measure
¶skimage.measure.
approximate_polygon
(coords, tolerance)¶Approximate a polygonal chain with the specified tolerance.
It is based on the DouglasPeucker algorithm.
Note that the approximated polygon is always within the convex hull of the original polygon.
Parameters:  coords : (N, 2) array
tolerance : float


Returns:  coords : (M, 2) array

References
[R117]  http://en.wikipedia.org/wiki/RamerDouglasPeucker_algorithm 
skimage.measure.
block_reduce
(image, block_size, func=<function sum>, cval=0)¶Downsample image by applying function to local blocks.
Parameters:  image : ndarray
block_size : array_like
func : callable
cval : float


Returns:  image : ndarray

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]]])
skimage.measure.
correct_mesh_orientation
(volume, verts, faces, spacing=(1.0, 1.0, 1.0), gradient_direction='descent')¶Correct orientations of mesh faces.
Parameters:  volume : (M, N, P) array of doubles
verts : (V, 3) array of floats
faces : (F, 3) array of ints
spacing : length3 tuple of floats
gradient_direction : string


Returns:  faces_corrected (F, 3) array of ints

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 misoriented faces.
Because marching cubes could be used to find isosurfaces either on
gradient descent (where the desired object has greater values than the
exterior) or ascent (where the desired object has lower values than the
exterior), the gradient_direction
kwarg allows the user to inform this
algorithm which is correct. If the resulting mesh appears to be oriented
completely incorrectly, try changing this option.
The arguments expected by this function are the exact outputs from skimage.measure.marching_cubes. Only faces is corrected and returned, as the vertices do not change; only the order in which they are referenced.
This algorithm assumes faces
provided are all triangles.
skimage.measure.
find_contours
(array, level, fully_connected='low', positive_orientation='low')¶Find isovalued contours in a 2D array for a given level value.
Uses the “marching squares” method to compute a the isovalued contours of the input 2D array for a particular level value. Array values are linearly interpolated to provide better precision for the output contours.
Parameters:  array : 2D ndarray of double
level : float
fully_connected : str, {‘low’, ‘high’}
positive_orientation : either ‘low’ or ‘high’


Returns:  contours : list of (n,2)ndarrays

Notes
The marching squares algorithm is a special case of the marching cubes algorithm [R118]. A simple explanation is available here:
http://www.essi.fr/~lingrand/MarchingCubes/algo.html
There is a single ambiguous case in the marching squares algorithm: when
a given 2 x 2
element square has two highvalued and two lowvalued
elements, each pair diagonally adjacent. (Where high and lowvalued is
with respect to the contour value sought.) In this case, either the
highvalued elements can be ‘connected together’ via a thin isthmus that
separates the lowvalued elements, or viceversa. When elements are
connected together across a diagonal, they are considered ‘fully
connected’ (also known as ‘face+vertexconnected’ or ‘8connected’). Only
highvalued or lowvalued elements can be fullyconnected, the other set
will be considered as ‘faceconnected’ or ‘4connected’. By default,
lowvalued elements are considered fullyconnected; this can be altered
with the ‘fully_connected’ parameter.
Output contours are not guaranteed to be closed: contours which intersect the array edge will be left open. All other contours will be closed. (The closedness of a contours can be tested by checking whether the beginning point is the same as the end point.)
Contours are oriented. By default, array values lower than the contour value are to the left of the contour and values greater than the contour value are to the right. This means that contours will wind counterclockwise (i.e. in ‘positive orientation’) around islands of lowvalued pixels. This behavior can be altered with the ‘positive_orientation’ parameter.
The order of the contours in the output list is determined by the position
of the smallest x,y
(in lexicographical order) coordinate in the
contour. This is a sideeffect of how the input array is traversed, but
can be relied upon.
Warning
Array coordinates/values are assumed to refer to the center of the
array element. Take a simple example input: [0, 1]
. The interpolated
position of 0.5 in this array is midway between the 0element (at
x=0
) and the 1element (at x=1
), and thus would fall at
x=0.5
.
This means that to find reasonable contours, it is best to find contours midway between the expected “light” and “dark” values. In particular, given a binarized array, do not choose to find contours at the low or high value of the array. This will often yield degenerate contours, especially around structures that are a single array element wide. Instead choose a middle value, as above.
References
[R118]  (1, 2) Lorensen, William and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163170). 
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. ]])]
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)
verts : (V, 2) array


Returns:  mask : (M, N) ndarray of bool

See also
skimage.measure.
label
(input, neighbors=None, background=None, return_num=False, connectivity=None)¶Label connected regions of an integer array.
Two pixels are connected when they are neighbors and have the same value. In 2D, they can be neighbors either in a 1 or 2connected sense. The value refers to the maximum number of orthogonal hops to consider a pixel/voxel a neighbor:
1connectivity 2connectivity diagonal connection closeup
[ ] [ ] [ ] [ ] [ ]
 \  /  < hop 2
[ ][x][ ] [ ][x][ ] [x][ ]
 /  \ hop 1
[ ] [ ] [ ] [ ]
Parameters:  input : ndarray of dtype int
neighbors : {4, 8}, int, optional
background : int, optional
return_num : bool, optional
connectivity : int, optional


Returns:  labels : ndarray of dtype int
num : int, optional

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))
[[0 1 1]
[2 3 1]
[2 2 4]]
>>> print(label(x, connectivity=2))
[[0 1 1]
[1 0 1]
[1 1 0]]
>>> x = np.array([[1, 0, 0],
... [1, 1, 5],
... [0, 0, 0]])
>>> print(label(x, background=0))
[[ 0 1 1]
[ 0 0 1]
[1 1 1]]
skimage.measure.
marching_cubes
(volume, level, spacing=(1.0, 1.0, 1.0))¶Marching cubes algorithm to find isovalued surfaces in 3d volumetric data
Parameters:  volume : (M, N, P) array of doubles
level : float
spacing : length3 tuple of floats


Returns:  verts : (V, 3) array
faces : (F, 3) array

Notes
The marching cubes algorithm is implemented as described in [R119]. 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 [R119], 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 [R119], but may be a good beginning for work with more recent techniques (Dual Marching Cubes, Extended Marching Cubes, Cubic Marching Squares, etc.).
Because of interactions between neighboring cubes, the isosurface(s) generated by this algorithm are NOT guaranteed to be closed, particularly for complicated contours. Furthermore, this algorithm does not guarantee a single contour will be returned. Indeed, ALL isosurfaces which cross level will be found, regardless of connectivity.
The output is a triangular mesh consisting of a set of unique vertices and
connecting triangles. The order of these vertices and triangles in the
output list is determined by the position of the smallest x,y,z
(in
lexicographical order) coordinate in the contour. This is a sideeffect
of how the input array is traversed, but can be relied upon.
The generated mesh does not guarantee coherent orientation because of how
symmetry is used in the algorithm. If this is required, e.g. due to a
particular visualization package or for generating 3D printing STL files,
the utility skimage.measure.correct_mesh_orientation
is available to
fix this in postprocessing.
To quantify the area of an isosurface generated by this algorithm, pass the outputs directly into skimage.measure.mesh_surface_area.
Regarding visualization of algorithm output, the mayavi
package
is recommended. To contour a volume named myvolume 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
[R119]  (1, 2, 3, 4) Lorensen, William and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163170). 
skimage.measure.
mesh_surface_area
(verts, faces)¶Compute surface area, given vertices & triangular faces
Parameters:  verts : (V, 3) array of floats
faces : (F, 3) array of ints


Returns:  area : float

Notes
The arguments expected by this function are the exact outputs from skimage.measure.marching_cubes. For unit correct output, ensure correct spacing was passed to skimage.measure.marching_cubes.
This algorithm works properly only if the faces
provided are all
triangles.
skimage.measure.
moments
(image, order=3)¶Calculate all raw image moments up to a certain order.
m[0, 0]
.m[0, 1] / m[0, 0]
, m[1, 0] / m[0, 0]
}.Note that raw moments are neither translation, scale nor rotation invariant.
Parameters:  image : 2D double or uint8 array
order : int, optional


Returns:  m : (

References
[R120]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R121]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R122]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R123]  http://en.wikipedia.org/wiki/Image_moment 
skimage.measure.
moments_central
(image, cr, cc, order=3)¶Calculate all central image moments up to a certain order.
Note that central moments are translation invariant but not scale and rotation invariant.
Parameters:  image : 2D double or uint8 array
cr : double
cc : double
order : int, optional


Returns:  mu : (

References
[R124]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R125]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R126]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R127]  http://en.wikipedia.org/wiki/Image_moment 
skimage.measure.
moments_hu
(nu)¶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


Returns:  nu : (7, 1) array

References
[R128]  M. K. Hu, “Visual Pattern Recognition by Moment Invariants”, IRE Trans. Info. Theory, vol. IT8, pp. 179187, 1962 
[R129]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R130]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R131]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R132]  http://en.wikipedia.org/wiki/Image_moment 
skimage.measure.
moments_normalized
(mu, order=3)¶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
order : int, optional


Returns:  nu : (

References
[R133]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R134]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R135]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R136]  http://en.wikipedia.org/wiki/Image_moment 
skimage.measure.
perimeter
(image, neighbourhood=4)¶Calculate total perimeter of all objects in binary image.
Parameters:  image : array
neighbourhood : 4 or 8, optional


Returns:  perimeter : float

References
[R137]  K. Benkrid, D. Crookes. Design and FPGA Implementation of a Perimeter Estimator. The Queen’s University of Belfast. http://www.cs.qub.ac.uk/~d.crookes/webpubs/papers/perimeter.doc 
skimage.measure.
points_in_poly
()¶Test whether points lie inside a polygon.
Parameters:  points : (N, 2) array
verts : (M, 2) array


Returns:  mask : (N,) array of bool

See also
skimage.measure.
profile_line
(img, src, dst, linewidth=1, order=1, mode='constant', cval=0.0)¶Return the intensity profile of an image measured along a scan line.
Parameters:  img : numeric array, shape (M, N[, C])
src : 2tuple of numeric scalar (float or int)
dst : 2tuple of numeric scalar (float or int)
linewidth : int, optional
order : int in {0, 1, 2, 3, 4, 5}, optional
mode : string, one of {‘constant’, ‘nearest’, ‘reflect’, ‘wrap’}, optional
cval : float, optional


Returns:  return_value : array

Notes
The destination point is included in the profile, in contrast to standard numpy indexing.
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.])
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)¶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:
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
model_class : object
min_samples : int
residual_threshold : float
is_data_valid : function, optional
is_model_valid : function, optional
max_trials : int, optional
stop_sample_num : int, optional
stop_residuals_sum : float, optional
stop_probability : float in range [0, 1], optional


Returns:  model : object
inliers : (N, ) array

References
[R138]  “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, 7.19203152e01])
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)
skimage.measure.
regionprops
(label_image, intensity_image=None, cache=True)¶Measure properties of labeled image regions.
Parameters:  label_image : (N, M) ndarray
intensity_image : (N, M) ndarray, optional
cache : bool, optional


Returns:  properties : list of RegionProperties

Notes
The following properties can be accessed as attributes or keys:
(min_row, min_col, max_row, max_col)
(row, col)
.(row, col)
of the region.area / (rows * cols)
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.
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.
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.
(row, col)
weighted with intensity
image.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.
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.
Normalized moments (translation and scale invariant) of intensity image up to 3rd order:
wnu_ji = wmu_ji / wm_00^[(i+j)/2 + 1]
where wm_00 is the zeroth spatial moment (intensityweighted area).
References
[R139]  Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. SpringerVerlag, London, 2009. 
[R140]  B. Jähne. Digital Image Processing. SpringerVerlag, BerlinHeidelberg, 6. edition, 2005. 
[R141]  T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. 
[R142]  http://en.wikipedia.org/wiki/Image_moment 
Examples
>>> from skimage import data, util
>>> from skimage.morphology import label
>>> img = util.img_as_ubyte(data.coins()) > 110
>>> label_img = label(img, connectivity=img.ndim)
>>> props = regionprops(label_img)
>>> # centroid of first labeled object
>>> props[0].centroid
(22.729879860483141, 81.912285234465827)
>>> # centroid of first labeled object
>>> props[0]['centroid']
(22.729879860483141, 81.912285234465827)
skimage.measure.
structural_similarity
(X, Y, win_size=None, gradient=False, dynamic_range=None, multichannel=False, gaussian_weights=False, full=False, **kwargs)¶Compute the mean structural similarity index between two images.
Parameters:  X, Y : ndarray
win_size : int or None
gradient : bool
dynamic_range : int
multichannel : int or None
gaussian_weights : bool
full : bool


Returns:  mssim : float or ndarray
grad : ndarray
S : ndarray

Other Parameters:  
use_sample_covariance : bool
K1 : float
K2 : float
sigma : float

Notes
To match the implementation of Wang et. al. [R143], set gaussian_weights to True, sigma to 1.5, and use_sample_covariance to False.
References
[R143]  (1, 2, 3, 4) Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13, 600612. https://ece.uwaterloo.ca/~z70wang/publications/ssim.pdf 
[R144]  (1, 2) Avanaki, A. N. (2009). Exact global histogram specification optimized for structural similarity. Optical Review, 16, 613621. http://arxiv.org/abs/0901.0065 
skimage.measure.
subdivide_polygon
(coords, degree=2, preserve_ends=False)¶Subdivision of polygonal curves using BSplines.
Note that the resulting curve is always within the convex hull of the original polygon. Circular polygons stay closed after subdivision.
Parameters:  coords : (N, 2) array
degree : {1, 2, 3, 4, 5, 6, 7}, optional
preserve_ends : bool, optional


Returns:  coords : (M, 2) array

References
[R145]  http://mrl.nyu.edu/publications/subdivcourse2000/coursenotes00.pdf 
CircleModel
¶skimage.measure.
CircleModel
¶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__
()¶estimate
(data)¶Estimate circle model from data using total least squares.
Parameters:  data : (N, 2) array


Returns:  success : bool

predict_xy
(t, params=None)¶Predict x and ycoordinates using the estimated model.
Parameters:  t : array
params : (3, ) array, optional


Returns:  xy : (..., 2) array

residuals
(data)¶Determine residuals of data to model.
For each point the shortest distance to the circle is returned.
Parameters:  data : (N, 2) array


Returns:  residuals : (N, ) array

EllipseModel
¶skimage.measure.
EllipseModel
¶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) for N + 5
unknowns (t_i,
xc, yc, a, b, theta), which gives you an effective redundancy of N  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__
()¶estimate
(data)¶Estimate circle model from data using total least squares.
Parameters:  data : (N, 2) array


Returns:  success : bool

predict_xy
(t, params=None)¶Predict x and ycoordinates using the estimated model.
Parameters:  t : array
params : (5, ) array, optional


Returns:  xy : (..., 2) array

residuals
(data)¶Determine residuals of data to model.
For each point the shortest distance to the ellipse is returned.
Parameters:  data : (N, 2) array


Returns:  residuals : (N, ) array

LineModel
¶skimage.measure.
LineModel
¶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.
Attributes
params  (tuple) Line model parameters in the following order dist, theta. 
__init__
()¶estimate
(data)¶Estimate line model from data using total least squares.
Parameters:  data : (N, 2) array


Returns:  success : bool

predict_x
(y, params=None)¶Predict xcoordinates using the estimated model.
Parameters:  y : array
params : (2, ) array, optional


Returns:  x : array

predict_y
(x, params=None)¶Predict ycoordinates using the estimated model.
Parameters:  x : array
params : (2, ) array, optional


Returns:  y : array

residuals
(data)¶Determine residuals of data to model.
For each point the shortest distance to the line is returned.
Parameters:  data : (N, 2) array


Returns:  residuals : (N, ) array
