# Module: metrics¶

 Compute Adapted Rand error as defined by the SNEMI3D contest. skimage.metrics.contingency_table(im_true, …) Return the contingency table for all regions in matched segmentations. skimage.metrics.hausdorff_distance(image0, …) Calculate the Hausdorff distance between nonzero elements of given images. skimage.metrics.mean_squared_error(image0, …) Compute the mean-squared error between two images. Compute the normalized root mean-squared error (NRMSE) between two images. Compute the peak signal to noise ratio (PSNR) for an image. Compute the mean structural similarity index between two images. Return symmetric conditional entropies associated with the VI.

skimage.metrics.adapted_rand_error(image_true=None, image_test=None, *, table=None, ignore_labels=(0, ))[source]

Compute Adapted Rand error as defined by the SNEMI3D contest. [1]

Parameters
image_truendarray of int

Ground-truth label image, same shape as im_test.

image_testndarray of int

Test image.

tablescipy.sparse array in crs format, optional

A contingency table built with skimage.evaluate.contingency_table. If None, it will be computed on the fly.

ignore_labelssequence of int, optional

Labels to ignore. Any part of the true image labeled with any of these values will not be counted in the score.

Returns
arefloat

The adapted Rand error; equal to $$1 - \frac{2pr}{p + r}$$, where p and r are the precision and recall described below.

precfloat

The adapted Rand precision: this is the number of pairs of pixels that have the same label in the test label image and in the true image, divided by the number in the test image.

recfloat

The adapted Rand recall: this is the number of pairs of pixels that have the same label in the test label image and in the true image, divided by the number in the true image.

Notes

Pixels with label 0 in the true segmentation are ignored in the score.

References

1

Arganda-Carreras I, Turaga SC, Berger DR, et al. (2015) Crowdsourcing the creation of image segmentation algorithms for connectomics. Front. Neuroanat. 9:142. DOI:10.3389/fnana.2015.00142

## contingency_table¶

skimage.metrics.contingency_table(im_true, im_test, *, ignore_labels=None, normalize=False)[source]

Return the contingency table for all regions in matched segmentations.

Parameters
im_truendarray of int

Ground-truth label image, same shape as im_test.

im_testndarray of int

Test image.

ignore_labelssequence of int, optional

Labels to ignore. Any part of the true image labeled with any of these values will not be counted in the score.

normalizebool

Determines if the contingency table is normalized by pixel count.

Returns
contscipy.sparse.csr_matrix

A contingency table. cont[i, j] will equal the number of voxels labeled i in im_true and j in im_test.

## hausdorff_distance¶

skimage.metrics.hausdorff_distance(image0, image1)[source]

Calculate the Hausdorff distance between nonzero elements of given images.

The Hausdorff distance [1] is the maximum distance between any point on image0 and its nearest point on image1, and vice-versa.

Parameters
image0, image1ndarray

Arrays where True represents a point that is included in a set of points. Both arrays must have the same shape.

Returns
distancefloat

The Hausdorff distance between coordinates of nonzero pixels in image0 and image1, using the Euclidian distance.

References

1

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

Examples

>>> points_a = (3, 0)
>>> points_b = (6, 0)
>>> shape = (7, 1)
>>> image_a = np.zeros(shape, dtype=bool)
>>> image_b = np.zeros(shape, dtype=bool)
>>> image_a[points_a] = True
>>> image_b[points_b] = True
>>> hausdorff_distance(image_a, image_b)
3.0


## mean_squared_error¶

skimage.metrics.mean_squared_error(image0, image1)[source]

Compute the mean-squared error between two images.

Parameters
image0, image1ndarray

Images. Any dimensionality, must have same shape.

Returns
msefloat

The mean-squared error (MSE) metric.

Notes

Changed in version 0.16: This function was renamed from skimage.measure.compare_mse to skimage.metrics.mean_squared_error.

## normalized_root_mse¶

skimage.metrics.normalized_root_mse(image_true, image_test, *, normalization='euclidean')[source]

Compute the normalized root mean-squared error (NRMSE) between two images.

Parameters
image_truendarray

Ground-truth image, same shape as im_test.

image_testndarray

Test image.

normalization{‘euclidean’, ‘min-max’, ‘mean’}, optional

Controls the normalization method to use in the denominator of the NRMSE. There is no standard method of normalization across the literature [1]. The methods available here are as follows:

• ‘euclidean’ : normalize by the averaged Euclidean norm of im_true:

NRMSE = RMSE * sqrt(N) / || im_true ||


where || . || denotes the Frobenius norm and N = im_true.size. This result is equivalent to:

NRMSE = || im_true - im_test || / || im_true ||.

• ‘min-max’ : normalize by the intensity range of im_true.

• ‘mean’ : normalize by the mean of im_true

Returns
nrmsefloat

The NRMSE metric.

Notes

Changed in version 0.16: This function was renamed from skimage.measure.compare_nrmse to skimage.metrics.normalized_root_mse.

References

1

https://en.wikipedia.org/wiki/Root-mean-square_deviation

## peak_signal_noise_ratio¶

skimage.metrics.peak_signal_noise_ratio(image_true, image_test, *, data_range=None)[source]

Compute the peak signal to noise ratio (PSNR) for an image.

Parameters
image_truendarray

Ground-truth image, same shape as im_test.

image_testndarray

Test image.

data_rangeint, optional

The data range of the input image (distance between minimum and maximum possible values). By default, this is estimated from the image data-type.

Returns
psnrfloat

The PSNR metric.

Notes

Changed in version 0.16: This function was renamed from skimage.measure.compare_psnr to skimage.metrics.peak_signal_noise_ratio.

References

1

https://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio

## structural_similarity¶

skimage.metrics.structural_similarity(im1, im2, *, win_size=None, gradient=False, data_range=None, multichannel=False, gaussian_weights=False, full=False, **kwargs)[source]

Compute the mean structural similarity index between two images.

Parameters
im1, im2ndarray

Images. Any dimensionality with same shape.

win_sizeint or None, optional

The side-length of the sliding window used in comparison. Must be an odd value. If gaussian_weights is True, this is ignored and the window size will depend on sigma.

If True, also return the gradient with respect to im2.

data_rangefloat, optional

The data range of the input image (distance between minimum and maximum possible values). By default, this is estimated from the image data-type.

multichannelbool, optional

If True, treat the last dimension of the array as channels. Similarity calculations are done independently for each channel then averaged.

gaussian_weightsbool, optional

If True, each patch has its mean and variance spatially weighted by a normalized Gaussian kernel of width sigma=1.5.

fullbool, optional

If True, also return the full structural similarity image.

Returns
mssimfloat

The mean structural similarity index over the image.

The gradient of the structural similarity between im1 and im2 [2]. This is only returned if gradient is set to True.

Sndarray

The full SSIM image. This is only returned if full is set to True.

Other Parameters
use_sample_covariancebool

If True, normalize covariances by N-1 rather than, N where N is the number of pixels within the sliding window.

K1float

Algorithm parameter, K1 (small constant, see [1]).

K2float

Algorithm parameter, K2 (small constant, see [1]).

sigmafloat

Standard deviation for the Gaussian when gaussian_weights is True.

Notes

To match the implementation of Wang et. al. [1], set gaussian_weights to True, sigma to 1.5, and use_sample_covariance to False.

Changed in version 0.16: This function was renamed from skimage.measure.compare_ssim to skimage.metrics.structural_similarity.

References

1(1,2,3)

Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13, 600-612. https://ece.uwaterloo.ca/~z70wang/publications/ssim.pdf, DOI:10.1109/TIP.2003.819861

2

Avanaki, A. N. (2009). Exact global histogram specification optimized for structural similarity. Optical Review, 16, 613-621. arXiv:0901.0065 DOI:10.1007/s10043-009-0119-z

## variation_of_information¶

skimage.metrics.variation_of_information(image0=None, image1=None, *, table=None, ignore_labels=())[source]

Return symmetric conditional entropies associated with the VI. [1]

The variation of information is defined as VI(X,Y) = H(X|Y) + H(Y|X). If X is the ground-truth segmentation, then H(X|Y) can be interpreted as the amount of under-segmentation and H(X|Y) as the amount of over-segmentation. In other words, a perfect over-segmentation will have H(X|Y)=0 and a perfect under-segmentation will have H(Y|X)=0.

Parameters
image0, image1ndarray of int

Label images / segmentations, must have same shape.

tablescipy.sparse array in csr format, optional

A contingency table built with skimage.evaluate.contingency_table. If None, it will be computed with skimage.evaluate.contingency_table. If given, the entropies will be computed from this table and any images will be ignored.

ignore_labelssequence of int, optional

Labels to ignore. Any part of the true image labeled with any of these values will not be counted in the score.

Returns
vindarray of float, shape (2,)

The conditional entropies of image1|image0 and image0|image1.

References

1

Marina Meilă (2007), Comparing clusterings—an information based distance, Journal of Multivariate Analysis, Volume 98, Issue 5, Pages 873-895, ISSN 0047-259X, DOI:10.1016/j.jmva.2006.11.013.