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Add ellipse estimator model
This commit is contained in:
Johannes Schönberger
committed by
Johannes Schönberger
Johannes Schönberger
parent
4a93f38395
commit
47c3ebac1c
+188
-11
@@ -27,6 +27,8 @@ class LineModel(BaseModel):
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dist, theta
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A minimum number of 2 points is required to solve for the parameters.
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'''
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def estimate(self, data):
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@@ -100,13 +102,15 @@ class LineModel(BaseModel):
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return data.shape[0] < 2
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def predict_x(self, y):
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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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@@ -115,16 +119,20 @@ class LineModel(BaseModel):
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'''
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dist, theta = self._params
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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 * np.cos(theta)) / np.cos(theta)
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def predict_y(self, x):
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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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@@ -133,7 +141,9 @@ class LineModel(BaseModel):
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'''
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dist, theta = self._params
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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 * np.cos(theta)) / np.sin(theta)
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@@ -154,6 +164,8 @@ class CircleModel(BaseModel):
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xc, yc, r
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A minimum number of 3 points is required to solve for the parameters.
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'''
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def estimate(self, data):
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@@ -199,7 +211,7 @@ class CircleModel(BaseModel):
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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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For each point the shortest distance to the circle is returned.
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Parameters
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----------
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@@ -236,16 +248,176 @@ class CircleModel(BaseModel):
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'''
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return data.shape[0] < 2
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return data.shape[0] < 3
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def predict_xy(self, theta):
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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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theta : array
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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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x : array
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Predicted x-coordinates.
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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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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 x, y
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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 d is the
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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, xc,
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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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'''
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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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x = data[:, 0]
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y = data[:, 1]
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N = data.shape[0]
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A = np.empty((5, 2 * N), dtype=np.double)
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def fun(params):
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xt, yt = self.predict_xy(params[5:], params[:5])
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fx = x - xt
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fy = y - yt
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return np.append(fx, fy)
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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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xc, yc, a, b, theta = self._params
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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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xt, yt = self.predict_xy(t)
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return (xi - xt)**2 + (yi - yt)**2
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def Dfun(t, xi, yi):
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xt, yt = self.predict_xy(t)
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dfx = - 2 * (xi - xt) * (- a * np.cos(theta) * np.sin(t)
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- b * np.sin(theta) * np.cos(t))
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dfy = - 2 * (yi - yt) * (- a * np.sin(theta) * np.sin(t)
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+ b * np.cos(theta) * np.cos(t))
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return dfx + dfy
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residuals = np.empty((N, ), dtype=np.double)
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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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t, _ = optimize.leastsq(fun, 0, args=(xi, yi), Dfun=Dfun,
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col_deriv=True)
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residuals[i] = np.sqrt(fun(t, xi, yi))
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return residuals
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@classmethod
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def is_degenerate(cls, data):
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'''Check whether set of points is degenerate.
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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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flag : bool
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Flag indicating if data is degenerate.
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'''
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return data.shape[0] < 5
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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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@@ -256,10 +428,15 @@ class CircleModel(BaseModel):
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'''
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xc, yc, r = self._params
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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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x = xc + r * np.cos(theta)
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y = yc + r * np.sin(theta)
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ct = np.cos(t)
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st = np.sin(t)
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x = xc + a * np.cos(theta) * ct - b * np.sin(theta) * st
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y = yc + a * np.sin(theta) * ct + b * np.cos(theta) * st
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return x, y
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