Module: measure¶

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.

References

Down-sample image by applying function to local blocks.

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]]])

Compute the mean-squared error between two images.

Compute the normalized root mean-squared error (NRMSE) between two images.

References

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

References

Compute the mean structural similarity index between two images.

Notes

To match the implementation of Wang et. al. [R663664], set gaussian_weights to True, sigma to 1.5, and use_sample_covariance to False.

References

Correct orientations of mesh faces.

Notes

Certain applications and mesh processing algorithms require all faces to be oriented in a consistent way. Generally, this means a normal vector points “out” of the meshed shapes. This algorithm corrects the output from skimage.measure.marching_cubes_classic by flipping the orientation of 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_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.

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.

Notes

The marching squares algorithm is a special case of the marching cubes algorithm [R667667]. 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.

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

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. ]])]

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.

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
     [ ]           [ ]  [ ]  [ ]

References

Examples

>>> import numpy as np
>>> x = np.eye(3).astype(int)
>>> print(x)
[[1 0 0]
 [0 1 0]
 [0 0 1]]
>>> print(label(x, connectivity=1))
[[1 0 0]
 [0 2 0]
 [0 0 3]]
>>> print(label(x, connectivity=2))
[[1 0 0]
 [0 1 0]
 [0 0 1]]
>>> print(label(x, background=-1))
[[1 2 2]
 [2 1 2]
 [2 2 1]]
>>> x = np.array([[1, 0, 0],
...               [1, 1, 5],
...               [0, 0, 0]])
>>> print(label(x))
[[1 0 0]
 [1 1 2]
 [0 0 0]]

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.

Notes

The algorithm [1] is an improved version of Chernyaev’s Marching Cubes 33 algorithm. It is an efficient algorithm that relies on heavy use of lookup tables to handle the many different cases, keeping the algorithm relatively easy. This implementation is written in Cython, ported from Lewiner’s C++ implementation.

To quantify the area of an isosurface generated by this algorithm, pass verts and faces to skimage.measure.mesh_surface_area.

Regarding visualization of algorithm output, to contour a volume named myvolume about the level 0.0, using the mayavi package:

>>> from mayavi import mlab 
>>> verts, faces, normals, values = marching_cubes(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() 

References

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.

Notes

The marching cubes algorithm is implemented as described in [R675675]. 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 [R675675], 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 [R675675], but may be a good beginning for work with more recent techniques (Dual Marching Cubes, Extended Marching Cubes, Cubic Marching Squares, etc.).

Because of interactions between neighboring cubes, the isosurface(s) generated by this algorithm are NOT guaranteed to be closed, particularly for complicated contours. Furthermore, this algorithm does not guarantee a single contour will be returned. Indeed, ALL isosurfaces which cross level will be found, regardless of connectivity.

The output is a triangular mesh consisting of a set of unique vertices and connecting triangles. The order of these vertices and triangles in the output list is determined by the position of the smallest x,y,z (in lexicographical order) coordinate in the contour. This is a side-effect of how the input array is traversed, but can be relied upon.

The generated mesh guarantees coherent orientation as of version 0.12.

To quantify the area of an isosurface generated by this algorithm, pass outputs directly into skimage.measure.mesh_surface_area.

References

Alias for marching_cubes().

Compute surface area, given vertices & triangular faces

Notes

The arguments expected by this function are the first two outputs from skimage.measure.marching_cubes. For unit correct output, ensure correct spacing was passed to skimage.measure.marching_cubes.

This algorithm works properly only if the faces provided are all triangles.

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]}.

Note that raw moments are neither translation, scale nor rotation invariant.

References

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)

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.

References

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.]])

Calculate Hu’s set of image moments.

Note that this set of moments is proofed to be translation, scale and rotation invariant.

References

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.

References

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.        ]])

Calculate total perimeter of all objects in binary image.

References

Test whether points lie inside a polygon.

Return the intensity profile of an image measured along a 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.])

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:

1. Select min_samples random samples from the original data and check whether the set of data is valid (see is_data_valid).
2. Estimate a model to the random subset (model_cls.estimate(*data[random_subset]) and check whether the estimated model is valid (see is_model_valid).
3. Classify all data as inliers or outliers by calculating the residuals to the estimated model (model_cls.residuals(*data)) - all data samples with residuals smaller than the residual_threshold are considered as inliers.
4. 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.

References

Examples

Generate ellipse data without tilt and add noise:

>>> t = np.linspace(0, 2 * np.pi, 50)
>>> xc, yc = 20, 30
>>> a, b = 5, 10
>>> x = xc + a * np.cos(t)
>>> y = yc + b * np.sin(t)
>>> data = np.column_stack([x, y])
>>> np.random.seed(seed=1234)
>>> data += np.random.normal(size=data.shape)

Add some faulty data:

>>> data[0] = (100, 100)
>>> data[1] = (110, 120)
>>> data[2] = (120, 130)
>>> data[3] = (140, 130)

Estimate ellipse model using all available data:

>>> model = EllipseModel()
>>> model.estimate(data)
True
>>> np.round(model.params)  
array([ 72.,  75.,  77.,  14.,   1.])

Estimate ellipse model using RANSAC:

>>> ransac_model, inliers = ransac(data, EllipseModel, 20, 3, max_trials=50)
>>> abs(np.round(ransac_model.params))
array([ 20.,  30.,   5.,  10.,   0.])
>>> inliers 
array([False, False, False, False,  True,  True,  True,  True,  True,
        True,  True,  True,  True,  True,  True,  True,  True,  True,
        True,  True,  True,  True,  True,  True,  True,  True,  True,
        True,  True,  True,  True,  True,  True,  True,  True,  True,
        True,  True,  True,  True,  True,  True,  True,  True,  True,
        True,  True,  True,  True,  True], dtype=bool)
>>> sum(inliers) > 40
True

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)

Measure properties of labeled image regions.

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

bbox_area : int

Number of pixels of bounding box.

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, and x_c and y_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 to pi/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, and x_c and y_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

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)

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.

Notes

The returned value is measured in bits or shannon (Sh) for base=2, natural unit (nat) for base=np.e and hartley (Hart) for base=10.

References

Examples

>>> from skimage import data
>>> shannon_entropy(data.camera())
17.732031303342747

Deprecated function. Use compare_ssim instead.

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.

References