measure
Module: measure
skimage.measure.approximate_polygon (coords, ...) | Approximate a polygonal chain with the specified tolerance. |
skimage.measure.block_reduce (image, block_size) | Down-sample image by applying function to local blocks. |
skimage.measure.compare_mse (im1, im2) | Compute the mean-squared error between two images. |
skimage.measure.compare_nrmse (im_true, im_test) | Compute the normalized root mean-squared 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 iso-valued contours in a 2D array for a given level value. |
skimage.measure.grid_points_in_poly | 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, level) | Marching cubes algorithm to find iso-valued surfaces in 3d volumetric data |
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 | 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.structural_similarity (*args, ...) |
Deprecated function. Use compare_ssim instead. |
skimage.measure.subdivide_polygon (coords[, ...]) | Subdivision of polygonal curves using B-Splines. |
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 N-dimensional lines. |
approximate_polygon
-
skimage.measure.approximate_polygon(coords, tolerance)
[source] -
Approximate a polygonal chain with the specified tolerance.
It is based on the Douglas-Peucker 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.
References
[R258] http://en.wikipedia.org/wiki/Ramer-Douglas-Peucker_algorithm
block_reduce
-
skimage.measure.block_reduce(image, block_size, func=<function sum>, cval=0)
[source] -
Down-sample image by applying function to local blocks.
Parameters: image : ndarray
N-dimensional input image.
block_size : array_like
Array containing down-sampling 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
axis
parameter such asnumpy.sum
ornumpy.min
.cval : float
Constant padding value if image is not perfectly divisible by the block size.
Returns: image : ndarray
Down-sampled 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 mean-squared error between two images.
Parameters: im1, im2 : ndarray
Image. Any dimensionality.
Returns: mse : float
The mean-squared error (MSE) metric.
compare_nrmse
-
skimage.measure.compare_nrmse(im_true, im_test, norm_type='Euclidean')
[source] -
Compute the normalized root mean-squared error (NRMSE) between two images.
Parameters: im_true : ndarray
Ground-truth image.
im_test : ndarray
Test image.
norm_type : {‘Euclidean’, ‘min-max’, ‘mean’}
Controls the normalization method to use in the denominator of the NRMSE. There is no standard method of normalization across the literature [R259]. The methods available here are as follows:
- ‘Euclidean’ : normalize by the Euclidean norm of
im_true
. - ‘min-max’ : normalize by the intensity range of
im_true
. - ‘mean’ : normalize by the mean of
im_true
.
Returns: nrmse : float
The NRMSE metric.
References
[R259] (1, 2) https://en.wikipedia.org/wiki/Root-mean-square_deviation - ‘Euclidean’ : normalize by the Euclidean norm of
compare_psnr
-
skimage.measure.compare_psnr(im_true, im_test, dynamic_range=None)
[source] -
Compute the peak signal to noise ratio (PSNR) for an image.
Parameters: im_true : ndarray
Ground-truth image.
im_test : ndarray
Test image.
dynamic_range : int
The dynamic range of the input image (distance between minimum and maximum possible values). By default, this is estimated from the image data-type.
Returns: psnr : float
The PSNR metric.
References
[R260] https://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio
compare_ssim
-
skimage.measure.compare_ssim(X, Y, win_size=None, gradient=False, dynamic_range=None, multichannel=False, gaussian_weights=False, full=False, **kwargs)
[source] -
Compute the mean structural similarity index between two images.
Parameters: X, Y : ndarray
Image. Any dimensionality.
win_size : int or None
The side-length of the sliding window used in comparison. Must be an odd value. If
gaussian_weights
is True, this is ignored and the window size will depend onsigma
.gradient : bool
If True, also return the gradient.
dynamic_range : int
The dynamic range of the input image (distance between minimum and maximum possible values). By default, this is estimated from the image data-type.
multichannel : int or None
If True, treat the last dimension of the array as channels. Similarity calculations are done independently for each channel then averaged.
gaussian_weights : bool
If True, each patch has its mean and variance spatially weighted by a normalized Gaussian kernel of width sigma=1.5.
full : bool
If True, return the full structural similarity image instead of the mean value
Returns: mssim : float or ndarray
The mean structural similarity over the image.
grad : ndarray
The gradient of the structural similarity index between X and Y [R262]. This is only returned if
gradient
is set to True.S : ndarray
The full SSIM image. This is only returned if
full
is set to True.Other Parameters: use_sample_covariance : bool
if True, normalize covariances by N-1 rather than, N where N is the number of pixels within the sliding window.
K1 : float
algorithm parameter, K1 (small constant, see [R261])
K2 : float
algorithm parameter, K2 (small constant, see [R261])
sigma : float
sigma for the Gaussian when
gaussian_weights
is True.Notes
To match the implementation of Wang et. al. [R261], set
gaussian_weights
to True,sigma
to 1.5, anduse_sample_covariance
to False.References
[R261] (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, 600-612. https://ece.uwaterloo.ca/~z70wang/publications/ssim.pdf [R262] (1, 2) Avanaki, A. N. (2009). Exact global histogram specification optimized for structural similarity. Optical Review, 16, 613-621. http://arxiv.org/abs/0901.0065
correct_mesh_orientation
-
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
np.float64
.verts : (V, 3) array of floats
Array containing (x, y, z) coordinates for V unique mesh vertices.
faces : (F, 3) array of ints
List of length-3 lists of integers, referencing vertex coordinates as provided in
verts
.spacing : length-3 tuple of floats
Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in
volume
.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
verts
.Notes
Certain applications and mesh processing algorithms require all faces to be oriented in a consistent way. Generally, this means a normal vector points “out” of the meshed shapes. This algorithm corrects the output from
skimage.measure.marching_cubes
by flipping the orientation of mis-oriented 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
. Onlyfaces
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.
find_contours
-
skimage.measure.find_contours(array, level, fully_connected='low', positive_orientation='low')
[source] -
Find iso-valued contours in a 2D array for a given level value.
Uses the “marching squares” method to compute a the iso-valued 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 fully-connected (and hence elements above the value will only be face connected), or vice-versa. (See notes below for details.)
positive_orientation : either ‘low’ or ‘high’
Indicates whether the output contours will produce positively-oriented polygons around islands of low- or high-valued elements. If ‘low’ then contours will wind counter- clockwise around elements below the iso-value. Alternately, this means that low-valued 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
(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 [R263]. 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 high-valued and two low-valued elements, each pair diagonally adjacent. (Where high- and low-valued is with respect to the contour value sought.) In this case, either the high-valued elements can be ‘connected together’ via a thin isthmus that separates the low-valued elements, or vice-versa. When elements are connected together across a diagonal, they are considered ‘fully connected’ (also known as ‘face+vertex-connected’ or ‘8-connected’). Only high-valued or low-valued elements can be fully-connected, the other set will be considered as ‘face-connected’ or ‘4-connected’. By default, low-valued elements are considered fully-connected; 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 closed-ness 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 counter-clockwise (i.e. in ‘positive orientation’) around islands of low-valued 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 side-effect 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 0-element (atx=0
) and the 1-element (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
[R263] (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. 163-170). 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()
-
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 anti-clockwise. 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
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 2-connected sense. The value refers to the maximum number of orthogonal hops to consider a pixel/voxel a neighbor:
1-connectivity 2-connectivity diagonal connection close-up [ ] [ ] [ ] [ ] [ ] | \ | / | <- 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, 0-valued 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
None
, a full connectivity ofinput.ndim
is used.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
True
.Examples
>>> import numpy as np >>> x = np.eye(3).astype(int) >>> print(x) [[1 0 0] [0 1 0] [0 0 1]] >>> from skimage.measure import label >>> 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, spacing=(1.0, 1.0, 1.0), gradient_direction='descent')
[source] -
Marching cubes algorithm to find iso-valued surfaces in 3d volumetric data
Parameters: volume : (M, N, P) array of doubles
Input data volume to find isosurfaces. Will be cast to
np.float64
.level : float
Contour value to search for isosurfaces in
volume
.spacing : length-3 tuple of floats
Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in
volume
.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
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 [R264]. 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 [R264], 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 [R264], but may be a good beginning for work with more recent techniques (Dual Marching Cubes, Extended Marching Cubes, Cubic Marching Squares, etc.).
Because of interactions between neighboring cubes, the isosurface(s) generated by this algorithm are NOT guaranteed to be closed, particularly for complicated contours. Furthermore, this algorithm does not guarantee a single contour will be returned. Indeed, ALL isosurfaces which cross
level
will be found, regardless of connectivity.The output is a triangular mesh consisting of a set of unique vertices and connecting triangles. The order of these vertices and triangles in the output list is determined by the position of the smallest
x,y,z
(in lexicographical order) coordinate in the contour. This is a side-effect of how the input array is traversed, but can be relied upon.The generated mesh guarantees coherent orientation as of version 0.12.
To quantify the area of an isosurface generated by this algorithm, pass outputs directly into
skimage.measure.mesh_surface_area
.Regarding visualization of algorithm output, the
mayavi
package is recommended. To contour a volume namedmyvolume
about the level 0.0:>>> from mayavi import mlab >>> verts, faces = marching_cubes(myvolume, 0.0, (1., 1., 2.)) >>> mlab.triangular_mesh([vert[0] for vert in verts], ... [vert[1] for vert in verts], ... [vert[2] for vert in verts], ... faces) >>> mlab.show()
References
[R264] (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. 163-170).
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 length-3 lists of integers, referencing vertex coordinates as provided in
verts
Returns: area : float
Surface area of mesh. Units now [coordinate units] ** 2.
Notes
The arguments expected by this function are the exact outputs from
skimage.measure.marching_cubes
. For unit correct output, ensure correctspacing
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[0, 1] / m[0, 0]
,m[1, 0] / m[0, 0]
}.
- Area as:
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 : (
order + 1
,order + 1
) arrayRaw image moments.
References
[R265] Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009. [R266] B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005. [R267] T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. [R268] http://en.wikipedia.org/wiki/Image_moment Examples
>>> image = np.zeros((20, 20), dtype=np.double) >>> image[13:17, 13:17] = 1 >>> m = moments(image) >>> cr = m[0, 1] / m[0, 0] >>> cc = m[1, 0] / m[0, 0] >>> cr, cc (14.5, 14.5)
moments_central
-
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 : (
order + 1
,order + 1
) arrayCentral image moments.
References
[R269] Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009. [R270] B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005. [R271] T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. [R272] http://en.wikipedia.org/wiki/Image_moment Examples
>>> image = np.zeros((20, 20), dtype=np.double) >>> image[13:17, 13:17] = 1 >>> m = moments(image) >>> cr = m[0, 1] / m[0, 0] >>> cc = m[1, 0] / m[0, 0] >>> moments_central(image, cr, cc) array([[ 16., 0., 20., 0.], [ 0., 0., 0., 0.], [ 20., 0., 25., 0.], [ 0., 0., 0., 0.]])
moments_hu
-
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.
References
[R273] M. K. Hu, “Visual Pattern Recognition by Moment Invariants”, IRE Trans. Info. Theory, vol. IT-8, pp. 179-187, 1962 [R274] Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009. [R275] B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005. [R276] T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. [R277] http://en.wikipedia.org/wiki/Image_moment
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 >
order
.order : int, optional
Maximum order of moments. Default is 3.
Returns: nu : (
order + 1
,order + 1
) arrayNormalized central image moments.
References
[R278] Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009. [R279] B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005. [R280] T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. [R281] http://en.wikipedia.org/wiki/Image_moment Examples
>>> image = np.zeros((20, 20), dtype=np.double) >>> image[13:17, 13:17] = 1 >>> m = moments(image) >>> cr = m[0, 1] / m[0, 0] >>> cc = m[1, 0] / m[0, 0] >>> mu = moments_central(image, cr, cc) >>> moments_normalized(mu) array([[ nan, nan, 0.078125 , 0. ], [ nan, 0. , 0. , 0. ], [ 0.078125 , 0. , 0.00610352, 0. ], [ 0. , 0. , 0. , 0. ]])
perimeter
-
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.
References
[R282] 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
points_in_poly
-
skimage.measure.points_in_poly()
-
Test whether points lie inside a polygon.
Parameters: points : (N, 2) array
Input points,
(x, y)
.verts : (M, 2) array
Vertices of the polygon, sorted either clockwise or anti-clockwise. The first point may (but does not need to be) duplicated.
Returns: mask : (N,) array of bool
True if corresponding point is inside the polygon.
See also
profile_line
-
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 : 2-tuple of numeric scalar (float or int)
The start point of the scan line.
dst : 2-tuple 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 non-integer coordinates. 0 means nearest-neighbor interpolation.
mode : {‘constant’, ‘nearest’, ‘reflect’, ‘mirror’, ‘wrap’}, optional
How to compute any values falling outside of the image.
cval : float, optional
If
mode
is ‘constant’, what constant value to use outside the image.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.
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.])
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)
[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 (seeis_data_valid
). - Estimate a model to the random subset (
model_cls.estimate(*data[random_subset]
) and check whether the estimated model is valid (seeis_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 theresidual_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, 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
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_class : object
Object with the following object methods:
success = estimate(*data)
residuals(*data)
where
success
indicates whether the model estimation succeeded (True
orNone
for success,False
for failure).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_data_valid(*random_data)
.is_model_valid : function, optional
This function is called with the estimated model and the randomly selected data:
is_model_valid(model, *random_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 outlier-free set of the training data is sampled with
probability >= stop_probability
, depending on the current best model’s inlier ratio and the number of trials. This requires to generate at least N samples (trials):N >= log(1 - probability) / log(1 - e**m)
where the probability (confidence) is typically set to a high value such as 0.99, and e is the current fraction of inliers w.r.t. the total number of samples.
Returns: model : object
Best model with largest consensus set.
inliers : (N, ) array
Boolean mask of inliers classified as
True
.References
[R283] “RANSAC”, Wikipedia, http://en.wikipedia.org/wiki/RANSAC Examples
Generate ellipse data without tilt and add noise:
>>> t = np.linspace(0, 2 * np.pi, 50) >>> a = 5 >>> b = 10 >>> xc = 20 >>> yc = 30 >>> 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 >>> model.params array([ -3.30354146e+03, -2.87791160e+03, 5.59062118e+03, 7.84365066e+00, 7.19203152e-01])
Estimate ellipse model using RANSAC:
>>> ransac_model, inliers = ransac(data, EllipseModel, 5, 3, max_trials=50) >>> ransac_model.params array([ 20.12762373, 29.73563063, 4.81499637, 10.4743584 , 0.05217117]) >>> 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)
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)
- Select
regionprops
-
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.
Notes
The following properties can be accessed as attributes or keys:
-
area : int
- Number of pixels of region.
-
bbox : tuple
- Bounding box
(min_row, min_col, max_row, max_col)
-
centroid : array
- Centroid coordinate tuple
(row, col)
. -
convex_area : int
- Number of pixels of convex hull image.
-
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. -
eccentricity : float
- Eccentricity of the ellipse that has the same second-moments as the region. The eccentricity is the ratio of the focal distance (distance between focal points) over the major axis length. The value is in the interval [0, 1). When it is 0, the ellipse becomes a circle.
-
equivalent_diameter : float
- The diameter of a circle with the same area as the region.
-
euler_number : int
- Euler characteristic of region. Computed as number of objects (= 1) subtracted by number of holes (8-connectivity).
-
extent : float
- Ratio of pixels in the region to pixels in the total bounding box. Computed as
area / (rows * cols)
-
filled_area : int
- Number of pixels of filled region.
-
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_tensor : (2, 2) ndarray
- Inertia tensor of the region for the rotation around its mass.
-
inertia_tensor_eigvals : tuple
- The two eigen values of the inertia tensor in decreasing order.
-
intensity_image : ndarray
- Image inside region bounding box.
-
label : int
- The label in the labeled input image.
-
local_centroid : array
- Centroid coordinate tuple
(row, col)
, relative to region bounding box. -
major_axis_length : float
- The length of the major axis of the ellipse that has the same normalized second central moments as the region.
-
max_intensity : float
- Value with the greatest intensity in the region.
-
mean_intensity : float
- Value with the mean intensity in the region.
-
min_intensity : float
- Value with the least intensity in the region.
-
minor_axis_length : float
- 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_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, andx_c
andy_c
are the coordinates of the region’s centroid. -
moments_hu : tuple
- Hu moments (translation, scale and rotation invariant).
-
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
- Angle between the X-axis and the major axis of the ellipse that has the same second-moments as the region. Ranging from
-pi/2
topi/2
in counter-clockwise direction. -
perimeter : float
- Perimeter of object which approximates the contour as a line through the centers of border pixels using a 4-connectivity.
-
solidity : float
- Ratio of pixels in the region to pixels of the convex hull image.
-
weighted_centroid : array
- Centroid coordinate tuple
(row, col)
weighted with intensity image. -
weighted_local_centroid : array
- 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_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, andx_c
andy_c
are the coordinates of the region’s weighted centroid. -
weighted_moments_hu : tuple
- 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_ji = wmu_ji / wm_00^[(i+j)/2 + 1]
where
wm_00
is the zeroth spatial moment (intensity-weighted area).
Each region also supports iteration, so that you can do:
for prop in region: print(prop, region[prop])
References
[R284] Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009. [R285] B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005. [R286] T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. [R287] http://en.wikipedia.org/wiki/Image_moment Examples
>>> from skimage import data, util >>> from skimage.measure import label >>> img = util.img_as_ubyte(data.coins()) > 110 >>> label_img = label(img, connectivity=img.ndim) >>> props = regionprops(label_img) >>> # centroid of first labeled object >>> props[0].centroid (22.729879860483141, 81.912285234465827) >>> # centroid of first labeled object >>> props[0]['centroid'] (22.729879860483141, 81.912285234465827)
-
structural_similarity
-
skimage.measure.structural_similarity(*args, **kwargs)
[source] -
Deprecated function. Use
compare_ssim
instead.
subdivide_polygon
-
skimage.measure.subdivide_polygon(coords, degree=2, preserve_ends=False)
[source] -
Subdivision of polygonal curves using B-Splines.
Note that the resulting curve is always within the convex hull of the original polygon. Circular polygons stay closed after subdivision.
Parameters: coords : (N, 2) array
Coordinate array.
degree : {1, 2, 3, 4, 5, 6, 7}, optional
Degree of B-Spline. Default is 2.
preserve_ends : bool, optional
Preserve first and last coordinate of non-circular polygon. Default is False.
Returns: coords : (M, 2) array
Subdivided coordinate array.
References
[R288] http://mrl.nyu.edu/publications/subdiv-course2000/coursenotes00.pdf
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.
Attributes
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
(x, y)
coordinates, respectively.Returns: success : bool
True, if model estimation succeeds.
-
predict_xy(t, params=None)
[source] -
Predict x- and y-coordinates using the estimated model.
Parameters: t : array
Angles in circle in radians. Angles start to count from positive x-axis to positive y-axis in a right-handed system.
params : (3, ) array, optional
Optional custom parameter set.
Returns: xy : (..., 2) array
Predicted x- and y-coordinates.
-
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
(x, y)
coordinates, respectively.Returns: residuals : (N, ) array
Residual for each data point.
-
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.This estimator minimizes the squared distances from all points to the ellipse:
min{ sum(d_i**2) } = min{ sum((x_i - xt)**2 + (y_i - yt)**2) }
Thus you have
2 * N
equations (x_i, y_i) forN + 5
unknowns (t_i, xc, yc, a, b, theta), which gives you an effective redundancy ofN - 5
.The
params
attribute contains the parameters in the following order:xc, yc, a, b, theta
A minimum number of 5 points is required to solve for the parameters.
Attributes
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
(x, y)
coordinates, respectively.Returns: success : bool
True, if model estimation succeeds.
-
predict_xy(t, params=None)
[source] -
Predict x- and y-coordinates using the estimated model.
Parameters: t : array
Angles in circle in radians. Angles start to count from positive x-axis to positive y-axis in a right-handed system.
params : (5, ) array, optional
Optional custom parameter set.
Returns: xy : (..., 2) array
Predicted x- and y-coordinates.
-
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
(x, y)
coordinates, respectively.Returns: residuals : (N, ) array
Residual for each data point.
-
LineModel
-
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.Attributes
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
(x, y)
coordinates, respectively.Returns: success : bool
True, if model estimation succeeds.
-
predict_x(y, params=None)
[source] -
Predict x-coordinates using the estimated model.
Parameters: y : array
y-coordinates.
params : (2, ) array, optional
Optional custom parameter set.
Returns: x : array
Predicted x-coordinates.
-
predict_y(x, params=None)
[source] -
Predict y-coordinates using the estimated model.
Parameters: x : array
x-coordinates.
params : (2, ) array, optional
Optional custom parameter set.
Returns: y : array
Predicted y-coordinates.
-
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
(x, y)
coordinates, respectively.Returns: residuals : (N, ) array
Residual for each data point.
-
LineModelND
-
class skimage.measure.LineModelND
[source] -
Bases:
skimage.measure.fit.BaseModel
Total least squares estimator for N-dimensional lines.
Lines are defined by a point (origin) and a unit vector (direction) according to the following vector equation:
X = origin + lambda * direction
Attributes
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 (
origin
,direction
).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 x-coordinates for 2D lines using the estimated model.
Alias for:
predict(y, axis=1)[:, 0]
Parameters: y : array
y-coordinates.
params : (2, ) array, optional
Optional custom parameter set in the form (
origin
,direction
).Returns: x : array
Predicted x-coordinates.
-
predict_y(x, params=None)
[source] -
Predict y-coordinates for 2D lines using the estimated model.
Alias for:
predict(x, axis=0)[:, 1]
Parameters: x : array
x-coordinates.
params : (2, ) array, optional
Optional custom parameter set in the form (
origin
,direction
).Returns: y : array
Predicted y-coordinates.
-
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 scikit-image team
Licensed under the BSD 3-clause License.
http://scikit-image.org/docs/0.12.x/api/skimage.measure.html