class skimage.measure.LineModelND [source]

Bases: skimage.measure.fit.BaseModel Total least squares estimator for N-dimensional 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.stack([x, y], axis=-1))
True
>>> tuple(np.round(lm.params, 5))
(array([1.5 , 5.25]), array([0.5547 , 0.83205]))
>>> res = lm.residuals(np.stack([x, y], axis=-1))
>>> 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

paramstuple

Line model parameters in the following order origin, direction.
__init__() [source]

Initialize self. See help(type(self)) for accurate signature.

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

data(N, dim) array

N points in a space of dimensionality dim >= 2.    Returns

successbool

True, if model estimation succeeds.

predict(x, axis=0, params=None) [source]

Predict intersection of the estimated line model with a hyperplane orthogonal to a given axis.  Parameters

x(n, 1) array

Coordinates along an axis.
axisint

Axis orthogonal to the hyperplane intersecting the line.
params(2, ) array, optional

Optional custom parameter set in the form (origin, direction).    Returns

data(n, m) array

Predicted coordinates.    Raises
 ValueError

If the line is parallel to the given axis.

predict_x(y, params=None) [source]

Predict x-coordinates for 2D lines using the estimated model. Alias for: predict(y, axis=1)[:, 0]
  Parameters

yarray

y-coordinates.
params(2, ) array, optional

Optional custom parameter set in the form (origin, direction).    Returns

xarray

Predicted x-coordinates.

predict_y(x, params=None) [source]

Predict y-coordinates for 2D lines using the estimated model. Alias for: predict(x, axis=0)[:, 1]
  Parameters

xarray

x-coordinates.
params(2, ) array, optional

Optional custom parameter set in the form (origin, direction).    Returns

yarray

Predicted y-coordinates.

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

data(N, dim) array

N points in a space of dimension dim.
params(2, ) array, optional

Optional custom parameter set in the form (origin, direction).    Returns

residuals(N, ) array

Residual for each data point.


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

data(N, dim) array

N points in a space of dimensionality dim >= 2.    Returns

successbool

True, if model estimation succeeds.


predict(x, axis=0, params=None) [source]

Predict intersection of the estimated line model with a hyperplane orthogonal to a given axis.  Parameters

x(n, 1) array

Coordinates along an axis.
axisint

Axis orthogonal to the hyperplane intersecting the line.
params(2, ) array, optional

Optional custom parameter set in the form (origin, direction).    Returns

data(n, m) array

Predicted coordinates.    Raises
 ValueError

If the line is parallel to the given axis.


predict_x(y, params=None) [source]

Predict x-coordinates for 2D lines using the estimated model. Alias for: predict(y, axis=1)[:, 0]
  Parameters

yarray

y-coordinates.
params(2, ) array, optional

Optional custom parameter set in the form (origin, direction).    Returns

xarray

Predicted x-coordinates.


predict_y(x, params=None) [source]

Predict y-coordinates for 2D lines using the estimated model. Alias for: predict(x, axis=0)[:, 1]
  Parameters

xarray

x-coordinates.
params(2, ) array, optional

Optional custom parameter set in the form (origin, direction).    Returns

yarray

Predicted y-coordinates.


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

data(N, dim) array

N points in a space of dimension dim.
params(2, ) array, optional

Optional custom parameter set in the form (origin, direction).    Returns

residuals(N, ) array

Residual for each data point.


__init__() [source]

Initialize self. See help(type(self)) for accurate signature.


skimage.measure.marching_cubes(volume, level=None, *, spacing=(1.0, 1.0, 1.0), gradient_direction='descent', step_size=1, allow_degenerate=True, method='lewiner', mask=None) [source]

Marching cubes algorithm to find surfaces in 3d volumetric data. In contrast with Lorensen et al. approach [2], Lewiner et al. algorithm is faster, resolves ambiguities, and guarantees topologically correct results. Therefore, this algorithm generally a better choice.  Parameters

volume(M, N, P) array

Input data volume to find isosurfaces. Will internally be converted to float32 if necessary.
levelfloat, optional

Contour value to search for isosurfaces in volume. If not given or None, the average of the min and max of vol is used.
spacinglength-3 tuple of floats, optional

Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in volume.
gradient_directionstring, optional

Controls if the mesh was generated from an isosurface with gradient descent toward objects of interest (the default), or the opposite, considering the left-hand rule. The two options are: * descent : Object was greater than exterior * ascent : Exterior was greater than object
step_sizeint, optional

Step size in voxels. Default 1. Larger steps yield faster but coarser results. The result will always be topologically correct though.
allow_degeneratebool, optional

Whether to allow degenerate (i.e. zero-area) triangles in the end-result. Default True. If False, degenerate triangles are removed, at the cost of making the algorithm slower.  method: str, optional

One of ‘lewiner’, ‘lorensen’ or ‘_lorensen’. Specify witch of Lewiner et al. or Lorensen et al. method will be used. The ‘_lorensen’ flag correspond to an old implementation that will be deprecated in version 0.19.
mask(M, N, P) array, optional

Boolean array. The marching cube algorithm will be computed only on True elements. This will save computational time when interfaces are located within certain region of the volume M, N, P-e.g. the top half of the cube-and also allow to compute finite surfaces-i.e. open surfaces that do not end at the border of the cube.    Returns

verts(V, 3) array

Spatial coordinates for V unique mesh vertices. Coordinate order matches input volume (M, N, P). If allow_degenerate is set to True, then the presence of degenerate triangles in the mesh can make this array have duplicate vertices.
faces(F, 3) array

Define triangular faces via referencing vertex indices from verts. This algorithm specifically outputs triangles, so each face has exactly three indices.
normals(V, 3) array

The normal direction at each vertex, as calculated from the data.
values(V, ) array

Gives a measure for the maximum value of the data in the local region near each vertex. This can be used by visualization tools to apply a colormap to the mesh.      See also
skimage.measure.mesh_surface_area

skimage.measure.find_contours

  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, _, _ = marching_cubes(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(myvolume, 0.0)
>> vv.mesh(np.fliplr(verts), faces, normals, values)
>> vv.use().Run()
 To reduce the number of triangles in the mesh for better performance, see this example using the mayavi package. 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. 1-15 (december 2003). DOI:10.1080/10867651.2003.10487582
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. 163-170). DOI:10.1145/37401.37422


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

volume(M, N, P) array of doubles

Input data volume to find isosurfaces. Will be cast to np.float64.
levelfloat

Contour value to search for isosurfaces in volume. If not given or None, the average of the min and max of vol is used.
spacinglength-3 tuple of floats

Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in volume.
gradient_directionstring

Controls if the mesh was generated from an isosurface with gradient descent toward objects of interest (the default), or the opposite. The two options are: * descent : Object was greater than exterior * ascent : Exterior was greater than object    Returns

verts(V, 3) array

Spatial coordinates for V unique mesh vertices. Coordinate order matches input volume (M, N, P). If allow_degenerate is set to True, then the presence of degenerate triangles in the mesh can make this array have duplicate vertices.
faces(F, 3) array

Define triangular faces via referencing vertex indices from verts. This algorithm specifically outputs triangles, so each face has exactly three indices.      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://users.polytech.unice.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 side-effect 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)
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. 163-170). DOI:10.1145/37401.37422


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, mask=None) [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

volume(M, N, P) array

Input data volume to find isosurfaces. Will internally be converted to float32 if necessary.
levelfloat

Contour value to search for isosurfaces in volume. If not given or None, the average of the min and max of vol is used.
spacinglength-3 tuple of floats

Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in volume.
gradient_directionstring

Controls if the mesh was generated from an isosurface with gradient descent toward objects of interest (the default), or the opposite, considering the left-hand rule. The two options are: * descent : Object was greater than exterior * ascent : Exterior was greater than object
step_sizeint

Step size in voxels. Default 1. Larger steps yield faster but coarser results. The result will always be topologically correct though.
allow_degeneratebool

Whether to allow degenerate (i.e. zero-area) triangles in the end-result. Default True. If False, degenerate triangles are removed, at the cost of making the algorithm slower.
use_classicbool

If given and True, the classic marching cubes by Lorensen (1987) is used. This option is included for reference purposes. Note that this algorithm has ambiguities and is not guaranteed to produce a topologically correct result. The results with using this option are not generally the same as the marching_cubes_classic() function.
mask(M, N, P) array

Boolean array. The marching cube algorithm will be computed only on True elements. This will save computational time when interfaces are located within certain region of the volume M, N, P-e.g. the top half of the cube-and also allow to compute finite surfaces-i.e. open surfaces that do not end at the border of the cube.    Returns

verts(V, 3) array

Spatial coordinates for V unique mesh vertices. Coordinate order matches input volume (M, N, P). If allow_degenerate is set to True, then the presence of degenerate triangles in the mesh can make this array have duplicate vertices.
faces(F, 3) array

Define triangular faces via referencing vertex indices from verts. This algorithm specifically outputs triangles, so each face has exactly three indices.
normals(V, 3) array

The normal direction at each vertex, as calculated from the data.
values(V, ) array

Gives a measure for the maximum value of the data in the local region near each vertex. This can be used by visualization tools to apply a colormap to the mesh.      See also
skimage.measure.marching_cubes

skimage.measure.mesh_surface_area

  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. 1-15 (december 2003). DOI:10.1080/10867651.2003.10487582


skimage.measure.mesh_surface_area(verts, faces) [source]

Compute surface area, given vertices & triangular faces  Parameters

verts(V, 3) array of floats

Array containing (x, y, z) coordinates for V unique mesh vertices.
faces(F, 3) array of ints

List of length-3 lists of integers, referencing vertex coordinates as provided in verts    Returns

areafloat

Surface area of mesh. Units now [coordinate units] ** 2.      See also
skimage.measure.marching_cubes

skimage.measure.marching_cubes_classic

  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.moments(image, order=3) [source]

Calculate all raw image moments up to a certain order.  The following properties can be calculated from raw image moments:

 Area as: M[0, 0]. Centroid as: {M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]}.    Note that raw moments are neither translation, scale nor rotation invariant.  Parameters

imagenD double or uint8 array

Rasterized shape as image.
orderint, optional

Maximum order of moments. Default is 3.    Returns

m(order + 1, order + 1) array

Raw image moments.     References
1
Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.
2
B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 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)
>>> centroid = (M[1, 0] / M[0, 0], M[0, 1] / M[0, 0])
>>> centroid
(14.5, 14.5)


skimage.measure.moments_central(image, center=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

imagenD double or uint8 array

Rasterized shape as image.
centertuple of float, optional

Coordinates of the image centroid. This will be computed if it is not provided.
orderint, optional

The maximum order of moments computed.    Returns

mu(order + 1, order + 1) array

Central image moments.     References
1
Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.
2
B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 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)
>>> centroid = (M[1, 0] / M[0, 0], M[0, 1] / M[0, 0])
>>> moments_central(image, centroid)
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.  The following properties can be calculated from raw image moments:

 Area as: M[0, 0]. Centroid as: {M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]}.    Note that raw moments are neither translation, scale nor rotation invariant.  Parameters

coords(N, D) double or uint8 array

Array of N points that describe an image of D dimensionality in Cartesian space.
orderint, optional

Maximum order of moments. Default is 3.    Returns

M(order + 1, order + 1, …) array

Raw image moments. (D dimensions)     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 = (M[1, 0] / M[0, 0], M[0, 1] / M[0, 0])
>>> centroid
(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.  The following properties can be calculated from raw image moments:

 Area as: M[0, 0]. Centroid as: {M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]}.    Note that raw moments are neither translation, scale nor rotation invariant.  Parameters

coords(N, D) double or uint8 array

Array of N points that describe an image of D dimensionality in Cartesian space. A tuple of coordinates as returned by np.nonzero is also accepted as input.
centertuple of float, optional

Coordinates of the image centroid. This will be computed if it is not provided.
orderint, optional

Maximum order of moments. Default is 3.    Returns

Mc(order + 1, order + 1, …) array

Central image moments. (D dimensions)     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, odd-order 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),
...          decimals=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_hu(nu) [source]

Calculate Hu’s set of image moments (2D-only). Note that this set of moments is proofed to be translation, scale and rotation invariant.  Parameters

nu(M, M) array

Normalized central image moments, where M must be >= 4.    Returns

nu(7,) array

Hu’s set of image moments.     References
1
M. K. Hu, “Visual Pattern Recognition by Moment Invariants”, IRE Trans. Info. Theory, vol. IT-8, pp. 179-187, 1962
2
Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.
3
B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 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   Examples >>> image = np.zeros((20, 20), dtype=np.double)
>>> image[13:17, 13:17] = 0.5
>>> image[10:12, 10:12] = 1
>>> mu = moments_central(image)
>>> nu = moments_normalized(mu)
>>> moments_hu(nu)
array([7.45370370e-01, 3.51165981e-01, 1.04049179e-01, 4.06442107e-02,
       2.64312299e-03, 2.40854582e-02, 4.33680869e-19])


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

mu(M,[ …,] M) array

Central image moments, where M must be greater than or equal to order.
orderint, optional

Maximum order of moments. Default is 3.    Returns

nu(order + 1,[ …,] order + 1) array

Normalized central image moments.     References
1
Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.
2
B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 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)
>>> centroid = (m[0, 1] / m[0, 0], m[1, 0] / m[0, 0])
>>> mu = moments_central(image, centroid)
>>> 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.perimeter(image, neighbourhood=4) [source]

Calculate total perimeter of all objects in binary image.  Parameters

image(N, M) ndarray

2D binary image.
neighbourhood4 or 8, optional

Neighborhood connectivity for border pixel determination. It is used to compute the contour. A higher neighbourhood widens the border on which the perimeter is computed.    Returns

perimeterfloat

Total perimeter of all objects in binary image.     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)
7796.867...
>>> perimeter(img_coins, neighbourhood=8)
8806.268...


skimage.measure.perimeter_crofton(image, directions=4) [source]

Calculate total Crofton perimeter of all objects in binary image.  Parameters

image(N, M) ndarray

2D image. If image is not binary, all values strictly greater than zero are considered as the object.
directions2 or 4, optional

Number of directions used to approximate the Crofton perimeter. By default, 4 is used: it should be more accurate than 2. Computation time is the same in both cases.    Returns

perimeterfloat

Total perimeter of all objects in binary image.     Notes This measure is based on Crofton formula [1], which is a measure from integral geometry. It is defined for general curve length evaluation via a double integral along all directions. In a discrete space, 2 or 4 directions give a quite good approximation, 4 being more accurate than 2 for more complex shapes. Similar to perimeter(), this function returns an approximation of the perimeter in continuous space. References
1
https://en.wikipedia.org/wiki/Crofton_formula
2
S. Rivollier. Analyse d’image geometrique et morphometrique par diagrammes de forme et voisinages adaptatifs generaux. PhD thesis, 2010. Ecole Nationale Superieure des Mines de Saint-Etienne. https://tel.archives-ouvertes.fr/tel-00560838   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_crofton(img_coins, directions=2)
8144.578...
>>> perimeter_crofton(img_coins, directions=4)
7837.077...


skimage.measure.points_in_poly(points, verts) [source]

Test whether points lie inside a polygon.  Parameters

points(N, 2) array

Input points, (x, y).
verts(M, 2) array

Vertices of the polygon, sorted either clockwise or anti-clockwise. The first point may (but does not need to be) duplicated.    Returns

mask(N,) array of bool

True if corresponding point is inside the polygon.      See also
grid_points_in_poly


skimage.measure.profile_line(image, src, dst, linewidth=1, order=None, mode=None, cval=0.0, *, reduce_func=<function mean>) [source]

Return the intensity profile of an image measured along a scan line.  Parameters

imagendarray, shape (M, N[, C])

The image, either grayscale (2D array) or multichannel (3D array, where the final axis contains the channel information).
srcarray_like, shape (2, )

The coordinates of the start point of the scan line.
dstarray_like, shape (2, )

The coordinates of the end point of the scan line. The destination point is included in the profile, in contrast to standard numpy indexing.
linewidthint, optional

Width of the scan, perpendicular to the line
orderint in {0, 1, 2, 3, 4, 5}, optional

The order of the spline interpolation, default is 0 if image.dtype is bool and 1 otherwise. The order has to be in the range 0-5. See skimage.transform.warp for detail.
mode{‘constant’, ‘nearest’, ‘reflect’, ‘mirror’, ‘wrap’}, optional

How to compute any values falling outside of the image.
cvalfloat, optional

If mode is ‘constant’, what constant value to use outside the image.
reduce_funccallable, optional

Function used to calculate the aggregation of pixel values perpendicular to the profile_line direction when linewidth > 1. If set to None the unreduced array will be returned.    Returns

return_valuearray

The intensity profile along the scan line. The length of the profile is the ceil of the computed length of the scan line.     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.])
 For different reduce_func inputs: >>> profile_line(img, (1, 0), (1, 3), linewidth=3, reduce_func=np.mean)
array([0.66666667, 0.66666667, 0.66666667, 1.33333333])
>>> profile_line(img, (1, 0), (1, 3), linewidth=3, reduce_func=np.max)
array([1, 1, 1, 2])
>>> profile_line(img, (1, 0), (1, 3), linewidth=3, reduce_func=np.sum)
array([2, 2, 2, 4])
 The unreduced array will be returned when reduce_func is None or when reduce_func acts on each pixel value individually. >>> profile_line(img, (1, 2), (4, 2), linewidth=3, order=0,
...     reduce_func=None)
array([[1, 1, 2],
       [1, 1, 2],
       [1, 1, 2],
       [0, 0, 0]])
>>> profile_line(img, (1, 0), (1, 3), linewidth=3, reduce_func=np.sqrt)
array([[1.        , 1.        , 0.        ],
       [1.        , 1.        , 0.        ],
       [1.        , 1.        , 0.        ],
       [1.41421356, 1.41421356, 0.        ]])


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, initial_inliers=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:  Select min_samples random samples from the original data and check whether the set of data is valid (see is_data_valid). Estimate a model to the random subset (model_cls.estimate(*data[random_subset]) and check whether the estimated model is valid (see is_model_valid). Classify all data as inliers or outliers by calculating the residuals to the estimated model (model_cls.residuals(*data)) - all data samples with residuals smaller than the residual_threshold are considered as inliers. Save estimated model as best model if number of inlier samples is maximal. In case the current estimated model has the same number of inliers, it is only considered as the best model if it has less sum of residuals.  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

data[list, tuple of] (N, …) array

Data set to which the model is fitted, where N is the number of data points and the remaining dimension are depending on model requirements. If the model class requires multiple input data arrays (e.g. source and destination coordinates of skimage.transform.AffineTransform), they can be optionally passed as tuple or list. Note, that in this case the functions estimate(*data), residuals(*data), is_model_valid(model, *random_data) and is_data_valid(*random_data) must all take each data array as separate arguments.
model_classobject

Object with the following object methods:  success = estimate(*data) residuals(*data)  where success indicates whether the model estimation succeeded (True or None for success, False for failure).
min_samplesint in range (0, N)

The minimum number of data points to fit a model to.
residual_thresholdfloat larger than 0

Maximum distance for a data point to be classified as an inlier.
is_data_validfunction, optional

This function is called with the randomly selected data before the model is fitted to it: is_data_valid(*random_data).
is_model_validfunction, optional

This function is called with the estimated model and the randomly selected data: is_model_valid(model, *random_data), .
max_trialsint, optional

Maximum number of iterations for random sample selection.
stop_sample_numint, optional

Stop iteration if at least this number of inliers are found.
stop_residuals_sumfloat, optional

Stop iteration if sum of residuals is less than or equal to this threshold.
stop_probabilityfloat in range [0, 1], optional

RANSAC iteration stops if at least one outlier-free set of the training data is sampled with probability >= stop_probability, depending on the current best model’s inlier ratio and the number of trials. This requires to generate at least N samples (trials): N >= log(1 - probability) / log(1 - e**m) where the probability (confidence) is typically set to a high value such as 0.99, e is the current fraction of inliers w.r.t. the total number of samples, and m is the min_samples value.
random_stateint, RandomState instance or None, optional

If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random.
initial_inliersarray-like of bool, shape (N,), optional

Initial samples selection for model estimation    Returns

modelobject

Best model with largest consensus set.
inliers(N, ) array

Boolean mask of inliers classified as True.     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)
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
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
 RANSAC can be used to robustly estimate a geometric transformation. In this section, we also show how to use a proportion of the total samples, rather than an absolute number. >>> 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)
>>> ratio = 0.5  # use half of the samples
>>> min_samples = int(ratio * len(src))
>>> model, inliers = ransac((src, dst), SimilarityTransform, min_samples, 10,
...                         initial_inliers=np.ones(len(src), dtype=bool))
>>> 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])


skimage.measure.regionprops(label_image, intensity_image=None, cache=True, coordinates=None, *, extra_properties=None) [source]

Measure properties of labeled image regions.  Parameters

label_image(M, N[, P]) ndarray

Labeled input image. Labels with value 0 are ignored.  Changed in version 0.14.1: Previously, label_image was processed by numpy.squeeze and so any number of singleton dimensions was allowed. This resulted in inconsistent handling of images with singleton dimensions. To recover the old behaviour, use regionprops(np.squeeze(label_image), ...).
intensity_image(M, N[, P][, C]) ndarray, optional

Intensity (i.e., input) image with same size as labeled image, plus optionally an extra dimension for multichannel data. Default is None.  Changed in version 0.18.0: The ability to provide an extra dimension for channels was added.
cachebool, optional

Determine whether to cache calculated properties. The computation is much faster for cached properties, whereas the memory consumption increases.
coordinatesDEPRECATED

This argument is deprecated and will be removed in a future version of scikit-image. See Coordinate conventions for more details.  Deprecated since version 0.16.0: Use “rc” coordinates everywhere. It may be sufficient to call numpy.transpose on your label image to get the same values as 0.15 and earlier. However, for some properties, the transformation will be less trivial. For example, the new orientation is \(\frac{\pi}{2}\) plus the old orientation.
extra_propertiesIterable of callables

Add extra property computation functions that are not included with skimage. The name of the property is derived from the function name, the dtype is inferred by calling the function on a small sample. If the name of an extra property clashes with the name of an existing property the extra property wil not be visible and a UserWarning is issued. A property computation function must take a region mask as its first argument. If the property requires an intensity image, it must accept the intensity image as the second argument.    Returns

propertieslist of RegionProperties

Each item describes one labeled region, and can be accessed using the attributes listed below.      See also
label

  Notes The following properties can be accessed as attributes or keys:
areaint

Number of pixels of the region.
bboxtuple

Bounding box (min_row, min_col, max_row, max_col). Pixels belonging to the bounding box are in the half-open interval [min_row; max_row) and [min_col; max_col).
bbox_areaint

Number of pixels of bounding box.
centroidarray

Centroid coordinate tuple (row, col).
convex_areaint

Number of pixels of convex hull image, which is the smallest convex polygon that encloses the region.
convex_image(H, J) ndarray

Binary convex hull image which has the same size as bounding box.
coords(N, 2) ndarray

Coordinate list (row, col) of the region.
eccentricityfloat

Eccentricity of the ellipse that has the same second-moments as the region. The eccentricity is the ratio of the focal distance (distance between focal points) over the major axis length. The value is in the interval [0, 1). When it is 0, the ellipse becomes a circle.
equivalent_diameterfloat

The diameter of a circle with the same area as the region.
euler_numberint

Euler characteristic of the set of non-zero pixels. Computed as number of connected components subtracted by number of holes (input.ndim connectivity). In 3D, number of connected components plus number of holes subtracted by number of tunnels.
extentfloat

Ratio of pixels in the region to pixels in the total bounding box. Computed as area / (rows * cols)
feret_diameter_maxfloat

Maximum Feret’s diameter computed as the longest distance between points around a region’s convex hull contour as determined by find_contours. [5]
filled_areaint

Number of pixels of the region will all the holes filled in. Describes the area of the filled_image.
filled_image(H, J) ndarray

Binary region image with filled holes which has the same size as bounding box.
image(H, J) ndarray

Sliced binary region image which has the same size as bounding box.
inertia_tensorndarray

Inertia tensor of the region for the rotation around its mass.
inertia_tensor_eigvalstuple

The eigenvalues of the inertia tensor in decreasing order.
intensity_imagendarray

Image inside region bounding box.
labelint

The label in the labeled input image.
local_centroidarray

Centroid coordinate tuple (row, col), relative to region bounding box.
major_axis_lengthfloat

The length of the major axis of the ellipse that has the same normalized second central moments as the region.
max_intensityfloat

Value with the greatest intensity in the region.
mean_intensityfloat

Value with the mean intensity in the region.
min_intensityfloat

Value with the least intensity in the region.
minor_axis_lengthfloat

The length of the minor axis of the ellipse that has the same normalized second central moments as the region.
moments(3, 3) ndarray

Spatial moments up to 3rd order: m_ij = sum{ array(row, col) * row^i * col^j }
 where the sum is over the row, col coordinates of the region.
moments_central(3, 3) ndarray

Central moments (translation invariant) up to 3rd order: mu_ij = sum{ array(row, col) * (row - row_c)^i * (col - col_c)^j }
 where the sum is over the row, col coordinates of the region, and row_c and col_c are the coordinates of the region’s centroid.
moments_hutuple

Hu moments (translation, scale and rotation invariant).
moments_normalized(3, 3) ndarray

Normalized moments (translation and scale invariant) up to 3rd order: nu_ij = mu_ij / m_00^[(i+j)/2 + 1]
 where m_00 is the zeroth spatial moment.
orientationfloat

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 counter-clockwise.
perimeterfloat

Perimeter of object which approximates the contour as a line through the centers of border pixels using a 4-connectivity.
perimeter_croftonfloat

Perimeter of object approximated by the Crofton formula in 4 directions.
slicetuple of slices

A slice to extract the object from the source image.
solidityfloat

Ratio of pixels in the region to pixels of the convex hull image.
weighted_centroidarray

Centroid coordinate tuple (row, col) weighted with intensity image.
weighted_local_centroidarray

Centroid coordinate tuple (row, col), relative to region bounding box, weighted with intensity image.
weighted_moments(3, 3) ndarray

Spatial moments of intensity image up to 3rd order: wm_ij = sum{ array(row, col) * row^i * col^j }
 where the sum is over the row, col coordinates of the region.
weighted_moments_central(3, 3) ndarray

Central moments (translation invariant) of intensity image up to 3rd order: wmu_ij = sum{ array(row, col) * (row - row_c)^i * (col - col_c)^j }
 where the sum is over the row, col coordinates of the region, and row_c and col_c are the coordinates of the region’s weighted centroid.
weighted_moments_hutuple

Hu moments (translation, scale and rotation invariant) of intensity image.
weighted_moments_normalized(3, 3) ndarray

Normalized moments (translation and scale invariant) of intensity image up to 3rd order: wnu_ij = wmu_ij / wm_00^[(i+j)/2 + 1]
 where wm_00 is the zeroth spatial moment (intensity-weighted 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. Springer-Verlag, London, 2009.
2
B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 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
5
W. Pabst, E. Gregorová. Characterization of particles and particle systems, pp. 27-28. ICT Prague, 2007. https://old.vscht.cz/sil/keramika/Characterization_of_particles/CPPS%20_English%20version_.pdf   Examples >>> from skimage import data, util
>>> from skimage.measure import label, regionprops
>>> 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
(22.72987986048314, 81.91228523446583)
>>> # centroid of first labeled object
>>> props[0]['centroid']
(22.72987986048314, 81.91228523446583)
 Add custom measurements by passing functions as extra_properties >>> from skimage import data, util
>>> from skimage.measure import label, regionprops
>>> import numpy as np
>>> img = util.img_as_ubyte(data.coins()) > 110
>>> label_img = label(img, connectivity=img.ndim)
>>> def pixelcount(regionmask):
...     return np.sum(regionmask)
>>> props = regionprops(label_img, extra_properties=(pixelcount,))
>>> props[0].pixelcount
7741
>>> props[1]['pixelcount']
42


skimage.measure.regionprops_table(label_image, intensity_image=None, properties=('label', 'bbox'), *, cache=True, separator='-', extra_properties=None) [source]

Compute image properties and return them as a pandas-compatible table. The table is a dictionary mapping column names to value arrays. See Notes section below for details.  New in version 0.16.   Parameters

label_image(N, M[, P]) ndarray

Labeled input image. Labels with value 0 are ignored.
intensity_image(M, N[, P][, C]) ndarray, optional

Intensity (i.e., input) image with same size as labeled image, plus optionally an extra dimension for multichannel data. Default is None.  Changed in version 0.18.0: The ability to provide an extra dimension for channels was added.
propertiestuple or list of str, optional

Properties that will be included in the resulting dictionary For a list of available properties, please see regionprops(). Users should remember to add “label” to keep track of region identities.
cachebool, optional

Determine whether to cache calculated properties. The computation is much faster for cached properties, whereas the memory consumption increases.
separatorstr, optional

For non-scalar properties not listed in OBJECT_COLUMNS, each element will appear in its own column, with the index of that element separated from the property name by this separator. For example, the inertia tensor of a 2D region will appear in four columns: inertia_tensor-0-0, inertia_tensor-0-1, inertia_tensor-1-0, and inertia_tensor-1-1 (where the separator is -). Object columns are those that cannot be split in this way because the number of columns would change depending on the object. For example, image and coords.
extra_propertiesIterable of callables

Add extra property computation functions that are not included with skimage. The name of the property is derived from the function name, the dtype is inferred by calling the function on a small sample. If the name of an extra property clashes with the name of an existing property the extra property wil not be visible and a UserWarning is issued. A property computation function must take a region mask as its first argument. If the property requires an intensity image, it must accept the intensity image as the second argument.    Returns

out_dictdict

Dictionary mapping property names to an array of values of that property, one value per region. This dictionary can be used as input to pandas DataFrame to map property names to columns in the frame and regions to rows. If the image has no regions, the arrays will have length 0, but the correct type.     Notes Each column contains either a scalar property, an object property, or an element in a multidimensional array. Properties with scalar values for each region, such as “eccentricity”, will appear as a float or int array with that property name as key. Multidimensional properties of fixed size for a given image dimension, such as “centroid” (every centroid will have three elements in a 3D image, no matter the region size), will be split into that many columns, with the name {property_name}{separator}{element_num} (for 1D properties), {property_name}{separator}{elem_num0}{separator}{elem_num1} (for 2D properties), and so on. For multidimensional properties that don’t have a fixed size, such as “image” (the image of a region varies in size depending on the region size), an object array will be used, with the corresponding property name as the key. Examples >>> from skimage import data, util, measure
>>> image = data.coins()
>>> label_image = measure.label(image > 110, connectivity=image.ndim)
>>> props = measure.regionprops_table(label_image, image,
...                           properties=['label', 'inertia_tensor',
...                                       'inertia_tensor_eigvals'])
>>> props
{'label': array([ 1,  2, ...]), ...
 'inertia_tensor-0-0': array([  4.012...e+03,   8.51..., ...]), ...
 ...,
 'inertia_tensor_eigvals-1': array([  2.67...e+02,   2.83..., ...])}
 The resulting dictionary can be directly passed to pandas, if installed, to obtain a clean DataFrame: >>> import pandas as pd
>>> data = pd.DataFrame(props)
>>> data.head()
   label  inertia_tensor-0-0  ...  inertia_tensor_eigvals-1
0      1         4012.909888  ...                267.065503
1      2            8.514739  ...                  2.834806
2      3            0.666667  ...                  0.000000
3      4            0.000000  ...                  0.000000
4      5            0.222222  ...                  0.111111
 [5 rows x 7 columns] If we want to measure a feature that does not come as a built-in property, we can define custom functions and pass them as extra_properties. For example, we can create a custom function that measures the intensity quartiles in a region: >>> from skimage import data, util, measure
>>> import numpy as np
>>> def quartiles(regionmask, intensity):
...     return np.percentile(intensity[regionmask], q=(25, 50, 75))
>>>
>>> image = data.coins()
>>> label_image = measure.label(image > 110, connectivity=image.ndim)
>>> props = measure.regionprops_table(label_image, intensity_image=image,
...                                   properties=('label',),
...                                   extra_properties=(quartiles,))
>>> import pandas as pd
>>> pd.DataFrame(props).head()
       label  quartiles-0  quartiles-1  quartiles-2
0      1       117.00        123.0        130.0
1      2       111.25        112.0        114.0
2      3       111.00        111.0        111.0
3      4       111.00        111.5        112.5
4      5       112.50        113.0        114.0


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

image(N, M) ndarray

Grayscale input image.
basefloat, optional

The logarithmic base to use.    Returns

entropyfloat
   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
>>> from skimage.measure import shannon_entropy
>>> shannon_entropy(data.camera())
7.231695011055706


skimage.measure.subdivide_polygon(coords, degree=2, preserve_ends=False) [source]

Subdivision of polygonal curves using B-Splines. Note that the resulting curve is always within the convex hull of the original polygon. Circular polygons stay closed after subdivision.  Parameters

coords(N, 2) array

Coordinate array.
degree{1, 2, 3, 4, 5, 6, 7}, optional

Degree of B-Spline. Default is 2.
preserve_endsbool, optional

Preserve first and last coordinate of non-circular polygon. Default is False.    Returns

coords(M, 2) array

Subdivided coordinate array.     References
1
http://mrl.nyu.edu/publications/subdiv-course2000/coursenotes00.pdf


Module: metrics
skimage.metrics.adapted_rand_error([…]) 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.
skimage.metrics.normalized_root_mse(…[, …]) Compute the normalized root mean-squared error (NRMSE) between two images.
skimage.metrics.peak_signal_noise_ratio(…) Compute the peak signal to noise ratio (PSNR) for an image.
skimage.metrics.structural_similarity(im1, …) Compute the mean structural similarity index between two images.
skimage.metrics.variation_of_information([…]) Return symmetric conditional entropies associated with the VI.   adapted_rand_error
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

 Examples using skimage.metrics.hausdorff_distance

  Hausdorff Distance   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.
gradientbool, optional

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.
gradndarray

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.


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


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.


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


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.


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


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


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.
gradientbool, optional

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.
gradndarray

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


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.