900 lines
30 KiB
Python
900 lines
30 KiB
Python
import math
|
|
from warnings import warn
|
|
|
|
import numpy as np
|
|
from numpy.linalg import inv
|
|
from scipy import optimize, spatial
|
|
|
|
_EPSILON = np.spacing(1)
|
|
|
|
|
|
def _check_data_dim(data, dim):
|
|
if data.ndim != 2 or data.shape[1] != dim:
|
|
raise ValueError('Input data must have shape (N, %d).' % dim)
|
|
|
|
|
|
def _check_data_atleast_2D(data):
|
|
if data.ndim < 2 or data.shape[1] < 2:
|
|
raise ValueError('Input data must be at least 2D.')
|
|
|
|
|
|
class BaseModel(object):
|
|
|
|
def __init__(self):
|
|
self.params = None
|
|
|
|
|
|
class LineModelND(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
|
|
|
|
Attributes
|
|
----------
|
|
params : tuple
|
|
Line model parameters in the following order `origin`, `direction`.
|
|
|
|
Examples
|
|
--------
|
|
>>> x = np.linspace(1, 2, 25)
|
|
>>> y = 1.5 * x + 3
|
|
>>> lm = LineModelND()
|
|
>>> lm.estimate(np.stack([x, y], axis=-1))
|
|
True
|
|
>>> tuple(np.round(lm.params, 5))
|
|
(array([1.5 , 5.25]), array([0.5547 , 0.83205]))
|
|
>>> res = lm.residuals(np.stack([x, y], axis=-1))
|
|
>>> 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])
|
|
|
|
"""
|
|
|
|
def estimate(self, data):
|
|
"""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
|
|
-------
|
|
success : bool
|
|
True, if model estimation succeeds.
|
|
"""
|
|
_check_data_atleast_2D(data)
|
|
|
|
origin = data.mean(axis=0)
|
|
data = data - origin
|
|
|
|
if data.shape[0] == 2: # well determined
|
|
direction = data[1] - data[0]
|
|
norm = np.linalg.norm(direction)
|
|
if norm != 0: # this should not happen to be norm 0
|
|
direction /= norm
|
|
elif data.shape[0] > 2: # over-determined
|
|
# Note: with full_matrices=1 Python dies with joblib parallel_for.
|
|
_, _, v = np.linalg.svd(data, full_matrices=False)
|
|
direction = v[0]
|
|
else: # under-determined
|
|
raise ValueError('At least 2 input points needed.')
|
|
|
|
self.params = (origin, direction)
|
|
|
|
return True
|
|
|
|
def residuals(self, data, params=None):
|
|
"""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.
|
|
"""
|
|
_check_data_atleast_2D(data)
|
|
if params is None:
|
|
if self.params is None:
|
|
raise ValueError('Parameters cannot be None')
|
|
params = self.params
|
|
if len(params) != 2:
|
|
raise ValueError('Parameters are defined by 2 sets.')
|
|
|
|
origin, direction = params
|
|
res = (data - origin) - \
|
|
((data - origin) @ direction)[..., np.newaxis] * direction
|
|
return np.linalg.norm(res, axis=1)
|
|
|
|
def predict(self, x, axis=0, params=None):
|
|
"""Predict intersection of the estimated line model with a hyperplane
|
|
orthogonal to a given axis.
|
|
|
|
Parameters
|
|
----------
|
|
x : (n, 1) array
|
|
Coordinates along an axis.
|
|
axis : int
|
|
Axis orthogonal to the hyperplane intersecting the line.
|
|
params : (2, ) array, optional
|
|
Optional custom parameter set in the form (`origin`, `direction`).
|
|
|
|
Returns
|
|
-------
|
|
data : (n, m) array
|
|
Predicted coordinates.
|
|
|
|
Raises
|
|
------
|
|
ValueError
|
|
If the line is parallel to the given axis.
|
|
"""
|
|
if params is None:
|
|
if self.params is None:
|
|
raise ValueError('Parameters cannot be None')
|
|
params = self.params
|
|
if len(params) != 2:
|
|
raise ValueError('Parameters are defined by 2 sets.')
|
|
|
|
origin, direction = params
|
|
|
|
if direction[axis] == 0:
|
|
# line parallel to axis
|
|
raise ValueError('Line parallel to axis %s' % axis)
|
|
|
|
l = (x - origin[axis]) / direction[axis]
|
|
data = origin + l[..., np.newaxis] * direction
|
|
return data
|
|
|
|
def predict_x(self, y, params=None):
|
|
"""Predict x-coordinates for 2D lines using the estimated model.
|
|
|
|
Alias for::
|
|
|
|
predict(y, axis=1)[:, 0]
|
|
|
|
Parameters
|
|
----------
|
|
y : array
|
|
y-coordinates.
|
|
params : (2, ) array, optional
|
|
Optional custom parameter set in the form (`origin`, `direction`).
|
|
|
|
Returns
|
|
-------
|
|
x : array
|
|
Predicted x-coordinates.
|
|
|
|
"""
|
|
x = self.predict(y, axis=1, params=params)[:, 0]
|
|
return x
|
|
|
|
def predict_y(self, x, params=None):
|
|
"""Predict y-coordinates for 2D lines using the estimated model.
|
|
|
|
Alias for::
|
|
|
|
predict(x, axis=0)[:, 1]
|
|
|
|
Parameters
|
|
----------
|
|
x : array
|
|
x-coordinates.
|
|
params : (2, ) array, optional
|
|
Optional custom parameter set in the form (`origin`, `direction`).
|
|
|
|
Returns
|
|
-------
|
|
y : array
|
|
Predicted y-coordinates.
|
|
|
|
"""
|
|
y = self.predict(x, axis=0, params=params)[:, 1]
|
|
return y
|
|
|
|
|
|
class CircleModel(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`.
|
|
|
|
Notes
|
|
-----
|
|
The estimation is carried out using a 2D version of the spherical
|
|
estimation given in [1]_.
|
|
|
|
References
|
|
----------
|
|
.. [1] Jekel, Charles F. Obtaining non-linear orthotropic material models
|
|
for pvc-coated polyester via inverse bubble inflation.
|
|
Thesis (MEng), Stellenbosch University, 2016. Appendix A, pp. 83-87.
|
|
https://hdl.handle.net/10019.1/98627
|
|
|
|
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.])
|
|
"""
|
|
|
|
def estimate(self, data):
|
|
"""Estimate circle model from data using total least squares.
|
|
|
|
Parameters
|
|
----------
|
|
data : (N, 2) array
|
|
N points with ``(x, y)`` coordinates, respectively.
|
|
|
|
Returns
|
|
-------
|
|
success : bool
|
|
True, if model estimation succeeds.
|
|
|
|
"""
|
|
|
|
_check_data_dim(data, dim=2)
|
|
|
|
# to prevent integer overflow, cast data to float, if it isn't already
|
|
float_type = np.promote_types(data.dtype, np.float32)
|
|
data = data.astype(float_type, copy=False)
|
|
|
|
# Adapted from a spherical estimator covered in a blog post by Charles
|
|
# Jeckel (see also reference 1 above):
|
|
# https://jekel.me/2015/Least-Squares-Sphere-Fit/
|
|
A = np.append(data * 2,
|
|
np.ones((data.shape[0], 1), dtype=float_type),
|
|
axis=1)
|
|
f = np.sum(data ** 2, axis=1)
|
|
C, _, rank, _ = np.linalg.lstsq(A, f, rcond=None)
|
|
|
|
if rank != 3:
|
|
warn("Input data does not contain enough significant data points. "
|
|
"In scikit-image 1.0, this warning will become a ValueError.")
|
|
|
|
center = C[0:2]
|
|
distances = spatial.minkowski_distance(center, data)
|
|
r = np.sqrt(np.mean(distances ** 2))
|
|
|
|
self.params = tuple(center) + (r,)
|
|
|
|
return True
|
|
|
|
def residuals(self, data):
|
|
"""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.
|
|
|
|
"""
|
|
|
|
_check_data_dim(data, dim=2)
|
|
|
|
xc, yc, r = self.params
|
|
|
|
x = data[:, 0]
|
|
y = data[:, 1]
|
|
|
|
return r - np.sqrt((x - xc)**2 + (y - yc)**2)
|
|
|
|
def predict_xy(self, t, params=None):
|
|
"""Predict x- and y-coordinates using the estimated model.
|
|
|
|
Parameters
|
|
----------
|
|
t : array
|
|
Angles in circle in radians. Angles start to count from positive
|
|
x-axis to positive y-axis in a right-handed system.
|
|
params : (3, ) array, optional
|
|
Optional custom parameter set.
|
|
|
|
Returns
|
|
-------
|
|
xy : (..., 2) array
|
|
Predicted x- and y-coordinates.
|
|
|
|
"""
|
|
if params is None:
|
|
params = self.params
|
|
xc, yc, r = params
|
|
|
|
x = xc + r * np.cos(t)
|
|
y = yc + r * np.sin(t)
|
|
|
|
return np.concatenate((x[..., None], y[..., None]), axis=t.ndim)
|
|
|
|
|
|
class EllipseModel(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
|
|
|
|
Attributes
|
|
----------
|
|
params : tuple
|
|
Ellipse model 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.])
|
|
"""
|
|
|
|
def estimate(self, data):
|
|
"""Estimate circle model from data using total least squares.
|
|
|
|
Parameters
|
|
----------
|
|
data : (N, 2) array
|
|
N points with ``(x, y)`` coordinates, respectively.
|
|
|
|
Returns
|
|
-------
|
|
success : bool
|
|
True, if model estimation succeeds.
|
|
|
|
|
|
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).
|
|
|
|
"""
|
|
# Original Implementation: Ben Hammel, Nick Sullivan-Molina
|
|
# another REFERENCE: [2] http://mathworld.wolfram.com/Ellipse.html
|
|
_check_data_dim(data, dim=2)
|
|
|
|
# to prevent integer overflow, cast data to float, if it isn't already
|
|
float_type = np.promote_types(data.dtype, np.float32)
|
|
data = data.astype(float_type, copy=False)
|
|
|
|
x = data[:, 0]
|
|
y = data[:, 1]
|
|
|
|
# Quadratic part of design matrix [eqn. 15] from [1]
|
|
D1 = np.vstack([x ** 2, x * y, y ** 2]).T
|
|
# Linear part of design matrix [eqn. 16] from [1]
|
|
D2 = np.vstack([x, y, np.ones_like(x)]).T
|
|
|
|
# forming scatter matrix [eqn. 17] from [1]
|
|
S1 = D1.T @ D1
|
|
S2 = D1.T @ D2
|
|
S3 = D2.T @ D2
|
|
|
|
# Constraint matrix [eqn. 18]
|
|
C1 = np.array([[0., 0., 2.], [0., -1., 0.], [2., 0., 0.]])
|
|
|
|
try:
|
|
# Reduced scatter matrix [eqn. 29]
|
|
M = inv(C1) @ (S1 - S2 @ inv(S3) @ S2.T)
|
|
except np.linalg.LinAlgError: # LinAlgError: Singular matrix
|
|
return False
|
|
|
|
# M*|a b c >=l|a b c >. Find eigenvalues and eigenvectors
|
|
# from this equation [eqn. 28]
|
|
eig_vals, eig_vecs = np.linalg.eig(M)
|
|
|
|
# eigenvector must meet constraint 4ac - b^2 to be valid.
|
|
cond = 4 * np.multiply(eig_vecs[0, :], eig_vecs[2, :]) \
|
|
- np.power(eig_vecs[1, :], 2)
|
|
a1 = eig_vecs[:, (cond > 0)]
|
|
# seeks for empty matrix
|
|
if 0 in a1.shape or len(a1.ravel()) != 3:
|
|
return False
|
|
a, b, c = a1.ravel()
|
|
|
|
# |d f g> = -S3^(-1)*S2^(T)*|a b c> [eqn. 24]
|
|
a2 = -inv(S3) @ S2.T @ a1
|
|
d, f, g = a2.ravel()
|
|
|
|
# eigenvectors are the coefficients of an ellipse in general form
|
|
# a*x^2 + 2*b*x*y + c*y^2 + 2*d*x + 2*f*y + g = 0 (eqn. 15) from [2]
|
|
b /= 2.
|
|
d /= 2.
|
|
f /= 2.
|
|
|
|
# finding center of ellipse [eqn.19 and 20] from [2]
|
|
x0 = (c * d - b * f) / (b ** 2. - a * c)
|
|
y0 = (a * f - b * d) / (b ** 2. - a * c)
|
|
|
|
# Find the semi-axes lengths [eqn. 21 and 22] from [2]
|
|
numerator = a * f ** 2 + c * d ** 2 + g * b ** 2 \
|
|
- 2 * b * d * f - a * c * g
|
|
term = np.sqrt((a - c) ** 2 + 4 * b ** 2)
|
|
denominator1 = (b ** 2 - a * c) * (term - (a + c))
|
|
denominator2 = (b ** 2 - a * c) * (- term - (a + c))
|
|
width = np.sqrt(2 * numerator / denominator1)
|
|
height = np.sqrt(2 * numerator / denominator2)
|
|
|
|
# angle of counterclockwise rotation of major-axis of ellipse
|
|
# to x-axis [eqn. 23] from [2].
|
|
phi = 0.5 * np.arctan((2. * b) / (a - c))
|
|
if a > c:
|
|
phi += 0.5 * np.pi
|
|
|
|
self.params = np.nan_to_num([x0, y0, width, height, phi]).tolist()
|
|
self.params = [float(np.real(x)) for x in self.params]
|
|
return True
|
|
|
|
def residuals(self, data):
|
|
"""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.
|
|
|
|
"""
|
|
|
|
_check_data_dim(data, dim=2)
|
|
|
|
xc, yc, a, b, theta = self.params
|
|
|
|
ctheta = math.cos(theta)
|
|
stheta = math.sin(theta)
|
|
|
|
x = data[:, 0]
|
|
y = data[:, 1]
|
|
|
|
N = data.shape[0]
|
|
|
|
def fun(t, xi, yi):
|
|
ct = math.cos(t)
|
|
st = math.sin(t)
|
|
xt = xc + a * ctheta * ct - b * stheta * st
|
|
yt = yc + a * stheta * ct + b * ctheta * st
|
|
return (xi - xt) ** 2 + (yi - yt) ** 2
|
|
|
|
# def Dfun(t, xi, yi):
|
|
# ct = math.cos(t)
|
|
# st = math.sin(t)
|
|
# xt = xc + a * ctheta * ct - b * stheta * st
|
|
# yt = yc + a * stheta * ct + b * ctheta * st
|
|
# dfx_t = - 2 * (xi - xt) * (- a * ctheta * st
|
|
# - b * stheta * ct)
|
|
# dfy_t = - 2 * (yi - yt) * (- a * stheta * st
|
|
# + b * ctheta * ct)
|
|
# return [dfx_t + dfy_t]
|
|
|
|
residuals = np.empty((N, ), dtype=np.double)
|
|
|
|
# initial guess for parameter t of closest point on ellipse
|
|
t0 = np.arctan2(y - yc, x - xc) - theta
|
|
|
|
# determine shortest distance to ellipse for each point
|
|
for i in range(N):
|
|
xi = x[i]
|
|
yi = y[i]
|
|
# faster without Dfun, because of the python overhead
|
|
t, _ = optimize.leastsq(fun, t0[i], args=(xi, yi))
|
|
residuals[i] = np.sqrt(fun(t, xi, yi))
|
|
|
|
return residuals
|
|
|
|
def predict_xy(self, t, params=None):
|
|
"""Predict x- and y-coordinates using the estimated model.
|
|
|
|
Parameters
|
|
----------
|
|
t : array
|
|
Angles in circle in radians. Angles start to count from positive
|
|
x-axis to positive y-axis in a right-handed system.
|
|
params : (5, ) array, optional
|
|
Optional custom parameter set.
|
|
|
|
Returns
|
|
-------
|
|
xy : (..., 2) array
|
|
Predicted x- and y-coordinates.
|
|
|
|
"""
|
|
|
|
if params is None:
|
|
params = self.params
|
|
|
|
xc, yc, a, b, theta = params
|
|
|
|
ct = np.cos(t)
|
|
st = np.sin(t)
|
|
ctheta = math.cos(theta)
|
|
stheta = math.sin(theta)
|
|
|
|
x = xc + a * ctheta * ct - b * stheta * st
|
|
y = yc + a * stheta * ct + b * ctheta * st
|
|
|
|
return np.concatenate((x[..., None], y[..., None]), axis=t.ndim)
|
|
|
|
|
|
def _dynamic_max_trials(n_inliers, n_samples, min_samples, probability):
|
|
"""Determine number trials such that at least one outlier-free subset is
|
|
sampled for the given inlier/outlier ratio.
|
|
|
|
Parameters
|
|
----------
|
|
n_inliers : int
|
|
Number of inliers in the data.
|
|
n_samples : int
|
|
Total number of samples in the data.
|
|
min_samples : int
|
|
Minimum number of samples chosen randomly from original data.
|
|
probability : float
|
|
Probability (confidence) that one outlier-free sample is generated.
|
|
|
|
Returns
|
|
-------
|
|
trials : int
|
|
Number of trials.
|
|
"""
|
|
if probability == 0:
|
|
return 0
|
|
if n_inliers == 0:
|
|
return np.inf
|
|
inlier_ratio = n_inliers / n_samples
|
|
nom = max(_EPSILON, 1 - probability)
|
|
denom = max(_EPSILON, 1 - inlier_ratio ** min_samples)
|
|
return np.ceil(np.log(nom) / np.log(denom))
|
|
|
|
|
|
def ransac(data, model_class, min_samples, residual_threshold,
|
|
is_data_valid=None, is_model_valid=None,
|
|
max_trials=100, stop_sample_num=np.inf, stop_residuals_sum=0,
|
|
stop_probability=1, random_state=None, initial_inliers=None):
|
|
"""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, ...) array
|
|
Data set to which the model is fitted, where N is the number of data
|
|
points and the remaining dimension are depending on model requirements.
|
|
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_class : object
|
|
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_samples : int in range (0, N)
|
|
The minimum number of data points to fit a model to.
|
|
residual_threshold : float larger than 0
|
|
Maximum distance for a data point to be classified as an inlier.
|
|
is_data_valid : function, optional
|
|
This function is called with the randomly selected data before the
|
|
model is fitted to it: `is_data_valid(*random_data)`.
|
|
is_model_valid : function, optional
|
|
This function is called with the estimated model and the randomly
|
|
selected data: `is_model_valid(model, *random_data)`, .
|
|
max_trials : int, optional
|
|
Maximum number of iterations for random sample selection.
|
|
stop_sample_num : int, optional
|
|
Stop iteration if at least this number of inliers are found.
|
|
stop_residuals_sum : float, optional
|
|
Stop iteration if sum of residuals is less than or equal to this
|
|
threshold.
|
|
stop_probability : float in range [0, 1], optional
|
|
RANSAC iteration stops if at least one outlier-free set of the
|
|
training data is sampled with ``probability >= stop_probability``,
|
|
depending on the current best model's inlier ratio and the number
|
|
of trials. This requires to generate at least N samples (trials):
|
|
|
|
N >= log(1 - probability) / log(1 - e**m)
|
|
|
|
where the probability (confidence) is typically set to a high value
|
|
such as 0.99, e is the current fraction of inliers w.r.t. the
|
|
total number of samples, and m is the min_samples value.
|
|
random_state : {None, int, `numpy.random.Generator`}, optional
|
|
If `random_state` is None the `numpy.random.Generator` singleton is
|
|
used.
|
|
If `random_state` is an int, a new ``Generator`` instance is used,
|
|
seeded with `random_state`.
|
|
If `random_state` is already a ``Generator`` instance then that
|
|
instance is used.
|
|
initial_inliers : array-like of bool, shape (N,), optional
|
|
Initial samples selection for model estimation
|
|
|
|
|
|
Returns
|
|
-------
|
|
model : object
|
|
Best model with largest consensus set.
|
|
inliers : (N, ) array
|
|
Boolean mask of inliers classified as ``True``.
|
|
|
|
References
|
|
----------
|
|
.. [1] "RANSAC", Wikipedia, https://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])
|
|
>>> rng = np.random.default_rng(203560) # do not copy this value
|
|
>>> data += rng.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., 10., 6., 2.])
|
|
>>> 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
|
|
|
|
RANSAC can be used to robustly estimate a geometric
|
|
transformation. In this section, we also show how to use a
|
|
proportion of the total samples, rather than an absolute number.
|
|
|
|
>>> from skimage.transform import SimilarityTransform
|
|
>>> rng = np.random.default_rng()
|
|
>>> src = 100 * rng.random((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)
|
|
>>> ratio = 0.5 # use half of the samples
|
|
>>> min_samples = int(ratio * len(src))
|
|
>>> model, inliers = ransac((src, dst), SimilarityTransform, min_samples,
|
|
... 10,
|
|
... initial_inliers=np.ones(len(src), dtype=bool))
|
|
>>> inliers # doctest: +SKIP
|
|
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])
|
|
|
|
"""
|
|
|
|
best_inlier_num = 0
|
|
best_inlier_residuals_sum = np.inf
|
|
best_inliers = []
|
|
validate_model = is_model_valid is not None
|
|
validate_data = is_data_valid is not None
|
|
|
|
random_state = np.random.default_rng(random_state)
|
|
|
|
# in case data is not pair of input and output, male it like it
|
|
if not isinstance(data, (tuple, list)):
|
|
data = (data, )
|
|
num_samples = len(data[0])
|
|
|
|
if not (0 < min_samples < num_samples):
|
|
raise ValueError(f"`min_samples` must be in range (0, {num_samples})")
|
|
|
|
if residual_threshold < 0:
|
|
raise ValueError("`residual_threshold` must be greater than zero")
|
|
|
|
if max_trials < 0:
|
|
raise ValueError("`max_trials` must be greater than zero")
|
|
|
|
if not (0 <= stop_probability <= 1):
|
|
raise ValueError("`stop_probability` must be in range [0, 1]")
|
|
|
|
if initial_inliers is not None and len(initial_inliers) != num_samples:
|
|
raise ValueError(
|
|
f"RANSAC received a vector of initial inliers (length "
|
|
f"{len(initial_inliers)}) that didn't match the number of "
|
|
f"samples ({num_samples}). The vector of initial inliers should "
|
|
f"have the same length as the number of samples and contain only "
|
|
f"True (this sample is an initial inlier) and False (this one "
|
|
f"isn't) values.")
|
|
|
|
# for the first run use initial guess of inliers
|
|
spl_idxs = (initial_inliers if initial_inliers is not None
|
|
else random_state.choice(num_samples, min_samples,
|
|
replace=False))
|
|
|
|
# estimate model for current random sample set
|
|
model = model_class()
|
|
|
|
num_trials = 0
|
|
# max_trials can be updated inside the loop, so this cannot be a for-loop
|
|
while num_trials < max_trials:
|
|
num_trials += 1
|
|
|
|
# do sample selection according data pairs
|
|
samples = [d[spl_idxs] for d in data]
|
|
|
|
# for next iteration choose random sample set and be sure that
|
|
# no samples repeat
|
|
spl_idxs = random_state.choice(num_samples, min_samples, replace=False)
|
|
|
|
# optional check if random sample set is valid
|
|
if validate_data and not is_data_valid(*samples):
|
|
continue
|
|
|
|
success = model.estimate(*samples)
|
|
# backwards compatibility
|
|
if success is not None and not success:
|
|
continue
|
|
|
|
# optional check if estimated model is valid
|
|
if validate_model and not is_model_valid(model, *samples):
|
|
continue
|
|
|
|
residuals = np.abs(model.residuals(*data))
|
|
# consensus set / inliers
|
|
inliers = residuals < residual_threshold
|
|
residuals_sum = residuals.dot(residuals)
|
|
|
|
# choose as new best model if number of inliers is maximal
|
|
inliers_count = np.count_nonzero(inliers)
|
|
if (
|
|
# more inliers
|
|
inliers_count > best_inlier_num
|
|
# same number of inliers but less "error" in terms of residuals
|
|
or (inliers_count == best_inlier_num
|
|
and residuals_sum < best_inlier_residuals_sum)):
|
|
best_inlier_num = inliers_count
|
|
best_inlier_residuals_sum = residuals_sum
|
|
best_inliers = inliers
|
|
max_trials = min(max_trials,
|
|
_dynamic_max_trials(best_inlier_num,
|
|
num_samples,
|
|
min_samples,
|
|
stop_probability))
|
|
if (best_inlier_num >= stop_sample_num
|
|
or best_inlier_residuals_sum <= stop_residuals_sum):
|
|
break
|
|
|
|
# estimate final model using all inliers
|
|
if any(best_inliers):
|
|
# select inliers for each data array
|
|
data_inliers = [d[best_inliers] for d in data]
|
|
model.estimate(*data_inliers)
|
|
if validate_model and not is_model_valid(model, *data_inliers):
|
|
warn("Estimated model is not valid. Try increasing max_trials.")
|
|
else:
|
|
model = None
|
|
best_inliers = None
|
|
warn("No inliers found. Model not fitted")
|
|
|
|
return model, best_inliers
|