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|
""" 2D Ellipse fitting
Fits an ellipse to a set of points (x_i, y_i) using the canonical
representation:
a * x^2 + b * x * y + c * y^2 + d * x + e * y + f = 0 (1)
Provided features
-----------------
The module provides several function related to ellipses:
`fit_ellipse`:
fits an ellipse from a set of points and return the parameters
of the canonical representation (see above)
`get_parameters`:
converts canonical parameters into intuitive representation i.e.
major and minor radii
let a_ be the vector a_ = [a, b, c, d, e, f]'
Let D be the (N x 6) design matrix:
D = [z_1 z_2 ... z_n]'
where
z_i = [ x_i^2, x_i * y_i, y_i^2, x_i, y_i, 1 ]'
We want to minimize
E = \sum_i (a_' * z_i)^2 = || D * a_ ||^2 = a_' * S * a
where
S = D' * D
If equation (1) corresponds to an ellipse we must have:
4 * a * c - b^2 > 0
Since equation (1) is unique up to a scaling factor we can impose:
4 * a * c - b^2 = 1
which can be written in matrix form as:
a_' * C * a_ = 1
with
::
C = | 0 0 2 0 0 0 |
| 0 -1 0 0 0 0 |
| 2 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
So the problem reduces to:
a_ = argmin a_' S a_
s.t. (2)
a_' * C * a_ = 1
which is equivalent to solving:
S a_ = l * C * a_ (3)
where l is a Lagrange multiplier.
Equation (2) is just a generalized eigen value problem. And the solution
to (1) is the eigen vector corresponding to the smallest positive eigen
value of (2).
It can be prooved that (3) has 2 negative eigen values and one positive.
The biggest eigen value the corresponds to the solution.
Since C has negative eigen values solvers in scipy/numpy are not able
to perform the eigen decomposition. To solve (2) we reduce the problem
to a 3x3 eigen value problem for wich we can solve the problem
analytically. For that, we split S and C into 3x3 blocks.
Let's define:
::
S = | A B |
| B' E |
C = | F 0 |
| 0 0 |
a_ = [x' y']'
Rewritting (3) we get:
A * x + B * y = l * F * x
B'* x + E * y = 0
which gives:
y = - E^(-1) * B' * x
(A - B * E^(-1) * B') * x = l * F * x (4)
Equation (4) is a 3x3 eigen value problem that we can solve analytically.
:Notes:
Detailed explanations can be found in:
*Direct Least square fitting of Ellipses*. A. Fitzgibbon, M. Pilu,
and R. B. Fisher. Pattern Analysis and Machine Intelligence. 1999
:Author: Alexis Mignon (c) 2012
:E-mail: alexis.mignon@gmail.com
"""
import numpy as np
from scipy.linalg import inv, eigh, solve
def _find_max_eigval(S):
"""
Finds the biggest generalized eigen value of the system
S * x = l * C * x
where
::
C = | 0 0 2 |
| 0 -1 0 |
| 2 0 1 |
Parameters:
-----------
S : 3x3 matrix
Returns:
--------
the highest eigen value
"""
a = S[0,0]
b = S[0,1]
c = S[0,2]
d = S[1,1]
e = S[1,2]
f = S[2,2]
# computes the coefficients of the caracteristique polynomial
# det(S - x * C) = 0
# Since the matrix is 3x3 we have a 3rd degree polynomial
# _a * x**3 + _b * x**2 + _c * x + _d
_a = -4
_b = 4 * (c - d)
_c = a * f - 4 * b * e + 4 * c * d - c * c
_d = a * d * f - b * b * f - a * e * e + 2 * b * c * e - c * c * d
# computes the roots of the polynomial
# there must be 2 negative roots and one
# positive, i.e. the biggest one.
x2, x1, x0 = sorted(np.roots([_a, _b, _c, _d] ))
return x0
def _find_max_eigvec(S):
"""
Computes the positive eigen value and the corresponding
eigen vector of the system:
S * x = l * C * x
where
::
C = | 0 0 2 |
| 0 -1 0 |
| 2 0 1 |
Parameters:
-----------
S : 3x3 matrix
Returns:
--------
(l, u)
l : float
the positive eigen value
u : the corresponding eigen vector
"""
l = _find_max_eigval(S)
a11 = S[0,0]
a12 = S[0,1]
a13 = S[0,2]
a22 = S[1,1]
a23 = S[1,2]
u = np.array([
a12 * a23 - (a13 - 2*l) * (a22 + l),
a12 * (a13 - 2*l) - a23 * a11,
a11 * (a22 + l) - a12 * a12
])
c = 4 * u[0] * u[2] - u[1] * u[1]
return l, u/np.sqrt(c)
def fit_ellipse(X):
""" Fit an ellipse.
Computes the best least squares parameters of an ellipse expressed as:
a * x^2 + b * x * y + c * y^2 + d * x + e * y + f = 0
Parameters
----------
X : N x 2 array
an array of N 2d points.
Returns:
--------
an array containing the parameters:
[ a , b, c, d, e, f]
"""
x = X[:,0]
y = X[:,1]
# building the design matrix
D = np.vstack([ x*x, x*y, y*y, x, y, np.ones(X.shape[0])]).T
S = np.dot(D.T, D)
S11 = S[:3][:,:3]
S12 = S[:3][:,3:]
S22 = S[3:][:,3:]
S22_inv = inv(S22)
S22_inv_S21 = np.dot(inv(S22), S12.T)
Sc = S11 - np.dot(S12, S22_inv_S21)
l, a = _find_max_eigvec(Sc)
b = - np.dot(S22_inv_S21, a)
return np.hstack([a,b])
def create_ellipse(r, xc, alpha, n=100, angle_range=(0,2*np.pi)):
""" Create points on an ellipse with uniform angle step
Parameters
----------
r: tuple
(rx, ry): major an minor radii of the ellipse. Radii are supposed to
be given in descending order. No check will be done.
xc : tuple
x and y coordinates of the center of the ellipse
alpha : float
angle between the x axis and the major axis of the ellipse
n : int, optional
The number of points to create
angle_range : tuple (a0, a1)
angles between which points are created.
Returns
-------
(n * 2) array of points
"""
R = np.array([
[np.cos(alpha), -np.sin(alpha)],
[np.sin(alpha), np.cos(alpha)]
])
a0,a1 = angle_range
angles = np.linspace(a0,a1,n)
X = np.vstack([ np.cos(angles) * r[0], np.sin(angles) * r[1]]).T
return np.dot(X,R.T) + xc
def get_parameters(x):
"""
Computes 'natural' parameters of an ellipse given the parameters
of the canonical equation:
a * x^2 + b * x * y + c * y^2 + d * x + e * y + f = 0
Parameters:
-----------
x : array_like
An array of 6 elements corresponding to the coefficients of the
canonical equation (see above)
Returns:
--------
tuple (rx, ry), (xc, yc), alpha
(rx, ry) : tuple
Radii of the major and minor axes
(xc, yc) : tuple
coordinates of the center
alpha : float
angle between the x axis and the major axis
:Note:
Computed the parameters of the ellipse when it is expressed as:
x'^2/rx^2 + y'/ry^2 = 1
where x' and y' correpsond to the rotated coordinates:
x' = cos(alpha)(x-xc) + sin(alpha)(y-yc)
y' = -sin(alpha)(x-xc) + cos(alpha)(y-yc)
Which can be put in matrix form as
(X-Xc)' R D R' (X-Xc) = 1
where
::
X = [x y] and Xc = [xc yc]
R = [ cos(alpha) -sin(alpha)]
[ sin(alpha) cos(alpha) ]
D = [ 1/rx^2 0 ]
[ 0 1/ry^2 ]
Parameters are given as the parameter of the conic:
a * x^2 + b * x * y + c * y^2 + d * x + e * y + f = 0
In matrix form we have:
X' A X + B' X + f = 0
where
::
X = [ x y ]'
A = [ a b/2]
[b/2 c ]
B = [ d e ]'
Any ellipse can be written as:
(X - Xc)' A (X - Xc) = r^2
which develops in:
X'A X - 2 * Xc' A X + Xc' A Xc - r^2 = 0
So we have:
B = - 2 * A Xc
and
f = Xc' A Xc - r^2
and thus:
Xc = -1/2 * A^(-1) B
r^2 = Xc' A Xc - f
We also see that
1/r^2 * (X - Xc)' A (X - Xc) = (X-Xc)' R D R' (X-Xc) = 1
By performing eigen decomposition on A = U L U', we obtain
R = U
and
lx / r^2 = 1/rx^2
ly / r^2 = 1/ry^2
hence
rx^2 = r^2 / lx
ry^2 = r^2 / ly
the angle alpha is finally determined using
::
U = | u11, u12 | = | cos(alpha) sin(alpha)|
|-u12, u22 | | -sin(alpha) cos(alpha)|
alpha = sign(u12) * arccos(u11)
"""
a,b,c,d,e,f = x
A = np.array([
[ a, b/2 ],
[b/2, c ]
])
B = np.array([d,e])
w,u = eigh(A)
Xc = solve(-2*A,B)
r2 = -0.5 * np.inner(Xc,B) - f
rr2 = r2 / w
alpha = np.arccos(u[0,0])
if alpha > np.pi/2:
alpha = alpha - np.pi
alpha *= np.sign(u[0,1])
return tuple(np.sqrt(rr2)), tuple(Xc), alpha |
[Python] Ellipse Fitting
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