1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
|
This example illustrates the class
:doxygen:`KullbackLeiblerDistanceImageFilter` for detecting changes
between pairs of images. This filter computes the Kullback-Leibler
distance between probability density functions (pdfs).
In fact, the Kullback-Leibler distance is itself approximated through
a cumulant-based expansion, since the pdfs are approximated through an
Edgeworth series.
The Kullback-Leibler distance is evaluated by:
.. math::
K_{\text{Edgeworth}}(X_1 | X_2) = \frac{1}{12} \frac{\kappa_{X_1; 3}^2}{\kappa_{X_1; 2}^2}
+ \frac{1}{2} \left( \log \frac{\kappa_{X_2; 2}}{\kappa_{X_1; 2}}
-1+\frac{1}{\kappa_{X_2; 2}}
\left( \kappa_{X_1; 1} - \kappa_{X_2; 1} + \kappa_{X_1; 2}^{1/2} \right)^2
\right)
- \left( \kappa_{X_2; 3} \frac{a_1}{6} + \kappa_{X_2; 4} \frac{a_2}{24}
+ \kappa_{X_2; 3}^2 \frac{a_3}{72} \right)
- \frac{1}{2} \frac{ \kappa_{X_2; 3}^2}{36}
\left(
c_6 - 6 \frac{c_4}{\kappa_{X_1; 2}} + 9 \frac{c_2}{\kappa_{X_2; 2}^2}
\right)
- 10 \frac{\kappa_{X_1; 3} \kappa_{X_2; 3}
\left( \kappa_{X_1; 1} - \kappa_{X_2; 1} \right)
\left( \kappa_{X_1; 2} - \kappa_{X_2; 2} \right)}{\kappa_{X_2; 2}^6} \qquad
where:
.. math::
a_1 = c_3 - 3 \frac{\alpha}{\kappa_{X_2; 2}}
a_2 = c_4 - 6 \frac{c_2}{\kappa_{X_2; 2}} + \frac{3}{\kappa_{X_2; 2}^2}
a_3 = c_6 - 15\frac{c_4}{\kappa_{X_2; 2}} + 45\frac{c_2}{\kappa_{X_2; 2}^2} - \frac{15}{\kappa_{X_2; 2}^3}
c_2 = \alpha^2 + \beta^2
c_3 = \alpha^3 + 3 \alpha \beta^2
c_4 = \alpha^4 + 6 \alpha^2 \beta^2 + 3 \beta^4
c_6 = \alpha^6 + 15\alpha^4 \beta^2 + 45 \alpha^2 \beta^4 + 15 \beta^6
\alpha = \frac{\kappa_{X_1; 1} - \kappa_{X_2; 1}}{\kappa_{X_2; 2}}
\beta = \frac{ \kappa_{X_1; 2}^{1/2} }{\kappa_{X_2; 2}}.
:math:`\kappa_{X_i; 1}`, :math:`\kappa_{X_i; 2}`, :math:`\kappa_{X_i; 3}` and :math:`\kappa_{X_i; 4}`
are the cumulants up to order 4 of the random variable :math:`X_i`.
These correspond to 2 Radarsat fine
mode acquisitions before and after a lava flow resulting from a
volcanic eruption.
The program itself is very similar to the ratio of means detector,
implemented in :doxygen:`MeanRatioImageFilter`.
Nevertheless the corresponding header file has to be used instead.
.. |image1| image:: /Input/GomaAvant.png
.. |image2| image:: /Input/GomaApres.png
.. |image3| image:: /Output/KLdistanceChDet.png
.. _Figure1:
+--------------------------+-------------------------+-------------------------+
| |image1| | |image2| | |image3| |
+--------------------------+-------------------------+-------------------------+
Result of the Kullback-Leibler change detector
|
