KullbackLeiblerDistanceChDet.cxx¶

This example illustrates the class otb::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:

$K_{\text{Edgeworth}}(X_1 | X_2) = \frac{1}{12} \frac{\kappa_{X_1; 3}^2}{\kappa_{X_1; 2}^2} + \frac{1}{2} \left( \log \frac{\kappa_{X_2; 2}}{\kappa_{X_1; 2}} -1+\frac{1}{\kappa_{X_2; 2}} \left( \kappa_{X_1; 1} - \kappa_{X_2; 1} + \kappa_{X_1; 2}^{1/2} \right)^2 \right) - \left( \kappa_{X_2; 3} \frac{a_1}{6} + \kappa_{X_2; 4} \frac{a_2}{24} + \kappa_{X_2; 3}^2 \frac{a_3}{72} \right) - \frac{1}{2} \frac{ \kappa_{X_2; 3}^2}{36} \left( c_6 - 6 \frac{c_4}{\kappa_{X_1; 2}} + 9 \frac{c_2}{\kappa_{X_2; 2}^2} \right) - 10 \frac{\kappa_{X_1; 3} \kappa_{X_2; 3} \left( \kappa_{X_1; 1} - \kappa_{X_2; 1} \right) \left( \kappa_{X_1; 2} - \kappa_{X_2; 2} \right)}{\kappa_{X_2; 2}^6} \qquad$

where:

$a_1 = c_3 - 3 \frac{\alpha}{\kappa_{X_2; 2}} a_2 = c_4 - 6 \frac{c_2}{\kappa_{X_2; 2}} + \frac{3}{\kappa_{X_2; 2}^2} a_3 = c_6 - 15\frac{c_4}{\kappa_{X_2; 2}} + 45\frac{c_2}{\kappa_{X_2; 2}^2} - \frac{15}{\kappa_{X_2; 2}^3} c_2 = \alpha^2 + \beta^2 c_3 = \alpha^3 + 3 \alpha \beta^2 c_4 = \alpha^4 + 6 \alpha^2 \beta^2 + 3 \beta^4 c_6 = \alpha^6 + 15\alpha^4 \beta^2 + 45 \alpha^2 \beta^4 + 15 \beta^6 \alpha = \frac{\kappa_{X_1; 1} - \kappa_{X_2; 1}}{\kappa_{X_2; 2}} \beta = \frac{ \kappa_{X_1; 2}^{1/2} }{\kappa_{X_2; 2}}.$

$\kappa_{X_i; 1}$ , $\kappa_{X_i; 2}$ , $\kappa_{X_i; 3}$ and $\kappa_{X_i; 4}$ are the cumulants up to order 4 of the random variable $X_i$ . 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 otb::MeanRatioImageFilter. Nevertheless the corresponding header file has to be used instead.

Result of the Kullback-Leibler change detector

Example usage:

./KullbackLeiblerDistanceChDet Input/GomaAvant.png Input/GomaApres.png Output/KLdistanceChDet.png 35

Example source code (KullbackLeiblerDistanceChDet.cxx):

#include "itkMacro.h"
#include "otbImage.h"
#include "otbImageFileReader.h"
#include "otbImageFileWriter.h"
#include "itkUnaryFunctorImageFilter.h"
#include "itkRescaleIntensityImageFilter.h"

#include "otbKullbackLeiblerDistanceImageFilter.h"

int main(int argc, char* argv[])
{
  if (argc != 5)
  {
    std::cerr << "Change detection through a Kullback-Leibler measure (which is a distance between local distributions)\n";
    std::cerr << "Kullback-Leibler measure is optimized by a Edgeworth series expansion\n";
    std::cerr << argv[0] << " imgAv imgAp imgResu winSize\n";
    return 1;
  }

  char* fileName1   = argv[1];
  char* fileName2   = argv[2];
  char* fileNameOut = argv[3];
  int   winSize     = atoi(argv[4]);

  const unsigned int Dimension = 2;
  using PixelType              = double;
  using OutputPixelType        = unsigned char;

  using ImageType       = otb::Image<PixelType, Dimension>;
  using OutputImageType = otb::Image<OutputPixelType, Dimension>;

  // The KullbackLeiblerDistanceImageFilter is templated over
  // the types of the two input images and the type of the generated change
  // image, in a similar way as the MeanRatioImageFilter. It is
  // the only line to be changed from the ratio of means change detection
  // example to perform a change detection through a distance between
  // distributions.
  using FilterType = otb::KullbackLeiblerDistanceImageFilter<ImageType, ImageType, ImageType>;

  // The different elements of the pipeline can now be instantiated. Follow the
  // ratio of means change detector example.
  using ReaderType = otb::ImageFileReader<ImageType>;
  using WriterType = otb::ImageFileWriter<OutputImageType>;

  ReaderType::Pointer reader1 = ReaderType::New();
  reader1->SetFileName(fileName1);

  ReaderType::Pointer reader2 = ReaderType::New();
  reader2->SetFileName(fileName2);

  // The only parameter for this change detector is the radius of
  // the window used for computing the cumulants.
  FilterType::Pointer filter = FilterType::New();
  filter->SetRadius((winSize - 1) / 2);

  // The pipeline is built by plugging all the elements together.
  filter->SetInput1(reader1->GetOutput());
  filter->SetInput2(reader2->GetOutput());

  using RescaleFilterType             = itk::RescaleIntensityImageFilter<ImageType, OutputImageType>;
  RescaleFilterType::Pointer rescaler = RescaleFilterType::New();

  rescaler->SetInput(filter->GetOutput());
  rescaler->SetOutputMinimum(0);
  rescaler->SetOutputMaximum(255);

  WriterType::Pointer writer = WriterType::New();
  writer->SetFileName(fileNameOut);
  writer->SetInput(rescaler->GetOutput());
  writer->Update();
}