. nansafe: bool, optional
If
False
(default) assumes that none of the values inimage
arenp.nan
, and uses a faster implementation. num_threads: int, optional
The maximum number of threads to use. If
None
use the OpenMP default value; typically equal to the maximum number of virtual cores. Note: This is an upper limit to the number of threads. The exact number is determined by the system’s OpenMP library.
 Returns
 backgroundndarray
The estimated background of the image.
Notes
For the pixel that has its background intensity estimated (without loss of generality at
center
) the rolling ball method centerskernel
under it and raises the kernel until the surface touches the image umbra at somepos=(y,x)
. The background intensity is then estimated using the image intensity at that position (image[pos]
) plus the difference ofkernel[center]  kernel[pos]
.This algorithm assumes that dark pixels correspond to the background. If you have a bright background, invert the image before passing it to the function, e.g., using utils.invert. See the gallery example for details.
This algorithm is sensitive to noise (in particular saltandpepper noise). If this is a problem in your image, you can apply mild gaussian smoothing before passing the image to this function.
References
 1
Sternberg, Stanley R. “Biomedical image processing.” Computer 1 (1983): 2234. DOI:10.1109/MC.1983.1654163
Examples
>>> import numpy as np >>> from skimage import data >>> from skimage.restoration import rolling_ball >>> image = data.coins() >>> background = rolling_ball(data.coins()) >>> filtered_image = image  background
>>> import numpy as np >>> from skimage import data >>> from skimage.restoration import rolling_ball, ellipsoid_kernel >>> image = data.coins() >>> kernel = ellipsoid_kernel((101, 101), 75) >>> background = rolling_ball(data.coins(), kernel=kernel) >>> filtered_image = image  background
Examples using skimage.restoration.rolling_ball
¶
unsupervised_wiener¶

skimage.restoration.
unsupervised_wiener
(image, psf, reg=None, user_params=None, is_real=True, clip=True)[source]¶ Unsupervised WienerHunt deconvolution.
Return the deconvolution with a WienerHunt 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.
 psfndarray
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)
). regndarray, optional
The regularisation operator. The Laplacian by default. It can be an impulse response or a transfer function, as for the psf.
 user_paramsdict, optional
Dictionary of parameters for the Gibbs sampler. See below.
 clipboolean, 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).
 chainsdict
The keys
noise
andprior
contain the chain list of noise and prior precision respectively.
 Other Parameters
 The keys of ``user_params`` are:
 thresholdfloat
The stopping criterion: the norm of the difference between to successive approximated solution (empirical mean of object samples, see Notes section). 1e4 by default.
 burninint
The number of sample to ignore to start computation of the mean. 15 by default.
 min_iterint
The minimum number of iterations. 30 by default.
 max_iterint
The maximum number of iterations if
threshold
is not satisfied. 200 by default. callbackcallable (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 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, JeanFrançois Giovannelli, and Thomas Rodet, “Bayesian estimation of regularization and point spread function parameters for WienerHunt deconvolution”, J. Opt. Soc. Am. A 27, 15931607 (2010)
https://www.osapublishing.org/josaa/abstract.cfm?URI=josaa2771593
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)
unwrap_phase¶

skimage.restoration.
unwrap_phase
(image, wrap_around=False, seed=None)[source]¶ Recover the original from a wrapped phase image.
From an image wrapped to lie in the interval [pi, pi), recover the original, unwrapped image.
 Parameters
 image1D, 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_aroundbool or sequence of bool, optional
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.
 seedint, optional
Unwrapping 2D or 3D images uses random initialization. This sets the seed of the PRNG to achieve deterministic behavior.
 Returns
 image_unwrappedarray_like, double
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
 1
Miguel Arevallilo Herraez, David R. Burton, Michael J. Lalor, and Munther A. Gdeisat, “Fast twodimensional phaseunwrapping algorithm based on sorting by reliability following a noncontinuous path”, Journal Applied Optics, Vol. 41, No. 35 (2002) 7437,
 2
AbdulRahman, H., Gdeisat, M., Burton, D., & Lalor, M., “Fast threedimensional phaseunwrapping algorithm based on sorting by reliability following a noncontinuous 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) < 1e6 # A constant offset is normal True
Examples using skimage.restoration.unwrap_phase
¶
wiener¶

skimage.restoration.
wiener
(image, psf, balance, reg=None, is_real=True, clip=True)[source]¶ WienerHunt deconvolution
Return the deconvolution with a WienerHunt approach (i.e. with Fourier diagonalisation).
 Parameters
 image(M, N) ndarray
Input degraded image
 psfndarray
Point Spread Function. This is assumed to be the impulse response (input image space) if the datatype is real, or the transfer function (Fourier space) if the datatype 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).
 balancefloat
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).
 regndarray, 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_realboolean, optional
True by default. Specify if
psf
andreg
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 ifpsf
and/orreg
are provided as transfer function. For the hermitian property seeuft
module ornp.fft.rfftn
. clipboolean, 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.
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
\[\hat x = F^\dagger (\Lambda_H^2 + \lambda \Lambda_D^2) \Lambda_H^\dagger F y\]where \(F\) and \(F^\dagger\) are the Fourier and inverse Fourier transforms respectively, \(\Lambda_H\) the transfer function (or the Fourier transform 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 highfrequency 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 \(H\), see [Hunt].
References
 1
François Orieux, JeanFrançois Giovannelli, and Thomas Rodet, “Bayesian estimation of regularization and point spread function parameters for WienerHunt deconvolution”, J. Opt. Soc. Am. A 27, 15931607 (2010)
https://www.osapublishing.org/josaa/abstract.cfm?URI=josaa2771593
 2
B. R. Hunt “A matrix theory proof of the discrete convolution theorem”, IEEE Trans. on Audio and Electroacoustics, vol. au19, no. 4, pp. 285288, dec. 1971
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)