Examples/Numerics/FourierDescriptors1.cxx

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//  Software Guide : BeginLatex
//
//  Fourier Descriptors provide a mechanism for representing a closed curve in
//  space.  The represented curve has infinite continuity because the
//  parametric coordinate of its points are computed from a Fourier Series.
//
//  In this example we illustrate how a curve that is initially defined by a
//  set of points in space can be represented in terms for Fourier
//  Descriptors. This representation is useful for several purposes. For
//  example, it provides a mechanism for interpolating values among the
//  points, it provides a way of analyzing the smoothness of the curve.  In
//  this particular example we will focus on this second application of the
//  Fourier Descriptors.
//
//  The first operation that we should use in this context is the computation
//  of the discrete fourier transform of the point coordinates. The
//  coordinates of the points are considered to be independent functions and
//  each one is decomposed in a Fourier Series. In this section we will use
//  $t$ as the parameter of the curve, and will assume that it goes from $0$
//  to $1$ and cycles as we go around the closed curve. // \begin{equation}
//  \textbf{V(t)} = \left( X(t), Y(t) \right)
//  \end{equation}
//
//  We take now the functions $X(t)$, $Y(t)$ and interpret them as the
//  components of a complex number for which we compute its discrete fourier
//  series in the form
//
//  \begin{equation}
//  V(t) = \sum_{k=0}^{N} \exp(-\frac{2 k \pi \textbf{i}}{N}) F_k
//  \end{equation}
//
//  Where the set of coefficients $F_k$ is the discrete spectrum of the
//  complex function $V(t)$. These coefficients are in general complex numbers
//  and both their magnitude and phase must be considered in any further
//  analysis of the spectrum.
//
//  Software Guide : EndLatex
 
//  Software Guide : BeginLatex
//
//  The class \code{vnl\_fft\_1d} is the VNL class that computes such
//  transform. In order to use it, we should include its header file first.
//
//  Software Guide : EndLatex
 
// Software Guide : BeginCodeSnippet
#include "vnl/algo/vnl_fft_1d.h"
// Software Guide : EndCodeSnippet
 
#include "itkPoint.h"
#include "itkVectorContainer.h"
 
#include <fstream>
 
int
main(int argc, char * argv[])
{
 
  if (argc < 2)
  {
    std::cerr << "Missing arguments" << std::endl;
    std::cerr << "Usage: " << std::endl;
    std::cerr << argv[0] << "inputFileWithPointCoordinates" << std::endl;
    return EXIT_FAILURE;
  }
 
  //  Software Guide : BeginLatex
  //
  //  We should now instantiate the filter that will compute the Fourier
  //  transform of the set of coordinates.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  using FFTCalculator = vnl_fft_1d<double>;
  // Software Guide : EndCodeSnippet
 
  //  Software Guide : BeginLatex
  //
  //  The points representing the curve are stored in a
  //  \doxygen{VectorContainer} of \doxygen{Point}.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  using PointType = itk::Point<double, 2>;
 
  using PointsContainer = itk::VectorContainer<unsigned int, PointType>;
 
  auto points = PointsContainer::New();
  // Software Guide : EndCodeSnippet
 
  //  Software Guide : BeginLatex
  //
  //  In this example we read the set of points from a text file.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  std::ifstream inputFile;
  inputFile.open(argv[1]);
 
  if (inputFile.fail())
  {
    std::cerr << "Problems opening file " << argv[1] << std::endl;
  }
 
  unsigned int numberOfPoints;
  inputFile >> numberOfPoints;
 
  points->Reserve(numberOfPoints);
 
  using PointIterator = PointsContainer::Iterator;
  PointIterator pointItr = points->Begin();
 
  PointType point;
  for (unsigned int pt = 0; pt < numberOfPoints; ++pt)
  {
    inputFile >> point[0] >> point[1];
    pointItr.Value() = point;
    ++pointItr;
  }
  // Software Guide : EndCodeSnippet
 
  //  Software Guide : BeginLatex
  //
  //  This class will compute the Fast Fourier transform of the input and it
  //  will return it in the same array. We must therefore copy the original
  //  data into an auxiliary array that will in its turn contain the results
  //  of the transform.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  using FFTCoefficientType = std::complex<double>;
  using FFTSpectrumType = std::vector<FFTCoefficientType>;
  // Software Guide : EndCodeSnippet
 
  //  Software Guide : BeginLatex
  //
  // The choice of the spectrum size is very important. Here we select to use
  // the next power of two that is larger than the number of points.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  const auto powerOfTwo = static_cast<unsigned int>(
    std::ceil(std::log(static_cast<double>(numberOfPoints)) / std::log(2.0)));
 
  const unsigned int spectrumSize = 1 << powerOfTwo;
 
  //  Software Guide : BeginLatex
  //
  //  The Fourier Transform type can now be used for constructing one of such
  //  filters. Note that this is a VNL class and does not follows ITK notation
  //  for construction and assignment to SmartPointers.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  FFTCalculator fftCalculator(spectrumSize);
  // Software Guide : EndCodeSnippet
 
  FFTSpectrumType signal(spectrumSize);
 
  pointItr = points->Begin();
  for (unsigned int p = 0; p < numberOfPoints; ++p)
  {
    signal[p] = FFTCoefficientType(pointItr.Value()[0], pointItr.Value()[1]);
    ++pointItr;
  }
  // Software Guide : EndCodeSnippet
 
  //  Software Guide : BeginLatex
  //
  // Fill in the rest of the input with zeros. This padding may have
  // undesirable effects on the spectrum if the signal is not attenuated to
  // zero close to their boundaries. Instead of zero-padding we could have
  // used repetition of the last value or mirroring of the signal.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  for (unsigned int pad = numberOfPoints; pad < spectrumSize; ++pad)
  {
    signal[pad] = 0.0;
  }
  // Software Guide : EndCodeSnippet
 
  //  Software Guide : BeginLatex
  //
  //  Now we print out the signal as it is passed to the transform calculator
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  std::cout << "Input to the FFT transform" << std::endl;
  for (unsigned int s = 0; s < spectrumSize; ++s)
  {
    std::cout << s << " : ";
    std::cout << signal[s] << std::endl;
  }
  // Software Guide : EndCodeSnippet
 
  //  Software Guide : BeginLatex
  //
  //  The actual transform is computed by invoking the \code{fwd_transform}
  //  method in the FFT calculator class.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  fftCalculator.fwd_transform(signal);
  // Software Guide : EndCodeSnippet
 
  //  Software Guide : BeginLatex
  //
  //  Now we print out the results of the transform.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  std::cout << std::endl;
  std::cout << "Result from the FFT transform" << std::endl;
  for (unsigned int k = 0; k < spectrumSize; ++k)
  {
    const double real = signal[k].real();
    const double imag = signal[k].imag();
    const double magnitude = std::sqrt(real * real + imag * imag);
    std::cout << k << "  " << magnitude << std::endl;
  }
  // Software Guide : EndCodeSnippet
 
  return EXIT_SUCCESS;
}