Module: measure
¶

Approximate a polygonal chain with the specified tolerance. 

Downsample image by applying function func to local blocks. 

Compute the meansquared error between two images. 

Compute the normalized root meansquared error (NRMSE) between two images. 

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

Compute the mean structural similarity index between two images. 

Find isovalued contours in a 2D array for a given level value. 

Test whether points on a specified grid are inside a polygon. 

Compute the inertia tensor of the input image. 
Compute the eigenvalues of the inertia tensor of the image. 


Label connected regions of an integer array. 

Marching cubes algorithm to find surfaces in 3d volumetric data. 
Classic marching cubes algorithm to find surfaces in 3d volumetric data. 

Lewiner marching cubes algorithm to find surfaces in 3d volumetric data. 


Compute surface area, given vertices & triangular faces 

Calculate all raw image moments up to a certain order. 

Calculate all central image moments up to a certain order. 

Calculate all raw image moments up to a certain order. 
Calculate all central image moments up to a certain order. 

Calculate Hu’s set of image moments (2Donly). 


Calculate all normalized central image moments up to a certain order. 

Calculate total perimeter of all objects in binary image. 

Test whether points lie inside a polygon. 

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

Fit a model to data with the RANSAC (random sample consensus) algorithm. 

Measure properties of labeled image regions. 

Compute image properties and return them as a pandascompatible table. 

Calculate the Shannon entropy of an image. 

Subdivision of polygonal curves using BSplines. 
Total least squares estimator for 2D circles. 

Total least squares estimator for 2D ellipses. 

Total least squares estimator for Ndimensional lines. 
approximate_polygon¶

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
 coords(N, 2) array
Coordinate array.
 tolerancefloat
Maximum distance from original points of polygon to approximated polygonal chain. If tolerance is 0, the original coordinate array is returned.
 Returns
 coords(M, 2) array
Approximated polygonal chain where M <= N.
References
Examples using skimage.measure.approximate_polygon
¶
block_reduce¶

skimage.measure.
block_reduce
(image, block_size, func=<function sum at 0x7f23a664cae8>, cval=0, func_kwargs=None)[source]¶ Downsample image by applying function func to local blocks.
This function is useful for max and mean pooling, for example.
 Parameters
 imagendarray
Ndimensional input image.
 block_sizearray_like
Array containing downsampling integer factor along each axis.
 funccallable
Function object which is used to calculate the return value for each local block. This function must implement an
axis
parameter. Primary functions arenumpy.sum
,numpy.min
,numpy.max
,numpy.mean
andnumpy.median
. See also func_kwargs. cvalfloat
Constant padding value if image is not perfectly divisible by the block size.
 func_kwargsdict
Keyword arguments passed to func. Notably useful for passing dtype argument to
np.mean
. Takes dictionary of inputs, e.g.:func_kwargs={'dtype': np.float16})
.
 Returns
 imagendarray
Downsampled image with same number of dimensions as input image.
Examples
>>> from skimage.measure import block_reduce >>> image = np.arange(3*3*4).reshape(3, 3, 4) >>> image 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 array([[[11]], [[23]], [[35]]]) >>> image_max2 = block_reduce(image, block_size=(3, 1, 4), func=np.max) >>> image_max2 array([[[27], [31], [35]]])
compare_mse¶

skimage.measure.
compare_mse
(im1, im2)[source]¶ Compute the meansquared error between two images.
 Parameters
 image0, image1ndarray
Images. Any dimensionality, must have same shape.
 Returns
 msefloat
The meansquared error (MSE) metric.
 Warns
 Deprecated:
New in version 0.16.
This function is deprecated and will be removed in scikitimage 0.18. Please use the function named
mean_squared_error
from themetrics
module instead.
See also
Notes
Changed in version 0.16: This function was renamed from
skimage.measure.compare_mse
toskimage.metrics.mean_squared_error
.
compare_nrmse¶

skimage.measure.
compare_nrmse
(im_true, im_test, norm_type='euclidean')[source]¶ Compute the normalized root meansquared error (NRMSE) between two images.
 Parameters
 image_truendarray
Groundtruth image, same shape as im_test.
 image_testndarray
Test image.
 normalization{‘euclidean’, ‘minmax’, ‘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 .
‘minmax’ : normalize by the intensity range of
im_true
.‘mean’ : normalize by the mean of
im_true
 Returns
 nrmsefloat
The NRMSE metric.
 Warns
 Deprecated:
New in version 0.16.
This function is deprecated and will be removed in scikitimage 0.18. Please use the function named
normalized_root_mse
from themetrics
module instead.
See also
Notes
Changed in version 0.16: This function was renamed from
skimage.measure.compare_nrmse
toskimage.metrics.normalized_root_mse
.References
compare_psnr¶

skimage.measure.
compare_psnr
(im_true, im_test, data_range=None)[source]¶ Compute the peak signal to noise ratio (PSNR) for an image.
 Parameters
 image_truendarray
Groundtruth 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 datatype.
 Returns
 psnrfloat
The PSNR metric.
 Warns
 Deprecated:
New in version 0.16.
This function is deprecated and will be removed in scikitimage 0.18. Please use the function named
peak_signal_noise_ratio
from themetrics
module instead.
Notes
Changed in version 0.16: This function was renamed from
skimage.measure.compare_psnr
toskimage.metrics.peak_signal_noise_ratio
.References
compare_ssim¶

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
 im1, im2ndarray
Images. Any dimensionality with same shape.
 win_sizeint or None, optional
The sidelength 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 datatype.
 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 N1 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.
 Warns
 Deprecated:
New in version 0.16.
This function is deprecated and will be removed in scikitimage 0.18. Please use the function named
structural_similarity
from themetrics
module instead.
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
toskimage.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, 600612. 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, 613621. arXiv:0901.0065 DOI:10.1007/s100430090119z
find_contours¶

skimage.measure.
find_contours
(array, level, fully_connected='low', positive_orientation='low', *, mask=None)[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
 array2D ndarray of double
Input data in which to find contours.
 levelfloat
Value along which to find contours in the array.
 fully_connectedstr, {‘low’, ‘high’}
Indicates whether array elements below the given level value are to be considered fullyconnected (and hence elements above the value will only be face connected), or viceversa. (See notes below for details.)
 positive_orientationeither ‘low’ or ‘high’
Indicates whether the output contours will produce positivelyoriented polygons around islands of low or highvalued elements. If ‘low’ then contours will wind counter clockwise around elements below the isovalue. Alternately, this means that lowvalued elements are always on the left of the contour. (See below for details.)
 mask2D ndarray of bool, or None
A boolean mask, True where we want to draw contours. Note that NaN values are always excluded from the considered region (
mask
is set toFalse
whereverarray
isNaN
).
 Returns
 contourslist of (n,2)ndarrays
Each contour is an ndarray of shape
(n, 2)
, consisting of n(row, column)
coordinates along the contour.
Notes
The marching squares algorithm is a special case of the marching cubes algorithm [1]. A simple explanation is available here:
http://users.polytech.unice.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 or a maskedoff region (either where mask is False or where array is NaN) 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 (atx=0
) and the 1element (atx=1
), and thus would fall atx=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
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. ]])]
grid_points_in_poly¶

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
 shapetuple (M, N)
Shape of the grid.
 verts(V, 2) array
Specify the V vertices of the polygon, sorted either clockwise or anticlockwise. The first point may (but does not need to be) duplicated.
 Returns
 mask(M, N) ndarray of bool
True where the grid falls inside the polygon.
See also
inertia_tensor¶

skimage.measure.
inertia_tensor
(image, mu=None)[source]¶ Compute the inertia tensor of the input image.
 Parameters
 imagearray
The input image.
 muarray, optional
The precomputed central moments of
image
. The inertia tensor computation requires the central moments of the image. If an application requires both the central moments and the inertia tensor (for example,skimage.measure.regionprops
), then it is more efficient to precompute them and pass them to the inertia tensor call.
 Returns
 Tarray, shape
(image.ndim, image.ndim)
The inertia tensor of the input image. \(T_{i, j}\) contains the covariance of image intensity along axes \(i\) and \(j\).
 Tarray, shape
References
 1
https://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_tensor
 2
Bernd Jähne. SpatioTemporal Image Processing: Theory and Scientific Applications. (Chapter 8: Tensor Methods) Springer, 1993.
inertia_tensor_eigvals¶

skimage.measure.
inertia_tensor_eigvals
(image, mu=None, T=None)[source]¶ Compute the eigenvalues of the inertia tensor of the image.
The inertia tensor measures covariance of the image intensity along the image axes. (See
inertia_tensor
.) The relative magnitude of the eigenvalues of the tensor is thus a measure of the elongation of a (bright) object in the image. Parameters
 imagearray
The input image.
 muarray, optional
The precomputed central moments of
image
. Tarray, shape
(image.ndim, image.ndim)
The precomputed inertia tensor. If
T
is given,mu
andimage
are ignored.
 Returns
 eigvalslist of float, length
image.ndim
The eigenvalues of the inertia tensor of
image
, in descending order.
 eigvalslist of float, length
Notes
Computing the eigenvalues requires the inertia tensor of the input image. This is much faster if the central moments (
mu
) are provided, or, alternatively, one can provide the inertia tensor (T
) directly.
label¶

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
 inputndarray of dtype int
Image to label.
 neighbors{4, 8}, int, optional
Whether to use 4 or 8“connectivity”. In 3D, 4“connectivity” means connected pixels have to share face, whereas with 8“connectivity”, they have to share only edge or vertex. Deprecated, use
connectivity
instead. backgroundint, optional
Consider all pixels with this value as background pixels, and label them as 0. By default, 0valued pixels are considered as background pixels.
 return_numbool, optional
Whether to return the number of assigned labels.
 connectivityint, optional
Maximum number of orthogonal hops to consider a pixel/voxel as a neighbor. Accepted values are ranging from 1 to input.ndim. If
None
, a full connectivity ofinput.ndim
is used.
 Returns
 labelsndarray of dtype int
Labeled array, where all connected regions are assigned the same integer value.
 numint, optional
Number of labels, which equals the maximum label index and is only returned if return_num is True.
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]]
marching_cubes¶

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
Contour value to search for isosurfaces in volume. If not given or None, the average of the min and max of vol is used.
 spacinglength3 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 lefthand 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. zeroarea) triangles in the endresult. Default True. If False, degenerate triangles are removed, at the cost of making the algorithm slower.
 method: str
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
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, Pe.g. the top half of the cubeand also allow to compute finite surfacesi.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).
 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
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()
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
 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). DOI:10.1145/37401.37422
Examples using skimage.measure.marching_cubes
¶
marching_cubes_classic¶

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.
 spacinglength3 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).
 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.
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 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)
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
marching_cubes_lewiner¶

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.
 spacinglength3 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 lefthand 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. zeroarea) triangles in the endresult. 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, Pe.g. the top half of the cubeand also allow to compute finite surfacesi.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).
 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.
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
mesh_surface_area¶

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 length3 lists of integers, referencing vertex coordinates as provided in verts
 Returns
 areafloat
Surface area of mesh. Units now [coordinate units] ** 2.
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 toskimage.measure.marching_cubes
.This algorithm works properly only if the
faces
provided are all triangles.
moments¶

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.
 m(
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
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)
moments_central¶

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.
 mu(
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
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.]])
moments_coords¶

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)
 M(
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)
moments_coords_central¶

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)
 Mc(
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), ... 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
moments_hu¶

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
 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. 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
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([7.45370370e01, 3.51165981e01, 1.04049179e01, 4.06442107e02, 2.64312299e03, 2.40854582e02, 4.33680869e19])
moments_normalized¶

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.
 nu(
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
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. ]])
perimeter¶

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...
points_in_poly¶

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 anticlockwise. 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
profile_line¶

skimage.measure.
profile_line
(image, src, dst, linewidth=1, order=None, mode=None, cval=0.0, *, reduce_func=<function mean at 0x7f23a664d950>)[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 05. 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. ]])
ransac¶

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 functionsestimate(*data)
,residuals(*data)
,is_model_valid(model, *random_data)
andis_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 outlierfree 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_inliersarraylike 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])
regionprops¶

skimage.measure.
regionprops
(label_image, intensity_image=None, cache=True, coordinates=None)[source]¶ Measure properties of labeled image regions.
 Parameters
 label_image(N, M) ndarray
Labeled input image. Labels with value 0 are ignored.
Changed in version 0.14.1: Previously,
label_image
was processed bynumpy.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, useregionprops(np.squeeze(label_image), ...)
. intensity_image(N, M) ndarray, optional
Intensity (i.e., input) image with same size as labeled image. Default is None.
 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 scikitimage.
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.
 Returns
 propertieslist of RegionProperties
Each item describes one labeled region, and can be accessed using the attributes listed below.
See also
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 halfopen 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 secondmoments 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 region. Computed as number of objects (= 1) subtracted by number of holes (8connectivity).
 extentfloat
Ratio of pixels in the region to pixels in the total bounding box. Computed as
area / (rows * cols)
 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 counterclockwise.
 perimeterfloat
Perimeter of object which approximates the contour as a line through the centers of border pixels using a 4connectivity.
 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 (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
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 (22.72987986048314, 81.91228523446583) >>> # centroid of first labeled object >>> props[0]['centroid'] (22.72987986048314, 81.91228523446583)
regionprops_table¶

skimage.measure.
regionprops_table
(label_image, intensity_image=None, properties=('label', 'bbox'), *, cache=True, separator='')[source]¶ Compute image properties and return them as a pandascompatible table.
The table is a dictionary mapping column names to value arrays. See Notes section below for details.
 Parameters
 label_image(N, M) ndarray
Labeled input image. Labels with value 0 are ignored.
 intensity_image(N, M) ndarray, optional
Intensity (i.e., input) image with same size as labeled image. Default is None.
 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 nonscalar 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_tensor00
,inertia_tensor01
,inertia_tensor10
, andinertia_tensor11
(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
andcoords
.
 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 = regionprops_table(label_image, image, ... properties=['label', 'inertia_tensor', ... 'inertia_tensor_eigvals']) >>> props {'label': array([ 1, 2, ...]), ... 'inertia_tensor00': array([ 4.012...e+03, 8.51..., ...]), ... ..., 'inertia_tensor_eigvals1': 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_tensor00 ... inertia_tensor_eigvals1 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]
shannon_entropy¶

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
 2
Examples
>>> from skimage import data >>> shannon_entropy(data.camera()) 7.047955232423086
subdivide_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
 coords(N, 2) array
Coordinate array.
 degree{1, 2, 3, 4, 5, 6, 7}, optional
Degree of BSpline. Default is 2.
 preserve_endsbool, optional
Preserve first and last coordinate of noncircular polygon. Default is False.
 Returns
 coords(M, 2) array
Subdivided coordinate array.
References
Examples using skimage.measure.subdivide_polygon
¶
CircleModel
¶

class
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
 paramstuple
Circle model parameters in the following order xc, yc, r.

estimate
(self, data)[source]¶ Estimate circle model from data using total least squares.
 Parameters
 data(N, 2) array
N points with
(x, y)
coordinates, respectively.
 Returns
 successbool
True, if model estimation succeeds.

predict_xy
(self, t, params=None)[source]¶ Predict x and ycoordinates using the estimated model.
 Parameters
 tarray
Angles in circle in radians. Angles start to count from positive xaxis to positive yaxis in a righthanded system.
 params(3, ) array, optional
Optional custom parameter set.
 Returns
 xy(…, 2) array
Predicted x and ycoordinates.
EllipseModel
¶

class
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
 paramstuple
Ellipse model parameters in the following order xc, yc, a, b, theta.

estimate
(self, data)[source]¶ Estimate circle model from data using total least squares.
 Parameters
 data(N, 2) array
N points with
(x, y)
coordinates, respectively.
 Returns
 successbool
True, if model estimation succeeds.
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
(self, t, params=None)[source]¶ Predict x and ycoordinates using the estimated model.
 Parameters
 tarray
Angles in circle in radians. Angles start to count from positive xaxis to positive yaxis in a righthanded system.
 params(5, ) array, optional
Optional custom parameter set.
 Returns
 xy(…, 2) array
Predicted x and ycoordinates.
LineModelND
¶

class
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
 paramstuple
Line model parameters in the following order origin, direction.

estimate
(self, 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
(self, 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
(self, y, params=None)[source]¶ Predict xcoordinates for 2D lines using the estimated model.
Alias for:
predict(y, axis=1)[:, 0]
 Parameters
 yarray
ycoordinates.
 params(2, ) array, optional
Optional custom parameter set in the form (origin, direction).
 Returns
 xarray
Predicted xcoordinates.

predict_y
(self, x, params=None)[source]¶ Predict ycoordinates for 2D lines using the estimated model.
Alias for:
predict(x, axis=0)[:, 1]
 Parameters
 xarray
xcoordinates.
 params(2, ) array, optional
Optional custom parameter set in the form (origin, direction).
 Returns
 yarray
Predicted ycoordinates.

residuals
(self, 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.