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blink1073
674 lines
20 KiB
Python
674 lines
20 KiB
Python
import math
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import warnings
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import numpy as np
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from scipy import optimize
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def _check_data_dim(data, dim):
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if data.ndim != 2 or data.shape[1] != dim:
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raise ValueError('Input data must have shape (N, %d).' % dim)
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class BaseModel(object):
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def __init__(self):
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self.params = None
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@property
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def _params(self):
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warnings.warn('`_params` attribute is deprecated, '
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'use `params` instead.')
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return self.params
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class LineModel(BaseModel):
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"""Total least squares estimator for 2D lines.
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Lines are parameterized using polar coordinates as functional model::
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dist = x * cos(theta) + y * sin(theta)
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This parameterization is able to model vertical lines in contrast to the
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standard line model ``y = a*x + b``.
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This estimator minimizes the squared distances from all points to the
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line::
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min{ sum((dist - x_i * cos(theta) + y_i * sin(theta))**2) }
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A minimum number of 2 points is required to solve for the parameters.
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Attributes
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----------
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params : tuple
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Line model parameters in the following order `dist`, `theta`.
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"""
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def estimate(self, data):
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"""Estimate line model from data using total least squares.
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Parameters
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----------
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data : (N, 2) array
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N points with ``(x, y)`` coordinates, respectively.
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"""
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_check_data_dim(data, dim=2)
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X0 = data.mean(axis=0)
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if data.shape[0] == 2: # well determined
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theta = np.arctan2(data[1, 1] - data[0, 1],
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data[1, 0] - data[0, 0])
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elif data.shape[0] > 2: # over-determined
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data = data - X0
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# first principal component
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_, _, v = np.linalg.svd(data)
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theta = np.arctan2(v[0, 1], v[0, 0])
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else: # under-determined
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raise ValueError('At least 2 input points needed.')
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# angle perpendicular to line angle
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theta = (theta + np.pi / 2) % np.pi
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# line always passes through mean
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dist = X0[0] * math.cos(theta) + X0[1] * math.sin(theta)
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self.params = (dist, theta)
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def residuals(self, data):
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"""Determine residuals of data to model.
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For each point the shortest distance to the line is returned.
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Parameters
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----------
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data : (N, 2) array
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N points with ``(x, y)`` coordinates, respectively.
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Returns
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-------
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residuals : (N, ) array
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Residual for each data point.
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"""
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_check_data_dim(data, dim=2)
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dist, theta = self.params
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x = data[:, 0]
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y = data[:, 1]
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return dist - (x * math.cos(theta) + y * math.sin(theta))
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def predict_x(self, y, params=None):
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"""Predict x-coordinates using the estimated model.
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Parameters
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----------
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y : array
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y-coordinates.
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params : (2, ) array, optional
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Optional custom parameter set.
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Returns
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-------
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x : array
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Predicted x-coordinates.
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"""
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if params is None:
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params = self.params
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dist, theta = params
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return (dist - y * math.sin(theta)) / math.cos(theta)
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def predict_y(self, x, params=None):
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"""Predict y-coordinates using the estimated model.
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Parameters
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----------
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x : array
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x-coordinates.
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params : (2, ) array, optional
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Optional custom parameter set.
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Returns
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-------
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y : array
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Predicted y-coordinates.
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"""
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if params is None:
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params = self.params
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dist, theta = params
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return (dist - x * math.cos(theta)) / math.sin(theta)
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class CircleModel(BaseModel):
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"""Total least squares estimator for 2D circles.
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The functional model of the circle is::
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r**2 = (x - xc)**2 + (y - yc)**2
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This estimator minimizes the squared distances from all points to the
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circle::
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min{ sum((r - sqrt((x_i - xc)**2 + (y_i - yc)**2))**2) }
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A minimum number of 3 points is required to solve for the parameters.
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Attributes
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----------
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params : tuple
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Circle model parameters in the following order `xc`, `yc`, `r`.
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"""
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def estimate(self, data):
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"""Estimate circle model from data using total least squares.
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Parameters
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----------
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data : (N, 2) array
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N points with ``(x, y)`` coordinates, respectively.
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"""
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_check_data_dim(data, dim=2)
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x = data[:, 0]
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y = data[:, 1]
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# pre-allocate jacobian for all iterations
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A = np.zeros((3, data.shape[0]), dtype=np.double)
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# same for all iterations: r
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A[2, :] = -1
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def dist(xc, yc):
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return np.sqrt((x - xc)**2 + (y - yc)**2)
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def fun(params):
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xc, yc, r = params
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return dist(xc, yc) - r
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def Dfun(params):
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xc, yc, r = params
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d = dist(xc, yc)
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A[0, :] = -(x - xc) / d
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A[1, :] = -(y - yc) / d
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# same for all iterations, so not changed in each iteration
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#A[2, :] = -1
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return A
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xc0 = x.mean()
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yc0 = y.mean()
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r0 = dist(xc0, yc0).mean()
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params0 = (xc0, yc0, r0)
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params, _ = optimize.leastsq(fun, params0, Dfun=Dfun, col_deriv=True)
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self.params = params
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def residuals(self, data):
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"""Determine residuals of data to model.
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For each point the shortest distance to the circle is returned.
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Parameters
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----------
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data : (N, 2) array
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N points with ``(x, y)`` coordinates, respectively.
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Returns
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-------
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residuals : (N, ) array
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Residual for each data point.
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"""
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_check_data_dim(data, dim=2)
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xc, yc, r = self.params
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x = data[:, 0]
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y = data[:, 1]
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return r - np.sqrt((x - xc)**2 + (y - yc)**2)
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def predict_xy(self, t, params=None):
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"""Predict x- and y-coordinates using the estimated model.
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Parameters
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----------
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t : array
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Angles in circle in radians. Angles start to count from positive
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x-axis to positive y-axis in a right-handed system.
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params : (3, ) array, optional
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Optional custom parameter set.
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Returns
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-------
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xy : (..., 2) array
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Predicted x- and y-coordinates.
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"""
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if params is None:
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params = self.params
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xc, yc, r = params
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x = xc + r * np.cos(t)
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y = yc + r * np.sin(t)
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return np.concatenate((x[..., None], y[..., None]), axis=t.ndim)
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class EllipseModel(BaseModel):
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"""Total least squares estimator for 2D ellipses.
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The functional model of the ellipse is::
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xt = xc + a*cos(theta)*cos(t) - b*sin(theta)*sin(t)
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yt = yc + a*sin(theta)*cos(t) + b*cos(theta)*sin(t)
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d = sqrt((x - xt)**2 + (y - yt)**2)
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where ``(xt, yt)`` is the closest point on the ellipse to ``(x, y)``. Thus
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d is the shortest distance from the point to the ellipse.
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This estimator minimizes the squared distances from all points to the
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ellipse::
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min{ sum(d_i**2) } = min{ sum((x_i - xt)**2 + (y_i - yt)**2) }
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Thus you have ``2 * N`` equations (x_i, y_i) for ``N + 5`` unknowns (t_i,
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xc, yc, a, b, theta), which gives you an effective redundancy of ``N - 5``.
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The ``params`` attribute contains the parameters in the following order::
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xc, yc, a, b, theta
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A minimum number of 5 points is required to solve for the parameters.
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Attributes
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----------
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params : tuple
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Ellipse model parameters in the following order `xc`, `yc`, `a`,
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`b`, `theta`.
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"""
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def estimate(self, data):
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"""Estimate circle model from data using total least squares.
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Parameters
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----------
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data : (N, 2) array
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N points with ``(x, y)`` coordinates, respectively.
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"""
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_check_data_dim(data, dim=2)
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x = data[:, 0]
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y = data[:, 1]
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N = data.shape[0]
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# pre-allocate jacobian for all iterations
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A = np.zeros((N + 5, 2 * N), dtype=np.double)
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# same for all iterations: xc, yc
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A[0, :N] = -1
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A[1, N:] = -1
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diag_idxs = np.diag_indices(N)
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def fun(params):
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xyt = self.predict_xy(params[5:], params[:5])
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fx = x - xyt[:, 0]
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fy = y - xyt[:, 1]
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return np.append(fx, fy)
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def Dfun(params):
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xc, yc, a, b, theta = params[:5]
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t = params[5:]
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ct = np.cos(t)
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st = np.sin(t)
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ctheta = math.cos(theta)
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stheta = math.sin(theta)
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# derivatives for fx, fy in the following order:
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# xc, yc, a, b, theta, t_i
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# fx
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A[2, :N] = - ctheta * ct
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A[3, :N] = stheta * st
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A[4, :N] = a * stheta * ct + b * ctheta * st
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A[5:, :N][diag_idxs] = a * ctheta * st + b * stheta * ct
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# fy
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A[2, N:] = - stheta * ct
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A[3, N:] = - ctheta * st
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A[4, N:] = - a * ctheta * ct + b * stheta * st
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A[5:, N:][diag_idxs] = a * stheta * st - b * ctheta * ct
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return A
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# initial guess of parameters using a circle model
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params0 = np.empty((N + 5, ), dtype=np.double)
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xc0 = x.mean()
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yc0 = y.mean()
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r0 = np.sqrt((x - xc0)**2 + (y - yc0)**2).mean()
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params0[:5] = (xc0, yc0, r0, 0, 0)
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params0[5:] = np.arctan2(y - yc0, x - xc0)
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params, _ = optimize.leastsq(fun, params0, Dfun=Dfun, col_deriv=True)
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self.params = params[:5]
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def residuals(self, data):
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"""Determine residuals of data to model.
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For each point the shortest distance to the ellipse is returned.
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Parameters
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----------
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data : (N, 2) array
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N points with ``(x, y)`` coordinates, respectively.
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Returns
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-------
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residuals : (N, ) array
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Residual for each data point.
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"""
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_check_data_dim(data, dim=2)
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xc, yc, a, b, theta = self.params
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ctheta = math.cos(theta)
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stheta = math.sin(theta)
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x = data[:, 0]
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y = data[:, 1]
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N = data.shape[0]
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def fun(t, xi, yi):
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ct = math.cos(t)
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st = math.sin(t)
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xt = xc + a * ctheta * ct - b * stheta * st
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yt = yc + a * stheta * ct + b * ctheta * st
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return (xi - xt)**2 + (yi - yt)**2
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# def Dfun(t, xi, yi):
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# ct = math.cos(t)
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# st = math.sin(t)
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# xt = xc + a * ctheta * ct - b * stheta * st
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# yt = yc + a * stheta * ct + b * ctheta * st
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# dfx_t = - 2 * (xi - xt) * (- a * ctheta * st
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# - b * stheta * ct)
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# dfy_t = - 2 * (yi - yt) * (- a * stheta * st
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# + b * ctheta * ct)
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# return [dfx_t + dfy_t]
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residuals = np.empty((N, ), dtype=np.double)
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# initial guess for parameter t of closest point on ellipse
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t0 = np.arctan2(y - yc, x - xc) - theta
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# determine shortest distance to ellipse for each point
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for i in range(N):
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xi = x[i]
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yi = y[i]
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# faster without Dfun, because of the python overhead
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t, _ = optimize.leastsq(fun, t0[i], args=(xi, yi))
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residuals[i] = np.sqrt(fun(t, xi, yi))
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return residuals
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def predict_xy(self, t, params=None):
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"""Predict x- and y-coordinates using the estimated model.
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Parameters
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----------
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t : array
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Angles in circle in radians. Angles start to count from positive
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x-axis to positive y-axis in a right-handed system.
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params : (5, ) array, optional
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Optional custom parameter set.
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Returns
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-------
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xy : (..., 2) array
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Predicted x- and y-coordinates.
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"""
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if params is None:
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params = self.params
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xc, yc, a, b, theta = params
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ct = np.cos(t)
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st = np.sin(t)
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ctheta = math.cos(theta)
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stheta = math.sin(theta)
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x = xc + a * ctheta * ct - b * stheta * st
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y = yc + a * stheta * ct + b * ctheta * st
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return np.concatenate((x[..., None], y[..., None]), axis=t.ndim)
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def ransac(data, model_class, min_samples, residual_threshold,
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is_data_valid=None, is_model_valid=None,
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max_trials=100, stop_sample_num=np.inf, stop_residuals_sum=0):
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"""Fit a model to data with the RANSAC (random sample consensus) algorithm.
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RANSAC is an iterative algorithm for the robust estimation of parameters
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from a subset of inliers from the complete data set. Each iteration
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performs the following tasks:
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1. Select `min_samples` random samples from the original data and check
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whether the set of data is valid (see `is_data_valid`).
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2. Estimate a model to the random subset
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(`model_cls.estimate(*data[random_subset]`) and check whether the
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estimated model is valid (see `is_model_valid`).
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3. Classify all data as inliers or outliers by calculating the residuals
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to the estimated model (`model_cls.residuals(*data)`) - all data samples
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with residuals smaller than the `residual_threshold` are considered as
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inliers.
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4. Save estimated model as best model if number of inlier samples is
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maximal. In case the current estimated model has the same number of
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inliers, it is only considered as the best model if it has less sum of
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residuals.
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These steps are performed either a maximum number of times or until one of
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the special stop criteria are met. The final model is estimated using all
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inlier samples of the previously determined best model.
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Parameters
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----------
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data : [list, tuple of] (N, D) array
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Data set to which the model is fitted, where N is the number of data
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points and D the dimensionality of the data.
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If the model class requires multiple input data arrays (e.g. source and
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destination coordinates of ``skimage.transform.AffineTransform``),
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they can be optionally passed as tuple or list. Note, that in this case
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the functions ``estimate(*data)``, ``residuals(*data)``,
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``is_model_valid(model, *random_data)`` and
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``is_data_valid(*random_data)`` must all take each data array as
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separate arguments.
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model_class : object
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Object with the following object methods:
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* ``estimate(*data)``
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* ``residuals(*data)``
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min_samples : int
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The minimum number of data points to fit a model to.
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residual_threshold : float
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Maximum distance for a data point to be classified as an inlier.
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is_data_valid : function, optional
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This function is called with the randomly selected data before the
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model is fitted to it: `is_data_valid(*random_data)`.
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is_model_valid : function, optional
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This function is called with the estimated model and the randomly
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selected data: `is_model_valid(model, *random_data)`, .
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max_trials : int, optional
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Maximum number of iterations for random sample selection.
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stop_sample_num : int, optional
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Stop iteration if at least this number of inliers are found.
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stop_residuals_sum : float, optional
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Stop iteration if sum of residuals is less equal than this threshold.
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Returns
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-------
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model : object
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Best model with largest consensus set.
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inliers : (N, ) array
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Boolean mask of inliers classified as ``True``.
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References
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----------
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.. [1] "RANSAC", Wikipedia, http://en.wikipedia.org/wiki/RANSAC
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Examples
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--------
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Generate ellipse data without tilt and add noise:
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>>> t = np.linspace(0, 2 * np.pi, 50)
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>>> a = 5
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>>> b = 10
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>>> xc = 20
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>>> yc = 30
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>>> x = xc + a * np.cos(t)
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>>> y = yc + b * np.sin(t)
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|
>>> data = np.column_stack([x, y])
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|
>>> np.random.seed(seed=1234)
|
|
>>> data += np.random.normal(size=data.shape)
|
|
|
|
Add some faulty data:
|
|
|
|
>>> data[0] = (100, 100)
|
|
>>> data[1] = (110, 120)
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|
>>> data[2] = (120, 130)
|
|
>>> data[3] = (140, 130)
|
|
|
|
Estimate ellipse model using all available data:
|
|
|
|
>>> model = EllipseModel()
|
|
>>> model.estimate(data)
|
|
>>> model.params # doctest: +SKIP
|
|
array([ -3.30354146e+03, -2.87791160e+03, 5.59062118e+03,
|
|
7.84365066e+00, 7.19203152e-01])
|
|
|
|
|
|
Estimate ellipse model using RANSAC:
|
|
|
|
>>> ransac_model, inliers = ransac(data, EllipseModel, 5, 3, max_trials=50)
|
|
>>> ransac_model.params
|
|
array([ 20.12762373, 29.73563063, 4.81499637, 10.4743584 , 0.05217117])
|
|
>>> inliers
|
|
array([False, False, False, False, True, True, True, True, True,
|
|
True, True, True, True, True, True, True, True, True,
|
|
True, True, True, True, True, True, True, True, True,
|
|
True, True, True, True, True, True, True, True, True,
|
|
True, True, True, True, True, True, True, True, True,
|
|
True, True, True, True, True], dtype=bool)
|
|
|
|
Robustly estimate geometric transformation:
|
|
|
|
>>> from skimage.transform import SimilarityTransform
|
|
>>> np.random.seed(0)
|
|
>>> src = 100 * np.random.rand(50, 2)
|
|
>>> model0 = SimilarityTransform(scale=0.5, rotation=1,
|
|
... translation=(10, 20))
|
|
>>> dst = model0(src)
|
|
>>> dst[0] = (10000, 10000)
|
|
>>> dst[1] = (-100, 100)
|
|
>>> dst[2] = (50, 50)
|
|
>>> model, inliers = ransac((src, dst), SimilarityTransform, 2, 10)
|
|
>>> inliers
|
|
array([False, False, False, True, True, True, True, True, True,
|
|
True, True, True, True, True, True, True, True, True,
|
|
True, True, True, True, True, True, True, True, True,
|
|
True, True, True, True, True, True, True, True, True,
|
|
True, True, True, True, True, True, True, True, True,
|
|
True, True, True, True, True], dtype=bool)
|
|
|
|
"""
|
|
|
|
best_model = None
|
|
best_inlier_num = 0
|
|
best_inlier_residuals_sum = np.inf
|
|
best_inliers = None
|
|
|
|
if not isinstance(data, list) and not isinstance(data, tuple):
|
|
data = [data]
|
|
|
|
# make sure data is list and not tuple, so it can be modified below
|
|
data = list(data)
|
|
# number of samples
|
|
N = data[0].shape[0]
|
|
|
|
for _ in range(max_trials):
|
|
|
|
# choose random sample set
|
|
samples = []
|
|
random_idxs = np.random.randint(0, N, min_samples)
|
|
for d in data:
|
|
samples.append(d[random_idxs])
|
|
|
|
# check if random sample set is valid
|
|
if is_data_valid is not None and not is_data_valid(*samples):
|
|
continue
|
|
|
|
# estimate model for current random sample set
|
|
sample_model = model_class()
|
|
sample_model.estimate(*samples)
|
|
|
|
# check if estimated model is valid
|
|
if is_model_valid is not None and not is_model_valid(sample_model,
|
|
*samples):
|
|
continue
|
|
|
|
sample_model_residuals = np.abs(sample_model.residuals(*data))
|
|
# consensus set / inliers
|
|
sample_model_inliers = sample_model_residuals < residual_threshold
|
|
sample_model_residuals_sum = np.sum(sample_model_residuals**2)
|
|
|
|
# choose as new best model if number of inliers is maximal
|
|
sample_inlier_num = np.sum(sample_model_inliers)
|
|
if (
|
|
# more inliers
|
|
sample_inlier_num > best_inlier_num
|
|
# same number of inliers but less "error" in terms of residuals
|
|
or (sample_inlier_num == best_inlier_num
|
|
and sample_model_residuals_sum < best_inlier_residuals_sum)
|
|
):
|
|
best_model = sample_model
|
|
best_inlier_num = sample_inlier_num
|
|
best_inlier_residuals_sum = sample_model_residuals_sum
|
|
best_inliers = sample_model_inliers
|
|
if (
|
|
best_inlier_num >= stop_sample_num
|
|
or best_inlier_residuals_sum <= stop_residuals_sum
|
|
):
|
|
break
|
|
|
|
# estimate final model using all inliers
|
|
if best_inliers is not None:
|
|
# select inliers for each data array
|
|
for i in range(len(data)):
|
|
data[i] = data[i][best_inliers]
|
|
best_model.estimate(*data)
|
|
|
|
return best_model, best_inliers
|