measure
¶skimage.measure.find_contours (array, level) 
Find isovalued contours in a 2D array for a given level value. 
skimage.measure.regionprops (label_image[, …]) 
Measure properties of labeled image regions. 
skimage.measure.perimeter (image[, neighbourhood]) 
Calculate total perimeter of all objects in binary image. 
skimage.measure.approximate_polygon (coords, …) 
Approximate a polygonal chain with the specified tolerance. 
skimage.measure.subdivide_polygon (coords[, …]) 
Subdivision of polygonal curves using BSplines. 
skimage.measure.ransac (data, model_class, …) 
Fit a model to data with the RANSAC (random sample consensus) algorithm. 
skimage.measure.block_reduce (image, block_size) 
Downsample image by applying function to local blocks. 
skimage.measure.moments (image[, order]) 
Calculate all raw image moments up to a certain order. 
skimage.measure.moments_central (image[, …]) 
Calculate all central image moments up to a certain order. 
skimage.measure.moments_coords (coords[, order]) 
Calculate all raw image moments up to a certain order. 
skimage.measure.moments_coords_central (coords) 
Calculate all central image moments up to a certain order. 
skimage.measure.moments_normalized (mu[, order]) 
Calculate all normalized central image moments up to a certain order. 
skimage.measure.moments_hu (nu) 
Calculate Hu’s set of image moments (2Donly). 
skimage.measure.marching_cubes_lewiner (volume) 
Lewiner marching cubes algorithm to find surfaces in 3d volumetric data. 
skimage.measure.marching_cubes_classic (volume) 
Classic marching cubes algorithm to find surfaces in 3d volumetric data. 
skimage.measure.mesh_surface_area (verts, faces) 
Compute surface area, given vertices & triangular faces 
skimage.measure.correct_mesh_orientation (…) 
Correct orientations of mesh faces. 
skimage.measure.profile_line (image, src, dst) 
Return the intensity profile of an image measured along a scan line. 
skimage.measure.label (input[, neighbors, …]) 
Label connected regions of an integer array. 
skimage.measure.points_in_poly (points, verts) 
Test whether points lie inside a polygon. 
skimage.measure.grid_points_in_poly (shape, verts) 
Test whether points on a specified grid are inside a polygon. 
skimage.measure.compare_ssim (X, Y[, …]) 
Compute the mean structural similarity index between two images. 
skimage.measure.compare_mse (im1, im2) 
Compute the meansquared error between two images. 
skimage.measure.compare_nrmse (im_true, im_test) 
Compute the normalized root meansquared error (NRMSE) between two images. 
skimage.measure.compare_psnr (im_true, im_test) 
Compute the peak signal to noise ratio (PSNR) for an image. 
skimage.measure.shannon_entropy (image[, base]) 
Calculate the Shannon entropy of an image. 
skimage.measure.LineModelND () 
Total least squares estimator for Ndimensional lines. 
skimage.measure.CircleModel () 
Total least squares estimator for 2D circles. 
skimage.measure.EllipseModel () 
Total least squares estimator for 2D ellipses. 
skimage.measure.
find_contours
(array, level, fully_connected='low', positive_orientation='low')[source]¶Find isovalued contours in a 2D array for a given level value.
Uses the “marching squares” method to compute a the isovalued contours of the input 2D array for a particular level value. Array values are linearly interpolated to provide better precision for the output contours.
Parameters: 


Returns: 

Notes
The marching squares algorithm is a special case of the marching cubes algorithm [1]. A simple explanation is available here:
http://www.essi.fr/~lingrand/MarchingCubes/algo.html
There is a single ambiguous case in the marching squares algorithm: when
a given 2 x 2
element square has two highvalued and two lowvalued
elements, each pair diagonally adjacent. (Where high and lowvalued is
with respect to the contour value sought.) In this case, either the
highvalued elements can be ‘connected together’ via a thin isthmus that
separates the lowvalued elements, or viceversa. When elements are
connected together across a diagonal, they are considered ‘fully
connected’ (also known as ‘face+vertexconnected’ or ‘8connected’). Only
highvalued or lowvalued elements can be fullyconnected, the other set
will be considered as ‘faceconnected’ or ‘4connected’. By default,
lowvalued elements are considered fullyconnected; this can be altered
with the ‘fully_connected’ parameter.
Output contours are not guaranteed to be closed: contours which intersect the array edge will be left open. All other contours will be closed. (The closedness of a contours can be tested by checking whether the beginning point is the same as the end point.)
Contours are oriented. By default, array values lower than the contour value are to the left of the contour and values greater than the contour value are to the right. This means that contours will wind counterclockwise (i.e. in ‘positive orientation’) around islands of lowvalued pixels. This behavior can be altered with the ‘positive_orientation’ parameter.
The order of the contours in the output list is determined by the position
of the smallest x,y
(in lexicographical order) coordinate in the
contour. This is a sideeffect of how the input array is traversed, but
can be relied upon.
Warning
Array coordinates/values are assumed to refer to the center of the
array element. Take a simple example input: [0, 1]
. The interpolated
position of 0.5 in this array is midway between the 0element (at
x=0
) and the 1element (at x=1
), and thus would fall at
x=0.5
.
This means that to find reasonable contours, it is best to find contours midway between the expected “light” and “dark” values. In particular, given a binarized array, do not choose to find contours at the low or high value of the array. This will often yield degenerate contours, especially around structures that are a single array element wide. Instead choose a middle value, as above.
References
[1]  (1, 2) Lorensen, William and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163170). 
Examples
>>> a = np.zeros((3, 3))
>>> a[0, 0] = 1
>>> a
array([[ 1., 0., 0.],
[ 0., 0., 0.],
[ 0., 0., 0.]])
>>> find_contours(a, 0.5)
[array([[ 0. , 0.5],
[ 0.5, 0. ]])]
skimage.measure.
regionprops
(label_image, intensity_image=None, cache=True, coordinates=None)[source]¶Measure properties of labeled image regions.
Parameters: 


Returns: 

See also
Notes
The following properties can be accessed as attributes or keys:
(min_row, min_col, max_row, max_col)
.
Pixels belonging to the bounding box are in the halfopen interval
[min_row; max_row)
and [min_col; max_col)
.(row, col)
.(row, col)
of the region.area / (rows * cols)
(row, col)
, relative to region bounding
box.Spatial moments up to 3rd order:
m_ji = sum{ array(x, y) * x^j * y^i }
where the sum is over the x, y coordinates of the region.
Central moments (translation invariant) up to 3rd order:
mu_ji = sum{ array(x, y) * (x  x_c)^j * (y  y_c)^i }
where the sum is over the x, y coordinates of the region, and x_c and y_c are the coordinates of the region’s centroid.
Normalized moments (translation and scale invariant) up to 3rd order:
nu_ji = mu_ji / m_00^[(i+j)/2 + 1]
where m_00 is the zeroth spatial moment.
In ‘rc’ coordinates, angle between the 0th axis (rows) and the major axis of the ellipse that has the same second moments as the region, ranging from pi/2 to pi/2 counterclockwise.
In xy coordinates, as above but the angle is now measured from the “x” or horizontal axis.
(row, col)
weighted with intensity
image.(row, col)
, relative to region bounding
box, weighted with intensity image.Spatial moments of intensity image up to 3rd order:
wm_ji = sum{ array(x, y) * x^j * y^i }
where the sum is over the x, y coordinates of the region.
Central moments (translation invariant) of intensity image up to 3rd order:
wmu_ji = sum{ array(x, y) * (x  x_c)^j * (y  y_c)^i }
where the sum is over the x, y coordinates of the region, and x_c and y_c are the coordinates of the region’s weighted centroid.
Normalized moments (translation and scale invariant) of intensity image up to 3rd order:
wnu_ji = wmu_ji / wm_00^[(i+j)/2 + 1]
where wm_00
is the zeroth spatial moment (intensityweighted area).
Each region also supports iteration, so that you can do:
for prop in region:
print(prop, region[prop])
References
[1]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[2]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[3]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[4]  https://en.wikipedia.org/wiki/Image_moment 
Examples
>>> from skimage import data, util
>>> from skimage.measure import label
>>> img = util.img_as_ubyte(data.coins()) > 110
>>> label_img = label(img, connectivity=img.ndim)
>>> props = regionprops(label_img)
>>> # centroid of first labeled object
>>> props[0].centroid
(22.729879860483141, 81.912285234465827)
>>> # centroid of first labeled object
>>> props[0]['centroid']
(22.729879860483141, 81.912285234465827)
skimage.measure.
perimeter
(image, neighbourhood=4)[source]¶Calculate total perimeter of all objects in binary image.
Parameters: 


Returns: 

References
[1]  K. Benkrid, D. Crookes. Design and FPGA Implementation of a Perimeter Estimator. The Queen’s University of Belfast. http://www.cs.qub.ac.uk/~d.crookes/webpubs/papers/perimeter.doc 
Examples
>>> from skimage import data, util
>>> from skimage.measure import label
>>> # coins image (binary)
>>> img_coins = data.coins() > 110
>>> # total perimeter of all objects in the image
>>> perimeter(img_coins, neighbourhood=4) # doctest: +ELLIPSIS
7796.867...
>>> perimeter(img_coins, neighbourhood=8) # doctest: +ELLIPSIS
8806.268...
skimage.measure.
approximate_polygon
(coords, tolerance)[source]¶Approximate a polygonal chain with the specified tolerance.
It is based on the DouglasPeucker algorithm.
Note that the approximated polygon is always within the convex hull of the original polygon.
Parameters: 


Returns: 

References
[1]  https://en.wikipedia.org/wiki/RamerDouglasPeucker_algorithm 
skimage.measure.approximate_polygon
¶skimage.measure.
subdivide_polygon
(coords, degree=2, preserve_ends=False)[source]¶Subdivision of polygonal curves using BSplines.
Note that the resulting curve is always within the convex hull of the original polygon. Circular polygons stay closed after subdivision.
Parameters: 


Returns: 

References
[1]  http://mrl.nyu.edu/publications/subdivcourse2000/coursenotes00.pdf 
skimage.measure.subdivide_polygon
¶skimage.measure.
ransac
(data, model_class, min_samples, residual_threshold, is_data_valid=None, is_model_valid=None, max_trials=100, stop_sample_num=inf, stop_residuals_sum=0, stop_probability=1, random_state=None)[source]¶Fit a model to data with the RANSAC (random sample consensus) algorithm.
RANSAC is an iterative algorithm for the robust estimation of parameters from a subset of inliers from the complete data set. Each iteration performs the following tasks:
These steps are performed either a maximum number of times or until one of the special stop criteria are met. The final model is estimated using all inlier samples of the previously determined best model.
Parameters: 


Returns: 

References
[1]  “RANSAC”, Wikipedia, https://en.wikipedia.org/wiki/RANSAC 
Examples
Generate ellipse data without tilt and add noise:
>>> t = np.linspace(0, 2 * np.pi, 50)
>>> xc, yc = 20, 30
>>> a, b = 5, 10
>>> x = xc + a * np.cos(t)
>>> y = yc + b * np.sin(t)
>>> data = np.column_stack([x, y])
>>> np.random.seed(seed=1234)
>>> data += np.random.normal(size=data.shape)
Add some faulty data:
>>> data[0] = (100, 100)
>>> data[1] = (110, 120)
>>> data[2] = (120, 130)
>>> data[3] = (140, 130)
Estimate ellipse model using all available data:
>>> model = EllipseModel()
>>> model.estimate(data)
True
>>> np.round(model.params) # doctest: +SKIP
array([ 72., 75., 77., 14., 1.])
Estimate ellipse model using RANSAC:
>>> ransac_model, inliers = ransac(data, EllipseModel, 20, 3, max_trials=50)
>>> abs(np.round(ransac_model.params))
array([ 20., 30., 5., 10., 0.])
>>> inliers # doctest: +SKIP
array([False, False, False, False, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True], dtype=bool)
>>> sum(inliers) > 40
True
Robustly estimate geometric transformation:
>>> from skimage.transform import SimilarityTransform
>>> np.random.seed(0)
>>> src = 100 * np.random.rand(50, 2)
>>> model0 = SimilarityTransform(scale=0.5, rotation=1,
... translation=(10, 20))
>>> dst = model0(src)
>>> dst[0] = (10000, 10000)
>>> dst[1] = (100, 100)
>>> dst[2] = (50, 50)
>>> model, inliers = ransac((src, dst), SimilarityTransform, 2, 10)
>>> inliers
array([False, False, False, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True], dtype=bool)
skimage.measure.
block_reduce
(image, block_size, func=<function sum>, cval=0)[source]¶Downsample image by applying function to local blocks.
Parameters: 


Returns: 

Examples
>>> from skimage.measure import block_reduce
>>> image = np.arange(3*3*4).reshape(3, 3, 4)
>>> image # doctest: +NORMALIZE_WHITESPACE
array([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]],
[[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23]],
[[24, 25, 26, 27],
[28, 29, 30, 31],
[32, 33, 34, 35]]])
>>> block_reduce(image, block_size=(3, 3, 1), func=np.mean)
array([[[ 16., 17., 18., 19.]]])
>>> image_max1 = block_reduce(image, block_size=(1, 3, 4), func=np.max)
>>> image_max1 # doctest: +NORMALIZE_WHITESPACE
array([[[11]],
[[23]],
[[35]]])
>>> image_max2 = block_reduce(image, block_size=(3, 1, 4), func=np.max)
>>> image_max2 # doctest: +NORMALIZE_WHITESPACE
array([[[27],
[31],
[35]]])
skimage.measure.
moments
(image, order=3)[source]¶Calculate all raw image moments up to a certain order.
M[0, 0]
.M[1, 0] / M[0, 0]
, M[0, 1] / M[0, 0]
}.Note that raw moments are neither translation, scale nor rotation invariant.
Parameters: 


Returns: 

References
[1]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[2]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[3]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[4]  https://en.wikipedia.org/wiki/Image_moment 
Examples
>>> image = np.zeros((20, 20), dtype=np.double)
>>> image[13:17, 13:17] = 1
>>> M = moments(image)
>>> cr = M[1, 0] / M[0, 0]
>>> cc = M[0, 1] / M[0, 0]
>>> cr, cc
(14.5, 14.5)
skimage.measure.
moments_central
(image, center=None, cc=None, order=3, **kwargs)[source]¶Calculate all central image moments up to a certain order.
The center coordinates (cr, cc) can be calculated from the raw moments as:
{M[1, 0] / M[0, 0]
, M[0, 1] / M[0, 0]
}.
Note that central moments are translation invariant but not scale and rotation invariant.
Parameters: 


Returns: 

Other Parameters:  

References
[1]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[2]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[3]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[4]  https://en.wikipedia.org/wiki/Image_moment 
Examples
>>> image = np.zeros((20, 20), dtype=np.double)
>>> image[13:17, 13:17] = 1
>>> M = moments(image)
>>> cr = M[1, 0] / M[0, 0]
>>> cc = M[0, 1] / M[0, 0]
>>> moments_central(image, (cr, cc))
array([[ 16., 0., 20., 0.],
[ 0., 0., 0., 0.],
[ 20., 0., 25., 0.],
[ 0., 0., 0., 0.]])
skimage.measure.
moments_coords
(coords, order=3)[source]¶Calculate all raw image moments up to a certain order.
M[0, 0]
.M[1, 0] / M[0, 0]
, M[0, 1] / M[0, 0]
}.Note that raw moments are neither translation, scale nor rotation invariant.
Parameters: 


Returns: 

References
[1]  Johannes Kilian. Simple Image Analysis By Moments. Durham University, version 0.2, Durham, 2001. 
Examples
>>> coords = np.array([[row, col]
... for row in range(13, 17)
... for col in range(14, 18)], dtype=np.double)
>>> M = moments_coords(coords)
>>> centroid_row = M[1, 0] / M[0, 0]
>>> centroid_col = M[0, 1] / M[0, 0]
>>> centroid_row, centroid_col
(14.5, 15.5)
skimage.measure.
moments_coords_central
(coords, center=None, order=3)[source]¶Calculate all central image moments up to a certain order.
M[0, 0]
.M[1, 0] / M[0, 0]
, M[0, 1] / M[0, 0]
}.Note that raw moments are neither translation, scale nor rotation invariant.
Parameters: 


Returns: 

References
[1]  Johannes Kilian. Simple Image Analysis By Moments. Durham University, version 0.2, Durham, 2001. 
Examples
>>> coords = np.array([[row, col]
... for row in range(13, 17)
... for col in range(14, 18)])
>>> moments_coords_central(coords)
array([[ 16., 0., 20., 0.],
[ 0., 0., 0., 0.],
[ 20., 0., 25., 0.],
[ 0., 0., 0., 0.]])
As seen above, for symmetric objects, oddorder moments (columns 1 and 3, rows 1 and 3) are zero when centered on the centroid, or center of mass, of the object (the default). If we break the symmetry by adding a new point, this no longer holds:
>>> coords2 = np.concatenate((coords, [[17, 17]]), axis=0)
>>> np.round(moments_coords_central(coords2), 2)
array([[ 17. , 0. , 22.12, 2.49],
[ 0. , 3.53, 1.73, 7.4 ],
[ 25.88, 6.02, 36.63, 8.83],
[ 4.15, 19.17, 14.8 , 39.6 ]])
Image moments and central image moments are equivalent (by definition) when the center is (0, 0):
>>> np.allclose(moments_coords(coords),
... moments_coords_central(coords, (0, 0)))
True
skimage.measure.
moments_normalized
(mu, order=3)[source]¶Calculate all normalized central image moments up to a certain order.
Note that normalized central moments are translation and scale invariant but not rotation invariant.
Parameters: 


Returns: 

References
[1]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[2]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[3]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[4]  https://en.wikipedia.org/wiki/Image_moment 
Examples
>>> image = np.zeros((20, 20), dtype=np.double)
>>> image[13:17, 13:17] = 1
>>> m = moments(image)
>>> cr = m[0, 1] / m[0, 0]
>>> cc = m[1, 0] / m[0, 0]
>>> mu = moments_central(image, cr, cc)
>>> moments_normalized(mu)
array([[ nan, nan, 0.078125 , 0. ],
[ nan, 0. , 0. , 0. ],
[ 0.078125 , 0. , 0.00610352, 0. ],
[ 0. , 0. , 0. , 0. ]])
skimage.measure.
moments_hu
(nu)[source]¶Calculate Hu’s set of image moments (2Donly).
Note that this set of moments is proofed to be translation, scale and rotation invariant.
Parameters: 


Returns: 

References
[1]  M. K. Hu, “Visual Pattern Recognition by Moment Invariants”, IRE Trans. Info. Theory, vol. IT8, pp. 179187, 1962 
[2]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[3]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[4]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[5]  https://en.wikipedia.org/wiki/Image_moment 
skimage.measure.
marching_cubes_lewiner
(volume, level=None, spacing=(1.0, 1.0, 1.0), gradient_direction='descent', step_size=1, allow_degenerate=True, use_classic=False)[source]¶Lewiner marching cubes algorithm to find surfaces in 3d volumetric data.
In contrast to marching_cubes_classic()
, this algorithm is faster,
resolves ambiguities, and guarantees topologically correct results.
Therefore, this algorithm generally a better choice, unless there
is a specific need for the classic algorithm.
Parameters: 


Returns: 

Notes
The algorithm [1] is an improved version of Chernyaev’s Marching Cubes 33 algorithm. It is an efficient algorithm that relies on heavy use of lookup tables to handle the many different cases, keeping the algorithm relatively easy. This implementation is written in Cython, ported from Lewiner’s C++ implementation.
To quantify the area of an isosurface generated by this algorithm, pass verts and faces to skimage.measure.mesh_surface_area.
Regarding visualization of algorithm output, to contour a volume
named myvolume about the level 0.0, using the mayavi
package:
>>> from mayavi import mlab
>>> verts, faces, normals, values = marching_cubes_lewiner(myvolume, 0.0)
>>> mlab.triangular_mesh([vert[0] for vert in verts],
... [vert[1] for vert in verts],
... [vert[2] for vert in verts],
... faces)
>>> mlab.show()
Similarly using the visvis
package:
>>> import visvis as vv
>>> verts, faces, normals, values = marching_cubes_lewiner(myvolume, 0.0)
>>> vv.mesh(np.fliplr(verts), faces, normals, values)
>>> vv.use().Run()
References
[1]  Thomas Lewiner, Helio Lopes, Antonio Wilson Vieira and Geovan Tavares. Efficient implementation of Marching Cubes’ cases with topological guarantees. Journal of Graphics Tools 8(2) pp. 115 (december 2003). DOI:10.1080/10867651.2003.10487582 
skimage.measure.marching_cubes_lewiner
¶skimage.measure.
marching_cubes_classic
(volume, level=None, spacing=(1.0, 1.0, 1.0), gradient_direction='descent')[source]¶Classic marching cubes algorithm to find surfaces in 3d volumetric data.
Note that the marching_cubes()
algorithm is recommended over
this algorithm, because it’s faster and produces better results.
Parameters: 


Returns: 

See also
skimage.measure.marching_cubes
, skimage.measure.mesh_surface_area
Notes
The marching cubes algorithm is implemented as described in [1]. A simple explanation is available here:
http://www.essi.fr/~lingrand/MarchingCubes/algo.html
There are several known ambiguous cases in the marching cubes algorithm. Using point labeling as in [1], Figure 4, as shown:
v8  v7
/  /  y
/  /  ^ z
v4  v3   /
 v5  v6 / (note: NOT right handed!)
 /  / > x
 /  /
v1  v2
Most notably, if v4, v8, v2, and v6 are all >= level (or any generalization of this case) two parallel planes are generated by this algorithm, separating v4 and v8 from v2 and v6. An equally valid interpretation would be a single connected thin surface enclosing all four points. This is the best known ambiguity, though there are others.
This algorithm does not attempt to resolve such ambiguities; it is a naive implementation of marching cubes as in [1], but may be a good beginning for work with more recent techniques (Dual Marching Cubes, Extended Marching Cubes, Cubic Marching Squares, etc.).
Because of interactions between neighboring cubes, the isosurface(s) generated by this algorithm are NOT guaranteed to be closed, particularly for complicated contours. Furthermore, this algorithm does not guarantee a single contour will be returned. Indeed, ALL isosurfaces which cross level will be found, regardless of connectivity.
The output is a triangular mesh consisting of a set of unique vertices and
connecting triangles. The order of these vertices and triangles in the
output list is determined by the position of the smallest x,y,z
(in
lexicographical order) coordinate in the contour. This is a sideeffect
of how the input array is traversed, but can be relied upon.
The generated mesh guarantees coherent orientation as of version 0.12.
To quantify the area of an isosurface generated by this algorithm, pass outputs directly into skimage.measure.mesh_surface_area.
References
[1]  (1, 2, 3, 4) Lorensen, William and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163170). DOI:10.1145/37401.37422 
skimage.measure.
mesh_surface_area
(verts, faces)[source]¶Compute surface area, given vertices & triangular faces
Parameters: 


Returns: 

See also
skimage.measure.marching_cubes
, skimage.measure.marching_cubes_classic
, skimage.measure.correct_mesh_orientation
Notes
The arguments expected by this function are the first two outputs from skimage.measure.marching_cubes. For unit correct output, ensure correct spacing was passed to skimage.measure.marching_cubes.
This algorithm works properly only if the faces
provided are all
triangles.
skimage.measure.
correct_mesh_orientation
(volume, verts, faces, spacing=(1.0, 1.0, 1.0), gradient_direction='descent')[source]¶Correct orientations of mesh faces.
Parameters: 


Returns: 

Notes
Certain applications and mesh processing algorithms require all faces to be oriented in a consistent way. Generally, this means a normal vector points “out” of the meshed shapes. This algorithm corrects the output from skimage.measure.marching_cubes_classic by flipping the orientation of misoriented faces.
Because marching cubes could be used to find isosurfaces either on
gradient descent (where the desired object has greater values than the
exterior) or ascent (where the desired object has lower values than the
exterior), the gradient_direction
kwarg allows the user to inform this
algorithm which is correct. If the resulting mesh appears to be oriented
completely incorrectly, try changing this option.
The arguments expected by this function are the exact outputs from skimage.measure.marching_cubes_classic. Only faces is corrected and returned, as the vertices do not change; only the order in which they are referenced.
This algorithm assumes faces
provided are all triangles.
skimage.measure.
profile_line
(image, src, dst, linewidth=1, order=1, mode='constant', cval=0.0)[source]¶Return the intensity profile of an image measured along a scan line.
Parameters: 


Returns: 

Examples
>>> x = np.array([[1, 1, 1, 2, 2, 2]])
>>> img = np.vstack([np.zeros_like(x), x, x, x, np.zeros_like(x)])
>>> img
array([[0, 0, 0, 0, 0, 0],
[1, 1, 1, 2, 2, 2],
[1, 1, 1, 2, 2, 2],
[1, 1, 1, 2, 2, 2],
[0, 0, 0, 0, 0, 0]])
>>> profile_line(img, (2, 1), (2, 4))
array([ 1., 1., 2., 2.])
>>> profile_line(img, (1, 0), (1, 6), cval=4)
array([ 1., 1., 1., 2., 2., 2., 4.])
The destination point is included in the profile, in contrast to standard numpy indexing. For example:
>>> profile_line(img, (1, 0), (1, 6)) # The final point is out of bounds
array([ 1., 1., 1., 2., 2., 2., 0.])
>>> profile_line(img, (1, 0), (1, 5)) # This accesses the full first row
array([ 1., 1., 1., 2., 2., 2.])
skimage.measure.
label
(input, neighbors=None, background=None, return_num=False, connectivity=None)[source]¶Label connected regions of an integer array.
Two pixels are connected when they are neighbors and have the same value. In 2D, they can be neighbors either in a 1 or 2connected sense. The value refers to the maximum number of orthogonal hops to consider a pixel/voxel a neighbor:
1connectivity 2connectivity diagonal connection closeup
[ ] [ ] [ ] [ ] [ ]
 \  /  < hop 2
[ ][x][ ] [ ][x][ ] [x][ ]
 /  \ hop 1
[ ] [ ] [ ] [ ]
Parameters: 


Returns: 

See also
References
[1]  Christophe Fiorio and Jens Gustedt, “Two linear time UnionFind strategies for image processing”, Theoretical Computer Science 154 (1996), pp. 165181. 
[2]  Kensheng Wu, Ekow Otoo and Arie Shoshani, “Optimizing connected component labeling algorithms”, Paper LBNL56864, 2005, Lawrence Berkeley National Laboratory (University of California), http://repositories.cdlib.org/lbnl/LBNL56864 
Examples
>>> import numpy as np
>>> x = np.eye(3).astype(int)
>>> print(x)
[[1 0 0]
[0 1 0]
[0 0 1]]
>>> print(label(x, connectivity=1))
[[1 0 0]
[0 2 0]
[0 0 3]]
>>> print(label(x, connectivity=2))
[[1 0 0]
[0 1 0]
[0 0 1]]
>>> print(label(x, background=1))
[[1 2 2]
[2 1 2]
[2 2 1]]
>>> x = np.array([[1, 0, 0],
... [1, 1, 5],
... [0, 0, 0]])
>>> print(label(x))
[[1 0 0]
[1 1 2]
[0 0 0]]
skimage.measure.
points_in_poly
(points, verts)[source]¶Test whether points lie inside a polygon.
Parameters: 


Returns: 

See also
skimage.measure.
grid_points_in_poly
(shape, verts)[source]¶Test whether points on a specified grid are inside a polygon.
For each (r, c)
coordinate on a grid, i.e. (0, 0)
, (0, 1)
etc.,
test whether that point lies inside a polygon.
Parameters: 


Returns: 

See also
skimage.measure.
compare_ssim
(X, Y, 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: 


Returns: 

Other Parameters:  

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.
References
[1]  (1, 2, 3, 4) 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, 600612. https://ece.uwaterloo.ca/~z70wang/publications/ssim.pdf, DOI:10.1109/TIP.2003.819861 
[2]  (1, 2) Avanaki, A. N. (2009). Exact global histogram specification optimized for structural similarity. Optical Review, 16, 613621. arXiv:0901.0065 DOI:10.1007/s100430090119z 
skimage.measure.compare_ssim
¶skimage.measure.
compare_nrmse
(im_true, im_test, norm_type='Euclidean')[source]¶Compute the normalized root meansquared error (NRMSE) between two images.
Parameters: 


Returns: 

References
[1]  (1, 2) https://en.wikipedia.org/wiki/Rootmeansquare_deviation 
skimage.measure.
compare_psnr
(im_true, im_test, data_range=None)[source]¶Compute the peak signal to noise ratio (PSNR) for an image.
Parameters: 


Returns: 

References
[1]  https://en.wikipedia.org/wiki/Peak_signaltonoise_ratio 
skimage.measure.
shannon_entropy
(image, base=2)[source]¶Calculate the Shannon entropy of an image.
The Shannon entropy is defined as S = sum(pk * log(pk)), where pk are frequency/probability of pixels of value k.
Parameters: 


Returns: 

Notes
The returned value is measured in bits or shannon (Sh) for base=2, natural unit (nat) for base=np.e and hartley (Hart) for base=10.
References
[1]  https://en.wikipedia.org/wiki/Entropy_(information_theory) 
[2]  https://en.wiktionary.org/wiki/Shannon_entropy 
Examples
>>> from skimage import data
>>> shannon_entropy(data.camera())
7.0479552324230861
LineModelND
¶skimage.measure.
LineModelND
[source]¶Bases: skimage.measure.fit.BaseModel
Total least squares estimator for Ndimensional lines.
In contrast to ordinary least squares line estimation, this estimator minimizes the orthogonal distances of points to the estimated line.
Lines are defined by a point (origin) and a unit vector (direction) according to the following vector equation:
X = origin + lambda * direction
Examples
>>> x = np.linspace(1, 2, 25)
>>> y = 1.5 * x + 3
>>> lm = LineModelND()
>>> lm.estimate(np.array([x, y]).T)
True
>>> tuple(np.round(lm.params, 5))
(array([ 1.5 , 5.25]), array([ 0.5547 , 0.83205]))
>>> res = lm.residuals(np.array([x, y]).T)
>>> np.abs(np.round(res, 9))
array([ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
>>> np.round(lm.predict_y(x[:5]), 3)
array([ 4.5 , 4.562, 4.625, 4.688, 4.75 ])
>>> np.round(lm.predict_x(y[:5]), 3)
array([ 1. , 1.042, 1.083, 1.125, 1.167])
Attributes: 


estimate
(data)[source]¶Estimate line model from data.
This minimizes the sum of shortest (orthogonal) distances from the given data points to the estimated line.
Parameters: 


Returns: 

predict
(x, axis=0, params=None)[source]¶Predict intersection of the estimated line model with a hyperplane orthogonal to a given axis.
Parameters: 


Returns: 

Raises: 

predict_x
(y, params=None)[source]¶Predict xcoordinates for 2D lines using the estimated model.
Alias for:
predict(y, axis=1)[:, 0]
Parameters: 


Returns: 

predict_y
(x, params=None)[source]¶Predict ycoordinates for 2D lines using the estimated model.
Alias for:
predict(x, axis=0)[:, 1]
Parameters: 


Returns: 

residuals
(data, params=None)[source]¶Determine residuals of data to model.
For each point, the shortest (orthogonal) distance to the line is returned. It is obtained by projecting the data onto the line.
Parameters: 


Returns: 

CircleModel
¶skimage.measure.
CircleModel
[source]¶Bases: skimage.measure.fit.BaseModel
Total least squares estimator for 2D circles.
The functional model of the circle is:
r**2 = (x  xc)**2 + (y  yc)**2
This estimator minimizes the squared distances from all points to the circle:
min{ sum((r  sqrt((x_i  xc)**2 + (y_i  yc)**2))**2) }
A minimum number of 3 points is required to solve for the parameters.
Examples
>>> t = np.linspace(0, 2 * np.pi, 25)
>>> xy = CircleModel().predict_xy(t, params=(2, 3, 4))
>>> model = CircleModel()
>>> model.estimate(xy)
True
>>> tuple(np.round(model.params, 5))
(2.0, 3.0, 4.0)
>>> res = model.residuals(xy)
>>> np.abs(np.round(res, 9))
array([ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
Attributes: 


estimate
(data)[source]¶Estimate circle model from data using total least squares.
Parameters: 


Returns: 

predict_xy
(t, params=None)[source]¶Predict x and ycoordinates using the estimated model.
Parameters: 


Returns: 

EllipseModel
¶skimage.measure.
EllipseModel
[source]¶Bases: skimage.measure.fit.BaseModel
Total least squares estimator for 2D ellipses.
The functional model of the ellipse is:
xt = xc + a*cos(theta)*cos(t)  b*sin(theta)*sin(t)
yt = yc + a*sin(theta)*cos(t) + b*cos(theta)*sin(t)
d = sqrt((x  xt)**2 + (y  yt)**2)
where (xt, yt)
is the closest point on the ellipse to (x, y)
. Thus
d is the shortest distance from the point to the ellipse.
The estimator is based on a least squares minimization. The optimal solution is computed directly, no iterations are required. This leads to a simple, stable and robust fitting method.
The params
attribute contains the parameters in the following order:
xc, yc, a, b, theta
Examples
>>> xy = EllipseModel().predict_xy(np.linspace(0, 2 * np.pi, 25),
... params=(10, 15, 4, 8, np.deg2rad(30)))
>>> ellipse = EllipseModel()
>>> ellipse.estimate(xy)
True
>>> np.round(ellipse.params, 2)
array([ 10. , 15. , 4. , 8. , 0.52])
>>> np.round(abs(ellipse.residuals(xy)), 5)
array([ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
Attributes: 


estimate
(data)[source]¶Estimate circle model from data using total least squares.
Parameters: 


Returns: 

References
[1]  Halir, R.; Flusser, J. “Numerically stable direct least squares fitting of ellipses”. In Proc. 6th International Conference in Central Europe on Computer Graphics and Visualization. WSCG (Vol. 98, pp. 125132). 
predict_xy
(t, params=None)[source]¶Predict x and ycoordinates using the estimated model.
Parameters: 


Returns: 
