lsst.gauss2d.utils Namespace Reference
Functions | |
| covar_to_ellipse (sigma_x_sq, sigma_y_sq, cov_xy, degrees=False) | |
| gauss2dint (xdivsigma) | |
Function Documentation
◆ covar_to_ellipse()
| lsst.gauss2d.utils.covar_to_ellipse | ( | sigma_x_sq, | |
| sigma_y_sq, | |||
| cov_xy, | |||
| degrees = False ) |
Convert covariance matrix terms to ellipse major axis, axis ratio and
position angle representation.
Parameters
----------
sigma_x_sq, sigma_y_sq : `float` or array-like
x- and y-axis squared standard deviations of a 2-dimensional normal
distribution (diagonal terms of its covariance matrix).
Must be scalar or identical length array-likes.
cov_xy : `float` or array-like
x-y covariance of a of a 2-dimensional normal distribution
(off-diagonal term of its covariance matrix).
Must be scalar or identical length array-likes.
degrees : `bool`
Whether to return the position angle in degrees instead of radians.
Returns
-------
r_major, axrat, angle : `float` or array-like
Converted major-axis radius, axis ratio and position angle
(counter-clockwise from the +x axis) of the ellipse defined by
each set of input covariance matrix terms.
Notes
-----
The eigenvalues from the determinant of a covariance matrix are:
|a-m b|
|b c-m|
det = (a-m)(c-m) - b^2 = ac - (a+c)m + m^2 - b^2 = m^2 - (a+c)m + (ac-b^2)
Solving:
m = ((a+c) +/- sqrt((a+c)^2 - 4(ac-b^2)))/2
...or equivalently:
m = ((a+c) +/- sqrt((a-c)^2 + 4b^2))/2
Unfortunately, the latter simplification is not as well-behaved
in floating point math, leading to square roots of negative numbers when
one of a or c is very close to zero.
The values from this function should match those from
`Ellipse.make_ellipse_major` to within rounding error, except in the
special case of sigma_x == sigma_y == 0, which returns a NaN axis ratio
here by default. This function mainly intended to be more convenient
(and possibly faster) for array-like inputs.
Definition at line 25 of file utils.py.
25def covar_to_ellipse(sigma_x_sq, sigma_y_sq, cov_xy, degrees=False):
26 """Convert covariance matrix terms to ellipse major axis, axis ratio and
27 position angle representation.
28
29 Parameters
30 ----------
31 sigma_x_sq, sigma_y_sq : `float` or array-like
32 x- and y-axis squared standard deviations of a 2-dimensional normal
33 distribution (diagonal terms of its covariance matrix).
34 Must be scalar or identical length array-likes.
35 cov_xy : `float` or array-like
36 x-y covariance of a of a 2-dimensional normal distribution
37 (off-diagonal term of its covariance matrix).
38 Must be scalar or identical length array-likes.
39 degrees : `bool`
40 Whether to return the position angle in degrees instead of radians.
41
42 Returns
43 -------
44 r_major, axrat, angle : `float` or array-like
45 Converted major-axis radius, axis ratio and position angle
46 (counter-clockwise from the +x axis) of the ellipse defined by
47 each set of input covariance matrix terms.
48
49 Notes
50 -----
51 The eigenvalues from the determinant of a covariance matrix are:
52 |a-m b|
53 |b c-m|
54 det = (a-m)(c-m) - b^2 = ac - (a+c)m + m^2 - b^2 = m^2 - (a+c)m + (ac-b^2)
55 Solving:
56 m = ((a+c) +/- sqrt((a+c)^2 - 4(ac-b^2)))/2
57 ...or equivalently:
58 m = ((a+c) +/- sqrt((a-c)^2 + 4b^2))/2
59
60 Unfortunately, the latter simplification is not as well-behaved
61 in floating point math, leading to square roots of negative numbers when
62 one of a or c is very close to zero.
63
64 The values from this function should match those from
65 `Ellipse.make_ellipse_major` to within rounding error, except in the
66 special case of sigma_x == sigma_y == 0, which returns a NaN axis ratio
67 here by default. This function mainly intended to be more convenient
68 (and possibly faster) for array-like inputs.
69 """
70 apc = sigma_x_sq + sigma_y_sq
71 x = apc/2
72 pm = np.sqrt(apc**2 - 4*(sigma_x_sq*sigma_y_sq - cov_xy**2))/2
73
74 r_major = x + pm
75 axrat = np.sqrt((x - pm)/r_major)
76 r_major = np.sqrt(r_major)
77 angle = np.arctan2(2*cov_xy, sigma_x_sq - sigma_y_sq)/2
78 return r_major, axrat, (np.degrees(angle) if degrees else angle)
79
80
◆ gauss2dint()
| lsst.gauss2d.utils.gauss2dint | ( | xdivsigma | ) |
Return the fraction of the total surface integral of a 2D Gaussian
contained with a given multiple of its dispersion.
Parameters
----------
xdivsigma : `float`
The multiple of the dispersion to integrate to.
Returns
-------
frac : `float`
The fraction of the surface integral contained within an ellipse of
size `xdivsigma`.
Notes
-----
This solution can be computed as follows:
https://www.wolframalpha.com/input/?i=
Integrate+2*pi*x*exp(-x%5E2%2F(2*s%5E2))%2F(s*sqrt(2*pi))+dx+from+0+to+r
Definition at line 81 of file utils.py.
81def gauss2dint(xdivsigma):
82 """Return the fraction of the total surface integral of a 2D Gaussian
83 contained with a given multiple of its dispersion.
84
85 Parameters
86 ----------
87 xdivsigma : `float`
88 The multiple of the dispersion to integrate to.
89
90 Returns
91 -------
92 frac : `float`
93 The fraction of the surface integral contained within an ellipse of
94 size `xdivsigma`.
95
96 Notes
97 -----
98 This solution can be computed as follows:
99
100 https://www.wolframalpha.com/input/?i=
101 Integrate+2*pi*x*exp(-x%5E2%2F(2*s%5E2))%2F(s*sqrt(2*pi))+dx+from+0+to+r
102 """
103 return 1 - np.exp(-xdivsigma**2/2.)
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