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# Module: measure¶

 skimage.measure.find_contours(array, level) Find iso-valued contours in a 2D array for a given level value. skimage.measure.regionprops(label_image[, …]) Measure properties of labeled image regions. skimage.measure.perimeter(image[, neighbourhood]) Calculate total perimeter of all objects in binary image. Approximate a polygonal chain with the specified tolerance. skimage.measure.subdivide_polygon(coords[, …]) Subdivision of polygonal curves using B-Splines. skimage.measure.ransac(data, model_class, …) Fit a model to data with the RANSAC (random sample consensus) algorithm. skimage.measure.block_reduce(image, block_size) Down-sample image by applying function to local blocks. skimage.measure.moments(image[, order]) Calculate all raw image moments up to a certain order. skimage.measure.moments_central(image[, …]) Calculate all central image moments up to a certain order. skimage.measure.moments_coords(coords[, order]) Calculate all raw image moments up to a certain order. Calculate all central image moments up to a certain order. skimage.measure.moments_normalized(mu[, order]) Calculate all normalized central image moments up to a certain order. Calculate Hu’s set of image moments (2D-only). Lewiner marching cubes algorithm to find surfaces in 3d volumetric data. Classic marching cubes algorithm to find surfaces in 3d volumetric data. skimage.measure.mesh_surface_area(verts, faces) Compute surface area, given vertices & triangular faces Correct orientations of mesh faces. skimage.measure.profile_line(image, src, dst) Return the intensity profile of an image measured along a scan line. skimage.measure.label(input[, neighbors, …]) Label connected regions of an integer array. skimage.measure.points_in_poly(points, verts) Test whether points lie inside a polygon. skimage.measure.grid_points_in_poly(shape, verts) Test whether points on a specified grid are inside a polygon. skimage.measure.compare_ssim(X, Y[, …]) Compute the mean structural similarity index between two images. 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.shannon_entropy(image[, base]) Calculate the Shannon entropy of an image. Total least squares estimator for N-dimensional lines. Total least squares estimator for 2D circles. Total least squares estimator for 2D ellipses.

## 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
array2D ndarray of double

Input data in which to find contours.

levelfloat

Value along which to find contours in the array.

fully_connectedstr, {‘low’, ‘high’}

Indicates whether array elements below the given level value are to be considered fully-connected (and hence elements above the value will only be face connected), or vice-versa. (See notes below for details.)

positive_orientationeither ‘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
contourslist of (n,2)-ndarrays

Each contour is an ndarray of shape (n, 2), consisting of n (row, column) coordinates along the contour.

Notes

The marching squares algorithm is a special case of the marching cubes algorithm [1]. A simple explanation is available here:

http://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 (at x=0) and the 1-element (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.

References

1

Lorensen, William and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 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. ]])]


## regionprops¶

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

Measure properties of labeled image regions.

Parameters
label_image(N, M) ndarray

Labeled input image. Labels with value 0 are ignored.

Changed in version 0.14.2: Previously, label_image was processed by numpy.squeeze and so any number of singleton dimensions was allowed. This resulted in inconsistent handling of images with singleton dimensions. To recover the old behaviour, use regionprops(np.squeeze(label_image), ...).

intensity_image(N, M) ndarray, optional

Intensity (i.e., input) image with same size as labeled image. Default is None.

cachebool, optional

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

coordinates‘rc’ or ‘xy’, optional

Coordinate conventions for 2D images. (Only ‘rc’ coordinates are supported for 3D images.)

Returns
propertieslist 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:

areaint

Number of pixels of region.

bboxtuple

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

bbox_areaint

Number of pixels of bounding box.

centroidarray

Centroid coordinate tuple (row, col).

convex_areaint

Number of pixels of convex hull image.

convex_image(H, J) ndarray

Binary convex hull image which has the same size as bounding box.

coords(N, 2) ndarray

Coordinate list (row, col) of the region.

eccentricityfloat

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

equivalent_diameterfloat

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

euler_numberint

Euler characteristic of region. Computed as number of objects (= 1) subtracted by number of holes (8-connectivity).

extentfloat

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

filled_areaint

Number of pixels of 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_eigvalstuple

The two eigen values of the inertia tensor in decreasing order.

intensity_imagendarray

Image inside region bounding box.

labelint

The label in the labeled input image.

local_centroidarray

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

major_axis_lengthfloat

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

max_intensityfloat

Value with the greatest intensity in the region.

mean_intensityfloat

Value with the mean intensity in the region.

min_intensityfloat

Value with the least intensity in the region.

minor_axis_lengthfloat

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

moments(3, 3) ndarray

Spatial moments up to 3rd order:

m_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_hutuple

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.

orientationfloat

In ‘rc’ coordinates, angle between the 0th axis (rows) and the major axis of the ellipse that has the same second moments as the region, ranging from -pi/2 to pi/2 counter-clockwise.

In xy coordinates, as above but the angle is now measured from the “x” or horizontal axis.

perimeterfloat

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

solidityfloat

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

weighted_centroidarray

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

weighted_local_centroidarray

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

weighted_moments(3, 3) ndarray

Spatial moments of intensity image up to 3rd order:

wm_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_hutuple

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

weighted_moments_normalized(3, 3) ndarray

Normalized moments (translation and scale invariant) of intensity image up to 3rd order:

wnu_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

1

Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.

2

B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005.

3

T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993.

4

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)


## perimeter¶

skimage.measure.perimeter(image, neighbourhood=4)[source]

Calculate total perimeter of all objects in binary image.

Parameters
imagearray

Binary image.

neighbourhood4 or 8, optional

Neighborhood connectivity for border pixel determination.

Returns
perimeterfloat

Total perimeter of all objects in binary image.

References

1

K. Benkrid, D. Crookes. Design and FPGA Implementation of a Perimeter Estimator. The Queen’s University of Belfast. http://www.cs.qub.ac.uk/~d.crookes/webpubs/papers/perimeter.doc

## 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.

tolerancefloat

Maximum distance from original points of polygon to approximated polygonal chain. If tolerance is 0, the original coordinate array is returned.

Returns
coords(M, 2) array

Approximated polygonal chain where M <= N.

References

1

http://en.wikipedia.org/wiki/Ramer-Douglas-Peucker_algorithm

## 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_endsbool, optional

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

Returns
coords(M, 2) array

Subdivided coordinate array.

References

1

http://mrl.nyu.edu/publications/subdiv-course2000/coursenotes00.pdf

## ransac¶

skimage.measure.ransac(data, model_class, min_samples, residual_threshold, is_data_valid=None, is_model_valid=None, max_trials=100, stop_sample_num=inf, stop_residuals_sum=0, stop_probability=1, random_state=None)[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:

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.

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 functions estimate(*data), residuals(*data), is_model_valid(model, *random_data) and is_data_valid(*random_data) must all take each data array as separate arguments.

model_classobject

Object with the following object methods:

• success = estimate(*data)

• residuals(*data)

where success indicates whether the model estimation succeeded (True or None for success, False for failure).

min_samplesint

The minimum number of data points to fit a model to.

residual_thresholdfloat

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

is_data_validfunction, optional

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

is_model_validfunction, optional

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

max_trialsint, optional

Maximum number of iterations for random sample selection.

stop_sample_numint, optional

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

stop_residuals_sumfloat, optional

Stop iteration if sum of residuals is less than or equal to this threshold.

stop_probabilityfloat in range [0, 1], optional

RANSAC iteration stops if at least one outlier-free set of the training data is sampled with probability >= stop_probability, depending on the current best model’s inlier ratio and the number of trials. This requires to generate at least N samples (trials):

N >= log(1 - probability) / log(1 - e**m)

where the probability (confidence) is typically set to a high value such as 0.99, and e is the current fraction of inliers w.r.t. the total number of samples.

random_stateint, RandomState instance or None, optional

If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random.

Returns
modelobject

Best model with largest consensus set.

inliers(N, ) array

Boolean mask of inliers classified as True.

References

1

“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)
>>> 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)  # doctest: +SKIP
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 # doctest: +SKIP
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)


## block_reduce¶

skimage.measure.block_reduce(image, block_size, func=<function sum at 0x7fe67bbd2b70>, cval=0)[source]

Down-sample image by applying function to local blocks.

Parameters
imagendarray

N-dimensional input image.

block_sizearray_like

Array containing down-sampling integer factor along each axis.

funccallable

Function object which is used to calculate the return value for each local block. This function must implement an axis parameter such as numpy.sum or numpy.min.

cvalfloat

Constant padding value if image is not perfectly divisible by the block size.

Returns
imagendarray

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 # doctest: +NORMALIZE_WHITESPACE
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 # doctest: +NORMALIZE_WHITESPACE
array([[[11]],
[[23]],
[[35]]])
>>> image_max2 = block_reduce(image, block_size=(3, 1, 4), func=np.max)
>>> image_max2 # doctest: +NORMALIZE_WHITESPACE
array([[[27],
[31],
[35]]])


## moments¶

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

Calculate all raw image moments up to a certain order.

The following properties can be calculated from raw image moments:
• Area as: M[0, 0].

• Centroid as: {M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]}.

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

Parameters
imagenD double or uint8 array

Rasterized shape as image.

orderint, optional

Maximum order of moments. Default is 3.

Returns
m(order + 1, order + 1) array

Raw image moments.

References

1

Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.

2

B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005.

3

T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993.

4

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[1, 0] / M[0, 0]
>>> cc = M[0, 1] / M[0, 0]
>>> cr, cc
(14.5, 14.5)


## moments_central¶

skimage.measure.moments_central(image, center=None, cc=None, order=3, **kwargs)[source]

Calculate all central image moments up to a certain order.

The center coordinates (cr, cc) can be calculated from the raw moments as: {M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]}.

Note that central moments are translation invariant but not scale and rotation invariant.

Parameters
imagenD double or uint8 array

Rasterized shape as image.

centertuple of float, optional

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

orderint, optional

The maximum order of moments computed.

Returns
mu(order + 1, order + 1) array

Central image moments.

Other Parameters
crdouble

DEPRECATED: Center row coordinate for 2D image.

ccdouble

DEPRECATED: Center column coordinate for 2D image.

References

1

Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.

2

B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005.

3

T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993.

4

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[1, 0] / M[0, 0]
>>> cc = M[0, 1] / 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_coords¶

skimage.measure.moments_coords(coords, order=3)[source]

Calculate all raw image moments up to a certain order.

The following properties can be calculated from raw image moments:
• Area as: M[0, 0].

• Centroid as: {M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]}.

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

Parameters
coords(N, D) double or uint8 array

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

orderint, optional

Maximum order of moments. Default is 3.

Returns
M(order + 1, order + 1, …) array

Raw image moments. (D dimensions)

References

1

Johannes Kilian. Simple Image Analysis By Moments. Durham University, version 0.2, Durham, 2001.

Examples

>>> coords = np.array([[row, col]
...                    for row in range(13, 17)
...                    for col in range(14, 18)], dtype=np.double)
>>> M = moments_coords(coords)
>>> centroid_row = M[1, 0] / M[0, 0]
>>> centroid_col = M[0, 1] / M[0, 0]
>>> centroid_row, centroid_col
(14.5, 15.5)


## moments_coords_central¶

skimage.measure.moments_coords_central(coords, center=None, order=3)[source]

Calculate all central image moments up to a certain order.

The following properties can be calculated from raw image moments:
• Area as: M[0, 0].

• Centroid as: {M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]}.

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

Parameters
coords(N, D) double or uint8 array

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

centertuple of float, optional

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

orderint, optional

Maximum order of moments. Default is 3.

Returns
Mc(order + 1, order + 1, …) array

Central image moments. (D dimensions)

References

1

Johannes Kilian. Simple Image Analysis By Moments. Durham University, version 0.2, Durham, 2001.

Examples

>>> coords = np.array([[row, col]
...                    for row in range(13, 17)
...                    for col in range(14, 18)])
>>> moments_coords_central(coords)
array([[ 16.,   0.,  20.,   0.],
[  0.,   0.,   0.,   0.],
[ 20.,   0.,  25.,   0.],
[  0.,   0.,   0.,   0.]])


As seen above, for symmetric objects, odd-order moments (columns 1 and 3, rows 1 and 3) are zero when centered on the centroid, or center of mass, of the object (the default). If we break the symmetry by adding a new point, this no longer holds:

>>> coords2 = np.concatenate((coords, [[17, 17]]), axis=0)
>>> np.round(moments_coords_central(coords2), 2)
array([[ 17.  ,   0.  ,  22.12,  -2.49],
[  0.  ,   3.53,   1.73,   7.4 ],
[ 25.88,   6.02,  36.63,   8.83],
[  4.15,  19.17,  14.8 ,  39.6 ]])


Image moments and central image moments are equivalent (by definition) when the center is (0, 0):

>>> np.allclose(moments_coords(coords),
...             moments_coords_central(coords, (0, 0)))
True


## moments_normalized¶

skimage.measure.moments_normalized(mu, order=3)[source]

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

Note that normalized central moments are translation and scale invariant but not rotation invariant.

Parameters
mu(M,[ …,] M) array

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

orderint, optional

Maximum order of moments. Default is 3.

Returns
nu(order + 1,[ …,] order + 1) array

Normalized central image moments.

References

1

Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.

2

B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005.

3

T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993.

4

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.        ]])


## moments_hu¶

skimage.measure.moments_hu(nu)[source]

Calculate Hu’s set of image moments (2D-only).

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

Parameters
nu(M, M) array

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

Returns
nu(7,) array

Hu’s set of image moments.

References

1

M. K. Hu, “Visual Pattern Recognition by Moment Invariants”, IRE Trans. Info. Theory, vol. IT-8, pp. 179-187, 1962

2

Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009.

3

B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005.

4

T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993.

5

http://en.wikipedia.org/wiki/Image_moment

## marching_cubes_lewiner¶

skimage.measure.marching_cubes_lewiner(volume, level=None, spacing=(1.0, 1.0, 1.0), gradient_direction='descent', step_size=1, allow_degenerate=True, use_classic=False)[source]

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

In contrast to marching_cubes_classic(), this algorithm is faster, resolves ambiguities, and guarantees topologically correct results. Therefore, this algorithm generally a better choice, unless there is a specific need for the classic algorithm.

Parameters
volume(M, N, P) array

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

levelfloat

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

spacinglength-3 tuple of floats

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

gradient_directionstring

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

step_sizeint

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

allow_degeneratebool

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

use_classicbool

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

Returns
verts(V, 3) array

Spatial coordinates for V unique mesh vertices. Coordinate order matches input volume (M, N, P).

faces(F, 3) array

Define triangular faces via referencing vertex indices from verts. This algorithm specifically outputs triangles, so each face has exactly three indices.

normals(V, 3) array

The normal direction at each vertex, as calculated from the data.

values(V, ) array

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

Notes

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

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

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

>>> from mayavi import mlab
>>> verts, faces, normals, values = marching_cubes_lewiner(myvolume, 0.0)
>>> mlab.triangular_mesh([vert[0] for vert in verts],
...                      [vert[1] for vert in verts],
...                      [vert[2] for vert in verts],
...                      faces)
>>> mlab.show()


Similarly using the visvis package:

>>> import visvis as vv
>>> verts, faces, normals, values = marching_cubes_lewiner(myvolume, 0.0)
>>> vv.mesh(np.fliplr(verts), faces, normals, values)
>>> vv.use().Run()


References

1

Thomas Lewiner, Helio Lopes, Antonio Wilson Vieira and Geovan Tavares. Efficient implementation of Marching Cubes’ cases with topological guarantees. Journal of Graphics Tools 8(2) pp. 1-15 (december 2003). DOI: 10.1080/10867651.2003.10487582

## marching_cubes_classic¶

skimage.measure.marching_cubes_classic(volume, level=None, spacing=(1.0, 1.0, 1.0), gradient_direction='descent')[source]

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

Note that the marching_cubes() algorithm is recommended over this algorithm, because it’s faster and produces better results.

Parameters
volume(M, N, P) array of doubles

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

levelfloat

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

spacinglength-3 tuple of floats

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

gradient_directionstring

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

Returns
verts(V, 3) array

Spatial coordinates for V unique mesh vertices. Coordinate order matches input volume (M, N, P).

faces(F, 3) array

Define triangular faces via referencing vertex indices from verts. This algorithm specifically outputs triangles, so each face has exactly three indices.

See also

skimage.measure.marching_cubes
skimage.measure.mesh_surface_area

Notes

The marching cubes algorithm is implemented as described in [1]. A simple explanation is available here:

http://www.essi.fr/~lingrand/MarchingCubes/algo.html


There are several known ambiguous cases in the marching cubes algorithm. Using point labeling as in [1], Figure 4, as shown:

    v8 ------ v7
/ |       / |        y
/  |      /  |        ^  z
v4 ------ v3   |        | /
|  v5 ----|- v6        |/          (note: NOT right handed!)
|  /      |  /          ----> x
| /       | /
v1 ------ v2


Most notably, if v4, v8, v2, and v6 are all >= level (or any generalization of this case) two parallel planes are generated by this algorithm, separating v4 and v8 from v2 and v6. An equally valid interpretation would be a single connected thin surface enclosing all four points. This is the best known ambiguity, though there are others.

This algorithm does not attempt to resolve such ambiguities; it is a naive implementation of marching cubes as in [1], but may be a good beginning for work with more recent techniques (Dual Marching Cubes, Extended Marching Cubes, Cubic Marching Squares, etc.).

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

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

The generated mesh guarantees coherent orientation as of version 0.12.

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

References

1(1,2,3)

Lorensen, William and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163-170). DOI: 10.1145/37401.37422

## 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
areafloat

Surface area of mesh. Units now [coordinate units] ** 2.

Notes

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

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

## 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.

spacinglength-3 tuple of floats

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

gradient_directionstring

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

Returns
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_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.

## profile_line¶

skimage.measure.profile_line(image, 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
imagenumeric array, shape (M, N[, C])

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

src2-tuple of numeric scalar (float or int)

The start point of the scan line.

dst2-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.

linewidthint, optional

Width of the scan, perpendicular to the line

orderint in {0, 1, 2, 3, 4, 5}, optional

The order of the spline interpolation 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.

cvalfloat, optional

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

Returns
return_valuearray

The intensity profile along the scan line. The length of the profile is the ceil of the computed length of the scan line.

Examples

>>> x = np.array([[1, 1, 1, 2, 2, 2]])
>>> img = np.vstack([np.zeros_like(x), x, x, x, np.zeros_like(x)])
>>> img
array([[0, 0, 0, 0, 0, 0],
[1, 1, 1, 2, 2, 2],
[1, 1, 1, 2, 2, 2],
[1, 1, 1, 2, 2, 2],
[0, 0, 0, 0, 0, 0]])
>>> profile_line(img, (2, 1), (2, 4))
array([ 1.,  1.,  2.,  2.])
>>> profile_line(img, (1, 0), (1, 6), cval=4)
array([ 1.,  1.,  1.,  2.,  2.,  2.,  4.])


The destination point is included in the profile, in contrast to standard numpy indexing. For example:

>>> profile_line(img, (1, 0), (1, 6))  # The final point is out of bounds
array([ 1.,  1.,  1.,  2.,  2.,  2.,  0.])
>>> profile_line(img, (1, 0), (1, 5))  # This accesses the full first row
array([ 1.,  1.,  1.,  2.,  2.,  2.])


## 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
inputndarray of dtype int

Image to label.

neighbors{4, 8}, int, optional

Whether to use 4- or 8-“connectivity”. In 3D, 4-“connectivity” means connected pixels have to share face, whereas with 8-“connectivity”, they have to share only edge or vertex. Deprecated, use connectivity instead.

backgroundint, optional

Consider all pixels with this value as background pixels, and label them as 0. By default, 0-valued pixels are considered as background pixels.

return_numbool, optional

Whether to return the number of assigned labels.

connectivityint, optional

Maximum number of orthogonal hops to consider a pixel/voxel as a neighbor. Accepted values are ranging from 1 to input.ndim. If None, a full connectivity of input.ndim is used.

Returns
labelsndarray of dtype int

Labeled array, where all connected regions are assigned the same integer value.

numint, optional

Number of labels, which equals the maximum label index and is only returned if return_num is True.

References

1

Christophe Fiorio and Jens Gustedt, “Two linear time Union-Find strategies for image processing”, Theoretical Computer Science 154 (1996), pp. 165-181.

2

Kensheng Wu, Ekow Otoo and Arie Shoshani, “Optimizing connected component labeling algorithms”, Paper LBNL-56864, 2005, Lawrence Berkeley National Laboratory (University of California), http://repositories.cdlib.org/lbnl/LBNL-56864

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]]


## points_in_poly¶

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

Test whether points lie inside a polygon.

Parameters
points(N, 2) array

Input points, (x, y).

verts(M, 2) array

Vertices of the polygon, sorted either clockwise or 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.

## grid_points_in_poly¶

skimage.measure.grid_points_in_poly(shape, verts)[source]

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

For each (r, c) coordinate on a grid, i.e. (0, 0), (0, 1) etc., test whether that point lies inside a polygon.

Parameters
shapetuple (M, N)

Shape of the grid.

verts(V, 2) array

Specify the V vertices of the polygon, sorted either clockwise or 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.

## compare_ssim¶

skimage.measure.compare_ssim(X, Y, win_size=None, gradient=False, data_range=None, multichannel=False, gaussian_weights=False, full=False, **kwargs)[source]

Compute the mean structural similarity index between two images.

Parameters
X, Yndarray

Image. Any dimensionality.

win_sizeint 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 on sigma.

gradientbool, optional

If True, also return the gradient with respect to Y.

data_rangefloat, optional

The data range of the input image (distance between minimum and maximum possible values). By default, this is estimated from the image data-type.

multichannelbool, optional

If True, treat the last dimension of the array as channels. Similarity calculations are done independently for each channel then averaged.

gaussian_weightsbool, optional

If True, each patch has its mean and variance spatially weighted by a normalized Gaussian kernel of width sigma=1.5.

fullbool, optional

If True, also return the full structural similarity image.

Returns
mssimfloat

The mean structural similarity over the image.

gradndarray

The gradient of the structural similarity index between X and Y [2]. This is only returned if gradient is set to True.

Sndarray

The full SSIM image. This is only returned if full is set to True.

Other Parameters
use_sample_covariancebool

if True, normalize covariances by N-1 rather than, N where N is the number of pixels within the sliding window.

K1float

algorithm parameter, K1 (small constant, see [1])

K2float

algorithm parameter, K2 (small constant, see [1])

sigmafloat

sigma for the Gaussian when gaussian_weights is True.

Notes

To match the implementation of Wang et. al. [1], set gaussian_weights to True, sigma to 1.5, and use_sample_covariance to False.

References

1(1,2,3)

Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13, 600-612. https://ece.uwaterloo.ca/~z70wang/publications/ssim.pdf, DOI:10.1109/TIP.2003.819861

2

Avanaki, A. N. (2009). Exact global histogram specification optimized for structural similarity. Optical Review, 16, 613-621. http://arxiv.org/abs/0901.0065, DOI:10.1007/s10043-009-0119-z

## compare_mse¶

skimage.measure.compare_mse(im1, im2)[source]

Compute the mean-squared error between two images.

Parameters
im1, im2ndarray

Image. Any dimensionality.

Returns
msefloat

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_truendarray

Ground-truth image.

im_testndarray

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 [1]. The methods available here are as follows:

• ‘Euclidean’ : normalize by the averaged Euclidean norm of im_true:

NRMSE = RMSE * sqrt(N) / || im_true ||


where || . || denotes the Frobenius norm and N = im_true.size. This result is equivalent to:

NRMSE = || im_true - im_test || / || im_true ||.

• ‘min-max’ : normalize by the intensity range of im_true.

• ‘mean’ : normalize by the mean of im_true

Returns
nrmsefloat

The NRMSE metric.

References

1

https://en.wikipedia.org/wiki/Root-mean-square_deviation

## compare_psnr¶

skimage.measure.compare_psnr(im_true, im_test, data_range=None)[source]

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

Parameters
im_truendarray

Ground-truth image.

im_testndarray

Test image.

data_rangeint

The data range of the input image (distance between minimum and maximum possible values). By default, this is estimated from the image data-type.

Returns
psnrfloat

The PSNR metric.

References

1

https://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio

## shannon_entropy¶

skimage.measure.shannon_entropy(image, base=2)[source]

Calculate the Shannon entropy of an image.

The Shannon entropy is defined as S = -sum(pk * log(pk)), where pk are frequency/probability of pixels of value k.

Parameters
image(N, M) ndarray

Grayscale input image.

basefloat, optional

The logarithmic base to use.

Returns
entropyfloat

Notes

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

References

1
2

https://en.wiktionary.org/wiki/Shannon_entropy

Examples

>>> from skimage import data
>>> shannon_entropy(data.camera())
7.0479552324230861


## LineModelND¶

class skimage.measure.LineModelND[source]

Bases: skimage.measure.fit.BaseModel

Total least squares estimator for N-dimensional lines.

In contrast to ordinary least squares line estimation, this estimator minimizes the orthogonal distances of points to the estimated line.

Lines are defined by a point (origin) and a unit vector (direction) according to the following vector equation:

X = origin + lambda * direction


Examples

>>> x = np.linspace(1, 2, 25)
>>> y = 1.5 * x + 3
>>> lm = LineModelND()
>>> lm.estimate(np.array([x, y]).T)
True
>>> tuple(np.round(lm.params, 5))
(array([ 1.5 ,  5.25]), array([ 0.5547 ,  0.83205]))
>>> res = lm.residuals(np.array([x, y]).T)
>>> np.abs(np.round(res, 9))
array([ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.])
>>> np.round(lm.predict_y(x[:5]), 3)
array([ 4.5  ,  4.562,  4.625,  4.688,  4.75 ])
>>> np.round(lm.predict_x(y[:5]), 3)
array([ 1.   ,  1.042,  1.083,  1.125,  1.167])

Attributes
paramstuple

Line model parameters in the following order origin, direction.

__init__(self)[source]

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

estimate(self, data)[source]

Estimate line model from data.

This minimizes the sum of shortest (orthogonal) distances from the given data points to the estimated line.

Parameters
data(N, dim) array

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

Returns
successbool

True, if model estimation succeeds.

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

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

Parameters
x(n, 1) array

Coordinates along an axis.

axisint

Axis orthogonal to the hyperplane intersecting the line.

params(2, ) array, optional

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

Returns
data(n, m) array

Predicted coordinates.

Raises
ValueError

If the line is parallel to the given axis.

predict_x(self, y, params=None)[source]

Predict x-coordinates for 2D lines using the estimated model.

Alias for:

predict(y, axis=1)[:, 0]

Parameters
yarray

y-coordinates.

params(2, ) array, optional

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

Returns
xarray

Predicted x-coordinates.

predict_y(self, x, params=None)[source]

Predict y-coordinates for 2D lines using the estimated model.

Alias for:

predict(x, axis=0)[:, 1]

Parameters
xarray

x-coordinates.

params(2, ) array, optional

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

Returns
yarray

Predicted y-coordinates.

residuals(self, data, params=None)[source]

Determine residuals of data to model.

For each point, the shortest (orthogonal) distance to the line is returned. It is obtained by projecting the data onto the line.

Parameters
data(N, dim) array

N points in a space of dimension dim.

params(2, ) array, optional

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

Returns
residuals(N, ) array

Residual for each data point.

## CircleModel¶

class skimage.measure.CircleModel[source]

Bases: skimage.measure.fit.BaseModel

Total least squares estimator for 2D circles.

The functional model of the circle is:

r**2 = (x - xc)**2 + (y - yc)**2


This estimator minimizes the squared distances from all points to the circle:

min{ sum((r - sqrt((x_i - xc)**2 + (y_i - yc)**2))**2) }


A minimum number of 3 points is required to solve for the parameters.

Examples

>>> t = np.linspace(0, 2 * np.pi, 25)
>>> xy = CircleModel().predict_xy(t, params=(2, 3, 4))
>>> model = CircleModel()
>>> model.estimate(xy)
True
>>> tuple(np.round(model.params, 5))
(2.0, 3.0, 4.0)
>>> res = model.residuals(xy)
>>> np.abs(np.round(res, 9))
array([ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.])

Attributes
paramstuple

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

__init__(self)[source]

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

estimate(self, data)[source]

Estimate circle model from data using total least squares.

Parameters
data(N, 2) array

N points with (x, y) coordinates, respectively.

Returns
successbool

True, if model estimation succeeds.

predict_xy(self, t, params=None)[source]

Predict x- and y-coordinates using the estimated model.

Parameters
tarray

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(self, 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.

The estimator is based on a least squares minimization. The optimal solution is computed directly, no iterations are required. This leads to a simple, stable and robust fitting method.

The params attribute contains the parameters in the following order:

xc, yc, a, b, theta


Examples

>>> xy = EllipseModel().predict_xy(np.linspace(0, 2 * np.pi, 25),
...                                params=(10, 15, 4, 8, np.deg2rad(30)))
>>> ellipse = EllipseModel()
>>> ellipse.estimate(xy)
True
>>> np.round(ellipse.params, 2)
array([ 10.  ,  15.  ,   4.  ,   8.  ,   0.52])
>>> np.round(abs(ellipse.residuals(xy)), 5)
array([ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.])

Attributes
paramstuple

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

__init__(self)[source]

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

estimate(self, data)[source]

Estimate circle model from data using total least squares.

Parameters
data(N, 2) array

N points with (x, y) coordinates, respectively.

Returns
successbool

True, if model estimation succeeds.

References

1

Halir, R.; Flusser, J. “Numerically stable direct least squares fitting of ellipses”. In Proc. 6th International Conference in Central Europe on Computer Graphics and Visualization. WSCG (Vol. 98, pp. 125-132).

predict_xy(self, t, params=None)[source]

Predict x- and y-coordinates using the estimated model.

Parameters
tarray

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(self, 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.