skimage.measure.approximate_polygon (coords, …)  Approximate a polygonal chain with the specified tolerance. 
skimage.measure.block_reduce (image, block_size)  Downsample image by applying function to local blocks. 
skimage.measure.compare_mse (im1, im2)  Compute the meansquared error between two images. 
skimage.measure.compare_nrmse (im_true, im_test)  Compute the normalized root meansquared error (NRMSE) between two images. 
skimage.measure.compare_psnr (im_true, im_test)  Compute the peak signal to noise ratio (PSNR) for an image. 
skimage.measure.compare_ssim (X, Y[, …])  Compute the mean structural similarity index between two images. 
skimage.measure.correct_mesh_orientation (…)  Correct orientations of mesh faces. 
skimage.measure.find_contours (array, level)  Find isovalued contours in a 2D array for a given level value. 
skimage.measure.grid_points_in_poly (shape, verts)  Test whether points on a specified grid are inside a polygon. 
skimage.measure.label (input[, neighbors, …])  Label connected regions of an integer array. 
skimage.measure.marching_cubes (volume[, …])  Lewiner marching cubes algorithm to find surfaces in 3d volumetric data. 
skimage.measure.marching_cubes_classic (volume)  Classic marching cubes algorithm to find surfaces in 3d volumetric data. 
skimage.measure.marching_cubes_lewiner (volume)  Alias for marching_cubes() . 
skimage.measure.mesh_surface_area (verts, faces)  Compute surface area, given vertices & triangular faces 
skimage.measure.moments (image[, order])  Calculate all raw image moments up to a certain order. 
skimage.measure.moments_central (image, cr, cc)  Calculate all central image moments up to a certain order. 
skimage.measure.moments_hu (nu)  Calculate Hu’s set of image moments. 
skimage.measure.moments_normalized (mu[, order])  Calculate all normalized central image moments up to a certain order. 
skimage.measure.perimeter (image[, neighbourhood])  Calculate total perimeter of all objects in binary image. 
skimage.measure.points_in_poly (points, verts)  Test whether points lie inside a polygon. 
skimage.measure.profile_line (img, src, dst)  Return the intensity profile of an image measured along a scan line. 
skimage.measure.ransac (data, model_class, …)  Fit a model to data with the RANSAC (random sample consensus) algorithm. 
skimage.measure.regionprops (label_image[, …])  Measure properties of labeled image regions. 
skimage.measure.shannon_entropy (image[, base])  Calculate the Shannon entropy of an image. 
skimage.measure.structural_similarity (X, Y) 
Deprecated function. Use compare_ssim instead. 
skimage.measure.subdivide_polygon (coords[, …])  Subdivision of polygonal curves using BSplines. 
skimage.measure.CircleModel ()  Total least squares estimator for 2D circles. 
skimage.measure.EllipseModel ()  Total least squares estimator for 2D ellipses. 
skimage.measure.LineModel ()  Total least squares estimator for 2D lines. 
skimage.measure.LineModelND ()  Total least squares estimator for Ndimensional lines. 
skimage.measure.block  
skimage.measure.entropy  
skimage.measure.fit  
skimage.measure.pnpoly  
skimage.measure.profile  
skimage.measure.simple_metrics 
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. tolerance : float 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. 
[R329]  http://en.wikipedia.org/wiki/RamerDouglasPeucker_algorithm 
skimage.measure.block_reduce(image, block_size, func=<function sum>, cval=0)
[source]
Downsample image by applying function to local blocks.
Parameters: 
image : ndarray Ndimensional input image. block_size : array_like Array containing downsampling integer factor along each axis. func : callable Function object which is used to calculate the return value for each local block. This function must implement an cval : float Constant padding value if image is not perfectly divisible by the block size. 

Returns: 
image : ndarray Downsampled image with same number of dimensions as input image. 
>>> 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]]])
skimage.measure.compare_mse(im1, im2)
[source]
Compute the meansquared error between two images.
Parameters: 
im1, im2 : ndarray Image. Any dimensionality. 

Returns: 
mse : float The meansquared error (MSE) metric. 
skimage.measure.compare_nrmse(im_true, im_test, norm_type='Euclidean')
[source]
Compute the normalized root meansquared error (NRMSE) between two images.
Parameters: 
im_true : ndarray Groundtruth image. im_test : ndarray Test image. norm_type : {‘Euclidean’, ‘minmax’, ‘mean’} Controls the normalization method to use in the denominator of the NRMSE. There is no standard method of normalization across the literature [R330]. The methods available here are as follows:


Returns: 
nrmse : float The NRMSE metric. 
[R330]  (1, 2) https://en.wikipedia.org/wiki/Rootmeansquare_deviation 
skimage.measure.compare_psnr(im_true, im_test, data_range=None, dynamic_range=None)
[source]
Compute the peak signal to noise ratio (PSNR) for an image.
Parameters: 
im_true : ndarray Groundtruth image. im_test : ndarray Test image. data_range : int The data range of the input image (distance between minimum and maximum possible values). By default, this is estimated from the image datatype. 

Returns: 
psnr : float The PSNR metric. 
[R331]  https://en.wikipedia.org/wiki/Peak_signaltonoise_ratio 
skimage.measure.compare_ssim(X, Y, win_size=None, gradient=False, data_range=None, multichannel=False, gaussian_weights=False, full=False, dynamic_range=None, **kwargs)
[source]
Compute the mean structural similarity index between two images.
Parameters: 
X, Y : ndarray Image. Any dimensionality. win_size : int or None The sidelength of the sliding window used in comparison. Must be an odd value. If gradient : bool, optional If True, also return the gradient. data_range : int, optional The data range of the input image (distance between minimum and maximum possible values). By default, this is estimated from the image datatype. multichannel : bool, optional If True, treat the last dimension of the array as channels. Similarity calculations are done independently for each channel then averaged. gaussian_weights : bool, optional If True, each patch has its mean and variance spatially weighted by a normalized Gaussian kernel of width sigma=1.5. full : bool, optional If True, return the full structural similarity image instead of the mean value. 

Returns: 
mssim : float The mean structural similarity over the image. grad : ndarray The gradient of the structural similarity index between X and Y [R333]. This is only returned if S : ndarray The full SSIM image. This is only returned if 
Other Parameters:  
use_sample_covariance : bool if True, normalize covariances by N1 rather than, N where N is the number of pixels within the sliding window. K1 : float algorithm parameter, K1 (small constant, see [R332]) K2 : float algorithm parameter, K2 (small constant, see [R332]) sigma : float sigma for the Gaussian when 
To match the implementation of Wang et. al. [R332], set gaussian_weights
to True, sigma
to 1.5, and use_sample_covariance
to False.
[R332]  (1, 2, 3, 4) Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13, 600612. https://ece.uwaterloo.ca/~z70wang/publications/ssim.pdf, DOI:10.1.1.11.2477 
[R333]  (1, 2) Avanaki, A. N. (2009). Exact global histogram specification optimized for structural similarity. Optical Review, 16, 613621. http://arxiv.org/abs/0901.0065, DOI:10.1007/s100430090119z 
skimage.measure.correct_mesh_orientation(volume, verts, faces, spacing=(1.0, 1.0, 1.0), gradient_direction='descent')
[source]
Correct orientations of mesh faces.
Parameters: 
volume : (M, N, P) array of doubles Input data volume to find isosurfaces. Will be cast to 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 spacing : length3 tuple of floats Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in gradient_direction : string 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: 
faces_corrected (F, 3) array of ints Corrected list of faces referencing vertex coordinates in 
Certain applications and mesh processing algorithms require all faces to be oriented in a consistent way. Generally, this means a normal vector points “out” of the meshed shapes. This algorithm corrects the output from skimage.measure.marching_cubes_classic
by flipping the orientation of misoriented faces.
Because marching cubes could be used to find isosurfaces either on gradient descent (where the desired object has greater values than the exterior) or ascent (where the desired object has lower values than the exterior), the gradient_direction
kwarg allows the user to inform this algorithm which is correct. If the resulting mesh appears to be oriented completely incorrectly, try changing this option.
The arguments expected by this function are the exact outputs from skimage.measure.marching_cubes_classic
. Only faces
is corrected and returned, as the vertices do not change; only the order in which they are referenced.
This algorithm assumes faces
provided are all triangles.
skimage.measure.find_contours(array, level, fully_connected='low', positive_orientation='low')
[source]
Find isovalued contours in a 2D array for a given level value.
Uses the “marching squares” method to compute a the isovalued contours of the input 2D array for a particular level value. Array values are linearly interpolated to provide better precision for the output contours.
Parameters: 
array : 2D ndarray of double Input data in which to find contours. level : float Value along which to find contours in the array. fully_connected : str, {‘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_orientation : either ‘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.) 

Returns: 
contours : list of (n,2)ndarrays Each contour is an ndarray of shape 
The marching squares algorithm is a special case of the marching cubes algorithm [R334]. A simple explanation is available here:
http://www.essi.fr/~lingrand/MarchingCubes/algo.html
There is a single ambiguous case in the marching squares algorithm: when a given 2 x 2
element square has two highvalued and two lowvalued elements, each pair diagonally adjacent. (Where high and lowvalued is with respect to the contour value sought.) In this case, either the highvalued elements can be ‘connected together’ via a thin isthmus that separates the lowvalued elements, or viceversa. When elements are connected together across a diagonal, they are considered ‘fully connected’ (also known as ‘face+vertexconnected’ or ‘8connected’). Only highvalued or lowvalued elements can be fullyconnected, the other set will be considered as ‘faceconnected’ or ‘4connected’. By default, lowvalued elements are considered fullyconnected; this can be altered with the ‘fully_connected’ parameter.
Output contours are not guaranteed to be closed: contours which intersect the array edge will be left open. All other contours will be closed. (The closedness of a contours can be tested by checking whether the beginning point is the same as the end point.)
Contours are oriented. By default, array values lower than the contour value are to the left of the contour and values greater than the contour value are to the right. This means that contours will wind counterclockwise (i.e. in ‘positive orientation’) around islands of lowvalued pixels. This behavior can be altered with the ‘positive_orientation’ parameter.
The order of the contours in the output list is determined by the position of the smallest x,y
(in lexicographical order) coordinate in the contour. This is a sideeffect of how the input array is traversed, but can be relied upon.
Warning
Array coordinates/values are assumed to refer to the center of the array element. Take a simple example input: [0, 1]
. The interpolated position of 0.5 in this array is midway between the 0element (at x=0
) and the 1element (at x=1
), and thus would fall at x=0.5
.
This means that to find reasonable contours, it is best to find contours midway between the expected “light” and “dark” values. In particular, given a binarized array, do not choose to find contours at the low or high value of the array. This will often yield degenerate contours, especially around structures that are a single array element wide. Instead choose a middle value, as above.
[R334]  (1, 2) Lorensen, William and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163170). 
>>> a = np.zeros((3, 3)) >>> a[0, 0] = 1 >>> a array([[ 1., 0., 0.], [ 0., 0., 0.], [ 0., 0., 0.]]) >>> find_contours(a, 0.5) [array([[ 0. , 0.5], [ 0.5, 0. ]])]
skimage.measure.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: 
shape : tuple (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
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: 
input : ndarray 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. background : int, 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_num : bool, optional Whether to return the number of assigned labels. connectivity : int, optional Maximum number of orthogonal hops to consider a pixel/voxel as a neighbor. Accepted values are ranging from 1 to input.ndim. If 

Returns: 
labels : ndarray of dtype int Labeled array, where all connected regions are assigned the same integer value. num : int, optional Number of labels, which equals the maximum label index and is only returned if return_num is 
See also
[R335]  Christophe Fiorio and Jens Gustedt, “Two linear time UnionFind strategies for image processing”, Theoretical Computer Science 154 (1996), pp. 165181. 
[R336]  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 
>>> import numpy as np >>> x = np.eye(3).astype(int) >>> print(x) [[1 0 0] [0 1 0] [0 0 1]] >>> print(label(x, connectivity=1)) [[1 0 0] [0 2 0] [0 0 3]] >>> print(label(x, connectivity=2)) [[1 0 0] [0 1 0] [0 0 1]] >>> print(label(x, background=1)) [[1 2 2] [2 1 2] [2 2 1]] >>> x = np.array([[1, 0, 0], ... [1, 1, 5], ... [0, 0, 0]]) >>> print(label(x)) [[1 0 0] [1 1 2] [0 0 0]]
skimage.measure.marching_cubes(volume, level=None, spacing=(1.0, 1.0, 1.0), gradient_direction='descent', step_size=1, allow_degenerate=True, use_classic=False)
[source]
Lewiner marching cubes algorithm to find surfaces in 3d volumetric data.
In contrast to marching_cubes_classic()
, this algorithm is faster, resolves ambiguities, and guarantees topologically correct results. Therefore, this algorithm generally a better choice, unless there is a specific need for the classic algorithm.
Parameters: 
volume : (M, N, P) array Input data volume to find isosurfaces. Will internally be converted to float32 if necessary. level : float Contour value to search for isosurfaces in spacing : length3 tuple of floats Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in gradient_direction : string 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_size : int Step size in voxels. Default 1. Larger steps yield faster but coarser results. The result will always be topologically correct though. allow_degenerate : bool 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_classic : bool 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 

Returns: 
verts : (V, 3) array Spatial coordinates for V unique mesh vertices. Coordinate order matches input faces : (F, 3) array Define triangular faces via referencing vertex indices from 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. 
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(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_classic(myvolume, 0.0) >>> vv.mesh(np.fliplr(verts), faces, normals, values) >>> vv.use().Run()
[R337]  Thomas Lewiner, Helio Lopes, Antonio Wilson Vieira and Geovan Tavares. Efficient implementation of Marching Cubes’ cases with topological guarantees. Journal of Graphics Tools 8(2) pp. 115 (december 2003). DOI: 10.1080/10867651.2003.10487582 
skimage.measure.marching_cubes_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 level : float Contour value to search for isosurfaces in spacing : length3 tuple of floats Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in gradient_direction : string 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 faces : (F, 3) array Define triangular faces via referencing vertex indices from 
The marching cubes algorithm is implemented as described in [R338]. A simple explanation is available here:
http://www.essi.fr/~lingrand/MarchingCubes/algo.html
There are several known ambiguous cases in the marching cubes algorithm. Using point labeling as in [R338], 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 [R338], 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
.
[R338]  (1, 2, 3, 4) Lorensen, William and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163170). DOI: 10.1145/37401.37422 
skimage.measure.marching_cubes_lewiner(volume, level=None, spacing=(1.0, 1.0, 1.0), gradient_direction='descent', step_size=1, allow_degenerate=True, use_classic=False)
[source]
Alias for marching_cubes()
.
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 

Returns: 
area : float Surface area of mesh. Units now [coordinate units] ** 2. 
See also
skimage.measure.marching_cubes
, skimage.measure.marching_cubes_classic
, skimage.measure.correct_mesh_orientation
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.
m[0, 0]
.m[0, 1] / m[0, 0]
, m[1, 0] / m[0, 0]
}.Note that raw moments are neither translation, scale nor rotation invariant.
Parameters: 
image : 2D double or uint8 array Rasterized shape as image. order : int, optional Maximum order of moments. Default is 3. 

Returns: 
m : ( Raw image moments. 
[R339]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R340]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R341]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R342]  http://en.wikipedia.org/wiki/Image_moment 
>>> image = np.zeros((20, 20), dtype=np.double) >>> image[13:17, 13:17] = 1 >>> m = moments(image) >>> cr = m[0, 1] / m[0, 0] >>> cc = m[1, 0] / m[0, 0] >>> cr, cc (14.5, 14.5)
skimage.measure.moments_central(image, cr, cc, order=3)
[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[0, 1] / m[0, 0]
, m[1, 0] / m[0, 0]
}.
Note that central moments are translation invariant but not scale and rotation invariant.
Parameters: 
image : 2D double or uint8 array Rasterized shape as image. cr : double Center row coordinate. cc : double Center column coordinate. order : int, optional Maximum order of moments. Default is 3. 

Returns: 
mu : ( Central image moments. 
[R343]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R344]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R345]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R346]  http://en.wikipedia.org/wiki/Image_moment 
>>> image = np.zeros((20, 20), dtype=np.double) >>> image[13:17, 13:17] = 1 >>> m = moments(image) >>> cr = m[0, 1] / m[0, 0] >>> cc = m[1, 0] / m[0, 0] >>> moments_central(image, cr, cc) array([[ 16., 0., 20., 0.], [ 0., 0., 0., 0.], [ 20., 0., 25., 0.], [ 0., 0., 0., 0.]])
skimage.measure.moments_hu(nu)
[source]
Calculate Hu’s set of image moments.
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, 1) array Hu’s set of image moments. 
[R347]  M. K. Hu, “Visual Pattern Recognition by Moment Invariants”, IRE Trans. Info. Theory, vol. IT8, pp. 179187, 1962 
[R348]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R349]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R350]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R351]  http://en.wikipedia.org/wiki/Image_moment 
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 > order : int, optional Maximum order of moments. Default is 3. 

Returns: 
nu : ( Normalized central image moments. 
[R352]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R353]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R354]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R355]  http://en.wikipedia.org/wiki/Image_moment 
>>> image = np.zeros((20, 20), dtype=np.double) >>> image[13:17, 13:17] = 1 >>> m = moments(image) >>> cr = m[0, 1] / m[0, 0] >>> cc = m[1, 0] / m[0, 0] >>> mu = moments_central(image, cr, cc) >>> moments_normalized(mu) array([[ nan, nan, 0.078125 , 0. ], [ nan, 0. , 0. , 0. ], [ 0.078125 , 0. , 0.00610352, 0. ], [ 0. , 0. , 0. , 0. ]])
skimage.measure.perimeter(image, neighbourhood=4)
[source]
Calculate total perimeter of all objects in binary image.
Parameters: 
image : array Binary image. neighbourhood : 4 or 8, optional Neighborhood connectivity for border pixel determination. 

Returns: 
perimeter : float Total perimeter of all objects in binary image. 
[R356]  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 
skimage.measure.points_in_poly(points, verts)
[source]
Test whether points lie inside a polygon.
Parameters: 
points : (N, 2) array Input points, 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
skimage.measure.profile_line(img, src, dst, linewidth=1, order=1, mode='constant', cval=0.0)
[source]
Return the intensity profile of an image measured along a scan line.
Parameters: 
img : numeric array, shape (M, N[, C]) The image, either grayscale (2D array) or multichannel (3D array, where the final axis contains the channel information). src : 2tuple of numeric scalar (float or int) The start point of the scan line. dst : 2tuple of numeric scalar (float or int) The end point of the scan line. The destination point is included in the profile, in constrast to standard numpy indexing. linewidth : int, optional Width of the scan, perpendicular to the line order : int in {0, 1, 2, 3, 4, 5}, optional The order of the spline interpolation to compute image values at noninteger coordinates. 0 means nearestneighbor interpolation. mode : {‘constant’, ‘nearest’, ‘reflect’, ‘mirror’, ‘wrap’}, optional How to compute any values falling outside of the image. cval : float, optional If 

Returns: 
return_value : array The intensity profile along the scan line. The length of the profile is the ceil of the computed length of the scan line. 
>>> x = np.array([[1, 1, 1, 2, 2, 2]]) >>> img = np.vstack([np.zeros_like(x), x, x, x, np.zeros_like(x)]) >>> img array([[0, 0, 0, 0, 0, 0], [1, 1, 1, 2, 2, 2], [1, 1, 1, 2, 2, 2], [1, 1, 1, 2, 2, 2], [0, 0, 0, 0, 0, 0]]) >>> profile_line(img, (2, 1), (2, 4)) array([ 1., 1., 2., 2.]) >>> profile_line(img, (1, 0), (1, 6), cval=4) array([ 1., 1., 1., 2., 2., 2., 4.])
The destination point is included in the profile, in contrast to standard numpy indexing. For example: >>> profile_line(img, (1, 0), (1, 6)) # The final point is out of bounds array([ 1., 1., 1., 2., 2., 2., 0.]) >>> profile_line(img, (1, 0), (1, 5)) # This accesses the full first row array([ 1., 1., 1., 2., 2., 2.])
skimage.measure.ransac(data, model_class, min_samples, residual_threshold, is_data_valid=None, is_model_valid=None, max_trials=100, stop_sample_num=inf, stop_residuals_sum=0, stop_probability=1, random_state=None)
[source]
Fit a model to data with the RANSAC (random sample consensus) algorithm.
RANSAC is an iterative algorithm for the robust estimation of parameters from a subset of inliers from the complete data set. Each iteration performs the following tasks:
min_samples
random samples from the original data and check whether the set of data is valid (see is_data_valid
).model_cls.estimate(*data[random_subset]
) and check whether the estimated model is valid (see is_model_valid
).model_cls.residuals(*data)
)  all data samples with residuals smaller than the residual_threshold
are considered as inliers.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, D) array Data set to which the model is fitted, where N is the number of data points and D the dimensionality of the data. If the model class requires multiple input data arrays (e.g. source and destination coordinates of model_class : object Object with the following object methods:
where min_samples : int The minimum number of data points to fit a model to. residual_threshold : float Maximum distance for a data point to be classified as an inlier. is_data_valid : function, optional This function is called with the randomly selected data before the model is fitted to it: is_model_valid : function, optional This function is called with the estimated model and the randomly selected data: max_trials : int, optional Maximum number of iterations for random sample selection. stop_sample_num : int, optional Stop iteration if at least this number of inliers are found. stop_residuals_sum : float, optional Stop iteration if sum of residuals is less than or equal to this threshold. stop_probability : float in range [0, 1], optional RANSAC iteration stops if at least one outlierfree set of the training data is sampled with N >= log(1  probability) / log(1  e**m) where the probability (confidence) is typically set to a high value such as 0.99, and e is the current fraction of inliers w.r.t. the total number of samples. random_state : int, 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 

Returns: 
model : object Best model with largest consensus set. inliers : (N, ) array Boolean mask of inliers classified as 
[R357]  “RANSAC”, Wikipedia, http://en.wikipedia.org/wiki/RANSAC 
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
Robustly estimate geometric transformation:
>>> from skimage.transform import SimilarityTransform >>> np.random.seed(0) >>> src = 100 * np.random.rand(50, 2) >>> model0 = SimilarityTransform(scale=0.5, rotation=1, ... translation=(10, 20)) >>> dst = model0(src) >>> dst[0] = (10000, 10000) >>> dst[1] = (100, 100) >>> dst[2] = (50, 50) >>> model, inliers = ransac((src, dst), SimilarityTransform, 2, 10) >>> inliers array([False, False, False, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True], dtype=bool)
skimage.measure.regionprops(label_image, intensity_image=None, cache=True)
[source]
Measure properties of labeled image regions.
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. cache : bool, optional Determine whether to cache calculated properties. The computation is much faster for cached properties, whereas the memory consumption increases. 

Returns: 
properties : list of RegionProperties Each item describes one labeled region, and can be accessed using the attributes listed below. 
See also
The following properties can be accessed as attributes or keys:
area : int
bbox : tuple
(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_area : int
centroid : array
(row, col)
.convex_area : int
convex_image : (H, J) ndarray
coords : (N, 2) ndarray
(row, col)
of the region.eccentricity : float
equivalent_diameter : float
euler_number : int
extent : float
area / (rows * cols)
filled_area : int
filled_image : (H, J) ndarray
image : (H, J) ndarray
inertia_tensor : (2, 2) ndarray
inertia_tensor_eigvals : tuple
intensity_image : ndarray
label : int
local_centroid : array
(row, col)
, relative to region bounding box.major_axis_length : float
max_intensity : float
mean_intensity : float
min_intensity : float
minor_axis_length : float
moments : (3, 3) ndarray
Spatial moments up to 3rd order:
m_ji = sum{ array(x, y) * x^j * y^i }
where the sum is over the x
, y
coordinates of the region.
moments_central : (3, 3) ndarray
Central moments (translation invariant) up to 3rd order:
mu_ji = sum{ array(x, y) * (x  x_c)^j * (y  y_c)^i }
where the sum is over the x
, y
coordinates of the region, and x_c
and y_c
are the coordinates of the region’s centroid.
moments_hu : tuple
moments_normalized : (3, 3) ndarray
Normalized moments (translation and scale invariant) up to 3rd order:
nu_ji = mu_ji / m_00^[(i+j)/2 + 1]
where m_00
is the zeroth spatial moment.
orientation : float
pi/2
to pi/2
in counterclockwise direction.perimeter : float
solidity : float
weighted_centroid : array
(row, col)
weighted with intensity image.weighted_local_centroid : array
(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_ji = sum{ array(x, y) * x^j * y^i }
where the sum is over the x
, y
coordinates of the region.
weighted_moments_central : (3, 3) ndarray
Central moments (translation invariant) of intensity image up to 3rd order:
wmu_ji = sum{ array(x, y) * (x  x_c)^j * (y  y_c)^i }
where the sum is over the x
, y
coordinates of the region, and x_c
and y_c
are the coordinates of the region’s weighted centroid.
weighted_moments_hu : tuple
weighted_moments_normalized : (3, 3) ndarray
Normalized moments (translation and scale invariant) of intensity image up to 3rd order:
wnu_ji = wmu_ji / wm_00^[(i+j)/2 + 1]
where wm_00
is the zeroth spatial moment (intensityweighted area).
Each region also supports iteration, so that you can do:
for prop in region: print(prop, region[prop])
[R358]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R359]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R360]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R361]  http://en.wikipedia.org/wiki/Image_moment 
>>> from skimage import data, util >>> from skimage.measure import label >>> img = util.img_as_ubyte(data.coins()) > 110 >>> label_img = label(img, connectivity=img.ndim) >>> props = regionprops(label_img) >>> # centroid of first labeled object >>> props[0].centroid (22.729879860483141, 81.912285234465827) >>> # centroid of first labeled object >>> props[0]['centroid'] (22.729879860483141, 81.912285234465827)
skimage.measure.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 the number of pixels of value k.
Parameters: 
image : (N, M) ndarray Grayscale input image. base : float, optional The logarithmic base to use. 

Returns: 
entropy : float 
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.
[R362]  https://en.wikipedia.org/wiki/Entropy_(information_theory) 
[R363]  https://en.wiktionary.org/wiki/Shannon_entropy 
>>> from skimage import data >>> shannon_entropy(data.camera()) 17.732031303342747
skimage.measure.structural_similarity(X, Y, win_size=None, gradient=False, dynamic_range=None, multichannel=False, gaussian_weights=False, full=False, **kwargs)
[source]
Deprecated function. Use compare_ssim
instead.
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_ends : bool, optional Preserve first and last coordinate of noncircular polygon. Default is False. 

Returns: 
coords : (M, 2) array Subdivided coordinate array. 
[R364]  http://mrl.nyu.edu/publications/subdivcourse2000/coursenotes00.pdf 
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.
params  (tuple) Circle model parameters in the following order xc , yc , r . 
__init__()
[source]
estimate(data)
[source]
Estimate circle model from data using total least squares.
Parameters: 
data : (N, 2) array N points with 

Returns: 
success : bool True, if model estimation succeeds. 
predict_xy(t, params=None)
[source]
Predict x and ycoordinates using the estimated model.
Parameters: 
t : array 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. 
residuals(data)
[source]
Determine residuals of data to model.
For each point the shortest distance to the circle is returned.
Parameters: 
data : (N, 2) array N points with 

Returns: 
residuals : (N, ) array Residual for each data point. 
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
>>> 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.])
params  (tuple) Ellipse model parameters in the following order xc , yc , a , b , theta . 
__init__()
[source]
estimate(data)
[source]
Estimate circle model from data using total least squares.
Parameters: 
data : (N, 2) array N points with 

Returns: 
success : bool True, if model estimation succeeds. 
[R365]  Halir, R.; Flusser, J. “Numerically stable direct least squares fitting of ellipses”. In Proc. 6th International Conference in Central Europe on Computer Graphics and Visualization. WSCG (Vol. 98, pp. 125132). 
predict_xy(t, params=None)
[source]
Predict x and ycoordinates using the estimated model.
Parameters: 
t : array 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. 
residuals(data)
[source]
Determine residuals of data to model.
For each point the shortest distance to the ellipse is returned.
Parameters: 
data : (N, 2) array N points with 

Returns: 
residuals : (N, ) array Residual for each data point. 
class skimage.measure.LineModel
[source]
Bases: skimage.measure.fit.BaseModel
Total least squares estimator for 2D lines.
Lines are parameterized using polar coordinates as functional model:
dist = x * cos(theta) + y * sin(theta)
This parameterization is able to model vertical lines in contrast to the standard line model y = a*x + b
.
This estimator minimizes the squared distances from all points to the line:
min{ sum((dist  x_i * cos(theta) + y_i * sin(theta))**2) }
A minimum number of 2 points is required to solve for the parameters.
Deprecated class. Use LineModelND
instead.
params  (tuple) Line model parameters in the following order dist , theta . 
__init__()
[source]
estimate(data)
[source]
Estimate line model from data using total least squares.
Parameters: 
data : (N, 2) array N points with 

Returns: 
success : bool True, if model estimation succeeds. 
predict_x(y, params=None)
[source]
Predict xcoordinates using the estimated model.
Parameters: 
y : array ycoordinates. params : (2, ) array, optional Optional custom parameter set. 

Returns: 
x : array Predicted xcoordinates. 
predict_y(x, params=None)
[source]
Predict ycoordinates using the estimated model.
Parameters: 
x : array xcoordinates. params : (2, ) array, optional Optional custom parameter set. 

Returns: 
y : array Predicted ycoordinates. 
residuals(data)
[source]
Determine residuals of data to model.
For each point the shortest distance to the line is returned.
Parameters: 
data : (N, 2) array N points with 

Returns: 
residuals : (N, ) array Residual for each data point. 
class skimage.measure.LineModelND
[source]
Bases: skimage.measure.fit.BaseModel
Total least squares estimator for Ndimensional lines.
Lines are defined by a point (origin) and a unit vector (direction) according to the following vector equation:
X = origin + lambda * direction
params  (tuple) Line model parameters in the following order origin , direction . 
__init__()
[source]
estimate(data)
[source]
Estimate line model from data.
Parameters: 
data : (N, dim) array N points in a space of dimensionality dim >= 2. 

Returns: 
success : bool 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 : array coordinates along an axis. axis : int axis orthogonal to the hyperplane intersecting the line. params : (2, ) array, optional Optional custom parameter set in the form ( 

Returns: 
y : array Predicted coordinates. If the line is parallel to the given axis, a ValueError is raised. 
predict_x(y, params=None)
[source]
Predict xcoordinates for 2D lines using the estimated model.
Alias for:
predict(y, axis=1)[:, 0]
Parameters: 
y : array ycoordinates. params : (2, ) array, optional Optional custom parameter set in the form ( 

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

Returns: 
y : array Predicted ycoordinates. 
residuals(data)
[source]
Determine residuals of data to model.
For each point the shortest 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. 

Returns: 
residuals : (N, ) array Residual for each data point. 
© 2011 the scikitimage team
Licensed under the BSD 3clause License.
http://scikitimage.org/docs/0.13.x/api/skimage.measure.html