Examples/Segmentation/GeodesicActiveContourImageFilter.cxx

/*=========================================================================
 *
 *  Copyright NumFOCUS
 *
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *  you may not use this file except in compliance with the License.
 *  You may obtain a copy of the License at
 *
 *         https://www.apache.org/licenses/LICENSE-2.0.txt
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 *
 *=========================================================================*/
 
//  Software Guide : BeginCommandLineArgs
//    INPUTS:  {BrainProtonDensitySlice.png}
//    OUTPUTS: {GeodesicActiveContourImageFilterOutput5.png}
//    ARGUMENTS:    81 114 5 1.0  -0.5  3.0   2
//  Software Guide : EndCommandLineArgs
//  Software Guide : BeginCommandLineArgs
//    INPUTS:  {BrainProtonDensitySlice.png}
//    OUTPUTS: {GeodesicActiveContourImageFilterOutput6.png}
//    ARGUMENTS:    99 114 5 1.0  -0.5  3.0  2
//  Software Guide : EndCommandLineArgs
//  Software Guide : BeginCommandLineArgs
//    INPUTS:  {BrainProtonDensitySlice.png}
//    OUTPUTS: {GeodesicActiveContourImageFilterOutput7.png}
//    ARGUMENTS:    56 92 5.0 1.0 -0.3 2.0 10.0
//  Software Guide : EndCommandLineArgs
//  Software Guide : BeginCommandLineArgs
//    INPUTS:  {BrainProtonDensitySlice.png}
//    OUTPUTS: {GeodesicActiveContourImageFilterOutput8.png}
//    ARGUMENTS:    40 90 5 0.5  -0.3  2.0 10
//  Software Guide : EndCommandLineArgs
//  Software Guide : BeginCommandLineArgs
//    INPUTS:  {BrainProtonDensitySlice.png}
//    OUTPUTS: {GeodesicActiveContourImageFilterOutput1.png}
//    ARGUMENTS:    56 92 5 1.0  -0.3  2.0 10
//    OUTPUTS: {GeodesicActiveContourImageFilterOutput2.png}
//    OUTPUTS: {GeodesicActiveContourImageFilterOutput3.png}
//  Software Guide : EndCommandLineArgs
 
// Software Guide : BeginLatex
//
// The use of the \doxygen{GeodesicActiveContourLevelSetImageFilter} is
// illustrated in the following example. The implementation of this filter in
// ITK is based on the paper by Caselles \cite{Caselles1997}.  This
// implementation extends the functionality of the
// \doxygen{ShapeDetectionLevelSetImageFilter} by the addition of a third
// advection term which attracts the level set to the object boundaries.
//
// GeodesicActiveContourLevelSetImageFilter expects two inputs.  The first is
// an initial level set in the form of an \doxygen{Image}. The second input
// is a feature image. For this algorithm, the feature image is an edge
// potential image that basically follows the same rules used for the
// ShapeDetectionLevelSetImageFilter discussed in
// Section~\ref{sec:ShapeDetectionLevelSetFilter}.  The configuration of this
// example is quite similar to the example on the use of the
// ShapeDetectionLevelSetImageFilter. We omit most of the redundant
// description. A look at the code will reveal the great degree of similarity
// between both examples.
//
// \begin{figure} \center
// \includegraphics[width=\textwidth]{GeodesicActiveContoursCollaborationDiagram1}
// \itkcaption[GeodesicActiveContourLevelSetImageFilter collaboration
// diagram]{Collaboration diagram for the
// GeodesicActiveContourLevelSetImageFilter applied to a segmentation task.}
// \label{fig:GeodesicActiveContoursCollaborationDiagram}
// \end{figure}
//
// Figure~\ref{fig:GeodesicActiveContoursCollaborationDiagram} shows the major
// components involved in the application of the
// GeodesicActiveContourLevelSetImageFilter to a segmentation task.
// This pipeline is quite similar to the one used by the
// ShapeDetectionLevelSetImageFilter in
// section~\ref{sec:ShapeDetectionLevelSetFilter}.
//
// The pipeline involves a first stage of smoothing using the\newline
// \doxygen{CurvatureAnisotropicDiffusionImageFilter}. The smoothed image is
// passed as the input to the
// \doxygen{GradientMagnitudeRecursiveGaussianImageFilter} and then to the
// \doxygen{SigmoidImageFilter} in order to produce the edge potential image.
// A set of user-provided seeds is passed to a
// \doxygen{FastMarchingImageFilter} in order to compute the distance map. A
// constant value is subtracted from this map in order to obtain a level set
// in which the \emph{zero set} represents the initial contour. This level
// set is also passed as input to the
// GeodesicActiveContourLevelSetImageFilter.
//
// Finally, the level set generated by the
// GeodesicActiveContourLevelSetImageFilter is passed to a
// \doxygen{BinaryThresholdImageFilter} in order to produce a binary mask
// representing the segmented object.
//
// Let's start by including the headers of the main filters involved in the
// preprocessing.
//
// Software Guide : EndLatex
 
 
// Software Guide : BeginCodeSnippet
#include "itkGeodesicActiveContourLevelSetImageFilter.h"
// Software Guide : EndCodeSnippet
 
 
#include "itkCurvatureAnisotropicDiffusionImageFilter.h"
#include "itkGradientMagnitudeRecursiveGaussianImageFilter.h"
#include "itkSigmoidImageFilter.h"
#include "itkFastMarchingImageFilter.h"
#include "itkRescaleIntensityImageFilter.h"
#include "itkBinaryThresholdImageFilter.h"
#include "itkImageFileReader.h"
#include "itkImageFileWriter.h"
 
 
int
main(int argc, char * argv[])
{
  if (argc < 10)
  {
    std::cerr << "Missing Parameters " << std::endl;
    std::cerr << "Usage: " << argv[0];
    std::cerr << " inputImage  outputImage";
    std::cerr << " seedX seedY InitialDistance";
    std::cerr << " Sigma SigmoidAlpha SigmoidBeta";
    std::cerr << " PropagationScaling" << std::endl;
    return EXIT_SUCCESS;
  }
 
  //  Software Guide : BeginLatex
  //
  //  We now define the image type using a particular pixel type and
  //  dimension. In this case the \code{float} type is used for the pixels
  //  due to the requirements of the smoothing filter.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  using InternalPixelType = float;
  constexpr unsigned int Dimension = 2;
  using InternalImageType = itk::Image<InternalPixelType, Dimension>;
  // Software Guide : EndCodeSnippet
 
 
  //  The following lines instantiate the thresholding filter that will
  //  process the final level set at the output of the
  //  GeodesicActiveContourLevelSetImageFilter.
  //
  using OutputPixelType = unsigned char;
  using OutputImageType = itk::Image<OutputPixelType, Dimension>;
  using ThresholdingFilterType =
    itk::BinaryThresholdImageFilter<InternalImageType, OutputImageType>;
 
  auto thresholder = ThresholdingFilterType::New();
 
  thresholder->SetLowerThreshold(-1000.0);
  thresholder->SetUpperThreshold(0.0);
 
  thresholder->SetOutsideValue(0);
  thresholder->SetInsideValue(255);
 
 
  // We instantiate reader and writer types in the following lines.
  //
  using ReaderType = itk::ImageFileReader<InternalImageType>;
  using WriterType = itk::ImageFileWriter<OutputImageType>;
 
  auto reader = ReaderType::New();
  auto writer = WriterType::New();
 
  reader->SetFileName(argv[1]);
  writer->SetFileName(argv[2]);
 
 
  //  The RescaleIntensityImageFilter type is declared below. This filter will
  //  renormalize image before sending them to writers.
  //
  using CastFilterType =
    itk::RescaleIntensityImageFilter<InternalImageType, OutputImageType>;
 
 
  //  The \doxygen{CurvatureAnisotropicDiffusionImageFilter} type is
  //  instantiated using the internal image type.
  //
  using SmoothingFilterType =
    itk::CurvatureAnisotropicDiffusionImageFilter<InternalImageType,
                                                  InternalImageType>;
 
  auto smoothing = SmoothingFilterType::New();
 
 
  //  The types of the
  //  GradientMagnitudeRecursiveGaussianImageFilter and
  //  SigmoidImageFilter are instantiated using the internal image
  //  type.
  //
  using GradientFilterType =
    itk::GradientMagnitudeRecursiveGaussianImageFilter<InternalImageType,
                                                       InternalImageType>;
  using SigmoidFilterType =
    itk::SigmoidImageFilter<InternalImageType, InternalImageType>;
 
  auto gradientMagnitude = GradientFilterType::New();
 
  auto sigmoid = SigmoidFilterType::New();
 
 
  //  The minimum and maximum values of the SigmoidImageFilter output
  //  are defined with the methods \code{SetOutputMinimum()} and
  //  \code{SetOutputMaximum()}. In our case, we want these two values to be
  //  $0.0$ and $1.0$ respectively in order to get a nice speed image to feed
  //  the \code{FastMarchingImageFilter}. Additional details on the user of
  //  the \doxygen{SigmoidImageFilter} are presented in
  //  section~\ref{sec:IntensityNonLinearMapping}.
 
  sigmoid->SetOutputMinimum(0.0);
  sigmoid->SetOutputMaximum(1.0);
 
 
  //  We declare now the type of the FastMarchingImageFilter that
  //  will be used to generate the initial level set in the form of a distance
  //  map.
  //
  using FastMarchingFilterType =
    itk::FastMarchingImageFilter<InternalImageType, InternalImageType>;
 
 
  //  Next we construct one filter of this class using the \code{New()}
  //  method.
  //
  auto fastMarching = FastMarchingFilterType::New();
 
  //  Software Guide : BeginLatex
  //
  //  In the following lines we instantiate the type of the
  //  GeodesicActiveContourLevelSetImageFilter and create an object of this
  //  type using the \code{New()} method.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  using GeodesicActiveContourFilterType =
    itk::GeodesicActiveContourLevelSetImageFilter<InternalImageType,
                                                  InternalImageType>;
  auto geodesicActiveContour = GeodesicActiveContourFilterType::New();
  // Software Guide : EndCodeSnippet
 
 
  //  Software Guide : BeginLatex
  //
  //  For the GeodesicActiveContourLevelSetImageFilter, scaling parameters
  //  are used to trade off between the propagation (inflation), the
  //  curvature (smoothing) and the advection terms. These parameters are set
  //  using methods \code{SetPropagationScaling()},
  //  \code{SetCurvatureScaling()} and \code{SetAdvectionScaling()}. In this
  //  example, we will set the curvature and advection scales to one and let
  //  the propagation scale be a command-line argument.
  //
  //  \index{itk::Geodesic\-Active\-Contour\-LevelSet\-Image\-Filter!SetPropagationScaling()}
  //  \index{itk::Segmentation\-Level\-Set\-Image\-Filter!SetPropagationScaling()}
  //  \index{itk::Geodesic\-Active\-Contour\-LevelSet\-Image\-Filter!SetCurvatureScaling()}
  //  \index{itk::Segmentation\-Level\-Set\-Image\-Filter!SetCurvatureScaling()}
  //  \index{itk::Geodesic\-Active\-Contour\-LevelSet\-Image\-Filter!SetAdvectionScaling()}
  //  \index{itk::Segmentation\-Level\-Set\-Image\-Filter!SetAdvectionScaling()}
  //
  //  Software Guide : EndLatex
 
  const double propagationScaling = std::stod(argv[9]);
 
  //  Software Guide : BeginCodeSnippet
  geodesicActiveContour->SetPropagationScaling(propagationScaling);
  geodesicActiveContour->SetCurvatureScaling(1.0);
  geodesicActiveContour->SetAdvectionScaling(1.0);
  //  Software Guide : EndCodeSnippet
 
  //  Once activated the level set evolution will stop if the convergence
  //  criteria or if the maximum number of iterations is reached.  The
  //  convergence criteria is defined in terms of the root mean squared (RMS)
  //  change in the level set function. The evolution is said to have
  //  converged if the RMS change is below a user specified threshold.  In a
  //  real application is desirable to couple the evolution of the zero set
  //  to a visualization module allowing the user to follow the evolution of
  //  the zero set. With this feedback, the user may decide when to stop the
  //  algorithm before the zero set leaks through the regions of low gradient
  //  in the contour of the anatomical structure to be segmented.
 
  geodesicActiveContour->SetMaximumRMSError(0.02);
  geodesicActiveContour->SetNumberOfIterations(800);
 
 
  //  Software Guide : BeginLatex
  //
  //  The filters are now connected in a pipeline indicated in
  //  Figure~\ref{fig:GeodesicActiveContoursCollaborationDiagram} using the
  //  following lines:
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  smoothing->SetInput(reader->GetOutput());
  gradientMagnitude->SetInput(smoothing->GetOutput());
  sigmoid->SetInput(gradientMagnitude->GetOutput());
 
  geodesicActiveContour->SetInput(fastMarching->GetOutput());
  geodesicActiveContour->SetFeatureImage(sigmoid->GetOutput());
 
  thresholder->SetInput(geodesicActiveContour->GetOutput());
  writer->SetInput(thresholder->GetOutput());
  // Software Guide : EndCodeSnippet
 
 
  //  The CurvatureAnisotropicDiffusionImageFilter requires a couple of
  //  parameter to be defined. The following are typical values for $2D$
  //  images. However they may have to be adjusted depending on the amount of
  //  noise present in the input image. This filter has been discussed in
  //  section~\ref{sec:GradientAnisotropicDiffusionImageFilter}.
 
  smoothing->SetTimeStep(0.125);
  smoothing->SetNumberOfIterations(5);
  smoothing->SetConductanceParameter(9.0);
 
 
  //  The GradientMagnitudeRecursiveGaussianImageFilter performs the
  //  equivalent of a convolution with a Gaussian kernel, followed by a
  //  derivative operator. The sigma of this Gaussian can be used to control
  //  the range of influence of the image edges. This filter has been
  //  discussed in
  //  Section~\ref{sec:GradientMagnitudeRecursiveGaussianImageFilter}.
 
  const double sigma = std::stod(argv[6]);
  gradientMagnitude->SetSigma(sigma);
 
 
  //  The SigmoidImageFilter requires two parameters that define the linear
  //  transformation to be applied to the sigmoid argument. This parameters
  //  have been discussed in Sections~\ref{sec:IntensityNonLinearMapping} and
  //  \ref{sec:FastMarchingImageFilter}.
 
  const double alpha = std::stod(argv[7]);
  const double beta = std::stod(argv[8]);
 
  sigmoid->SetAlpha(alpha);
  sigmoid->SetBeta(beta);
 
 
  //  The FastMarchingImageFilter requires the user to provide a seed
  //  point from which the level set will be generated. The user can actually
  //  pass not only one seed point but a set of them. Note the the
  //  FastMarchingImageFilter is used here only as a helper in the
  //  determination of an initial level set. We could have used the
  //  \doxygen{DanielssonDistanceMapImageFilter} in the same way.
  //
  //  The seeds are passed stored in a container. The type of this
  //  container is defined as \code{NodeContainer} among the
  //  FastMarchingImageFilter traits.
  //
  using NodeContainer = FastMarchingFilterType::NodeContainer;
  using NodeType = FastMarchingFilterType::NodeType;
 
  auto seeds = NodeContainer::New();
 
  InternalImageType::IndexType seedPosition;
 
  seedPosition[0] = std::stoi(argv[3]);
  seedPosition[1] = std::stoi(argv[4]);
 
 
  //  Nodes are created as stack variables and initialized with a value and an
  //  \doxygen{Index} position. Note that here we assign the value of minus
  //  the user-provided distance to the unique node of the seeds passed to the
  //  FastMarchingImageFilter. In this way, the value will increment
  //  as the front is propagated, until it reaches the zero value
  //  corresponding to the contour. After this, the front will continue
  //  propagating until it fills up the entire image. The initial distance is
  //  taken here from the command line arguments. The rule of thumb for the
  //  user is to select this value as the distance from the seed points at
  //  which she want the initial contour to be.
  const double initialDistance = std::stod(argv[5]);
 
  NodeType node;
 
  const double seedValue = -initialDistance;
 
  node.SetValue(seedValue);
  node.SetIndex(seedPosition);
 
 
  //  The list of nodes is initialized and then every node is inserted using
  //  the \code{InsertElement()}.
 
  seeds->Initialize();
  seeds->InsertElement(0, node);
 
 
  //  The set of seed nodes is passed now to the
  //  FastMarchingImageFilter with the method
  //  \code{SetTrialPoints()}.
  //
  fastMarching->SetTrialPoints(seeds);
 
 
  //  Since the FastMarchingImageFilter is used here just as a
  //  Distance Map generator. It does not require a speed image as input.
  //  Instead the constant value $1.0$ is passed using the
  //  \code{SetSpeedConstant()} method.
  //
  fastMarching->SetSpeedConstant(1.0);
 
 
  //  Here we configure all the writers required to see the intermediate
  //  outputs of the pipeline. This is added here only for
  //  pedagogical/debugging purposes. These intermediate output are normally
  //  not required. Only the output of the final thresholding filter should be
  //  relevant.  Observing intermediate output is helpful in the process of
  //  fine tuning the parameters of filters in the pipeline.
  //
  auto caster1 = CastFilterType::New();
  auto caster2 = CastFilterType::New();
  auto caster3 = CastFilterType::New();
  auto caster4 = CastFilterType::New();
 
  auto writer1 = WriterType::New();
  auto writer2 = WriterType::New();
  auto writer3 = WriterType::New();
  auto writer4 = WriterType::New();
 
  caster1->SetInput(smoothing->GetOutput());
  writer1->SetInput(caster1->GetOutput());
  writer1->SetFileName("GeodesicActiveContourImageFilterOutput1.png");
  caster1->SetOutputMinimum(0);
  caster1->SetOutputMaximum(255);
  writer1->Update();
 
  caster2->SetInput(gradientMagnitude->GetOutput());
  writer2->SetInput(caster2->GetOutput());
  writer2->SetFileName("GeodesicActiveContourImageFilterOutput2.png");
  caster2->SetOutputMinimum(0);
  caster2->SetOutputMaximum(255);
  writer2->Update();
 
  caster3->SetInput(sigmoid->GetOutput());
  writer3->SetInput(caster3->GetOutput());
  writer3->SetFileName("GeodesicActiveContourImageFilterOutput3.png");
  caster3->SetOutputMinimum(0);
  caster3->SetOutputMaximum(255);
  writer3->Update();
 
  caster4->SetInput(fastMarching->GetOutput());
  writer4->SetInput(caster4->GetOutput());
  writer4->SetFileName("GeodesicActiveContourImageFilterOutput4.png");
  caster4->SetOutputMinimum(0);
  caster4->SetOutputMaximum(255);
 
 
  //  The FastMarchingImageFilter requires the user to specify the
  //  size of the image to be produced as output. This is done using the
  //  \code{SetOutputSize()}. Note that the size is obtained here from the
  //  output image of the smoothing filter. The size of this image is valid
  //  only after the \code{Update()} methods of this filter has been called
  //  directly or indirectly.
  //
  fastMarching->SetOutputSize(
    reader->GetOutput()->GetBufferedRegion().GetSize());
 
 
  //  Software Guide : BeginLatex
  //
  //  The invocation of the \code{Update()} method on the writer triggers the
  //  execution of the pipeline.  As usual, the call is placed in a
  //  \code{try/catch} block should any errors occur or exceptions be thrown.
  //
  //  Software Guide : EndLatex
 
  // Software Guide : BeginCodeSnippet
  try
  {
    writer->Update();
  }
  catch (const itk::ExceptionObject & excep)
  {
    std::cerr << "Exception caught !" << std::endl;
    std::cerr << excep << std::endl;
    return EXIT_FAILURE;
  }
  // Software Guide : EndCodeSnippet
 
  // Print out some useful information
  std::cout << std::endl;
  std::cout << "Max. no. iterations: "
            << geodesicActiveContour->GetNumberOfIterations() << std::endl;
  std::cout << "Max. RMS error: "
            << geodesicActiveContour->GetMaximumRMSError() << std::endl;
  std::cout << std::endl;
  std::cout << "No. elpased iterations: "
            << geodesicActiveContour->GetElapsedIterations() << std::endl;
  std::cout << "RMS change: " << geodesicActiveContour->GetRMSChange()
            << std::endl;
 
  writer4->Update();
 
 
  // The following writer type is used to save the output of the time-crossing
  // map in a file with appropriate pixel representation. The advantage of
  // saving this image in native format is that it can be used with a viewer
  // to help determine an appropriate threshold to be used on the output of
  // the fastmarching filter.
  //
  using InternalWriterType = itk::ImageFileWriter<InternalImageType>;
 
  auto mapWriter = InternalWriterType::New();
  mapWriter->SetInput(fastMarching->GetOutput());
  mapWriter->SetFileName("GeodesicActiveContourImageFilterOutput4.mha");
  mapWriter->Update();
 
  auto speedWriter = InternalWriterType::New();
  speedWriter->SetInput(sigmoid->GetOutput());
  speedWriter->SetFileName("GeodesicActiveContourImageFilterOutput3.mha");
  speedWriter->Update();
 
  auto gradientWriter = InternalWriterType::New();
  gradientWriter->SetInput(gradientMagnitude->GetOutput());
  gradientWriter->SetFileName("GeodesicActiveContourImageFilterOutput2.mha");
  gradientWriter->Update();
 
 
  //  Software Guide : BeginLatex
  //
  //  Let's now run this example using as input the image
  //  \code{BrainProtonDensitySlice.png} provided in the directory
  //  \code{Examples/Data}. We can easily segment the major anatomical
  //  structures by providing seeds in the appropriate locations.
  //  Table~\ref{tab:GeodesicActiveContourImageFilterOutput2} presents the
  //  parameters used for some structures.
  //
  //  \begin{table}
  //  \begin{center}
  //  \begin{tabular}{|l|c|c|c|c|c|c|c|c|}
  //  \hline
  //  Structure    & Seed Index &  Distance   &   $\sigma$  &
  //  $\alpha$     &  $\beta$   & Propag. & Output Image \\  \hline
  //  Left Ventricle  & $(81,114)$ & 5.0 & 1.0 & -0.5 & 3.0  &  2.0 &
  //  First  \\  \hline  Right Ventricle & $(99,114)$ & 5.0 & 1.0 & -0.5 &
  //  3.0  &  2.0 & Second  \\  \hline White matter    & $(56, 92)$ & 5.0 &
  //  1.0 & -0.3 & 2.0 & 10.0 & Third   \\  \hline Gray matter     &
  //  $(40, 90)$ & 5.0 & 0.5 & -0.3 & 2.0  & 10.0 & Fourth  \\  \hline
  //  \end{tabular} \end{center}
  //  \itkcaption[GeodesicActiveContour segmentation example
  //  parameters]{Parameters used for segmenting some brain structures shown
  //  in Figure~\ref{fig:GeodesicActiveContourImageFilterOutput2} using the
  //  filter GeodesicActiveContourLevelSetImageFilter.
  //  \label{tab:GeodesicActiveContourImageFilterOutput2}}
  //  \end{table}
  //
  //  Figure~\ref{fig:GeodesicActiveContourImageFilterOutput} presents the
  //  intermediate outputs of the pipeline illustrated in
  //  Figure~\ref{fig:GeodesicActiveContoursCollaborationDiagram}. They are
  //  from left to right: the output of the anisotropic diffusion filter, the
  //  gradient magnitude of the smoothed image and the sigmoid of the gradient
  //  magnitude which is finally used as the edge potential for the
  //  GeodesicActiveContourLevelSetImageFilter.
  //
  //  \begin{figure} \center
  //  \includegraphics[height=0.40\textheight]{BrainProtonDensitySlice}
  //  \includegraphics[height=0.40\textheight]{GeodesicActiveContourImageFilterOutput1}
  //  \includegraphics[height=0.40\textheight]{GeodesicActiveContourImageFilterOutput2}
  //  \includegraphics[height=0.40\textheight]{GeodesicActiveContourImageFilterOutput3}
  //  \itkcaption[GeodesicActiveContourLevelSetImageFilter intermediate
  //  output]{Images generated by the segmentation process based on the
  //  GeodesicActiveContourLevelSetImageFilter. From left to right and top to
  //  bottom: input image to be segmented, image smoothed with an
  //  edge-preserving smoothing filter, gradient magnitude of the smoothed
  //  image, sigmoid of the gradient magnitude. This last image, the sigmoid,
  //  is used to compute the speed term for the front propagation.}
  //  \label{fig:GeodesicActiveContourImageFilterOutput} \end{figure}
  //
  //  Segmentations of the main brain structures are presented in
  //  Figure~\ref{fig:GeodesicActiveContourImageFilterOutput2}. The results
  //  are quite similar to those obtained with the
  //  ShapeDetectionLevelSetImageFilter in
  //  Section~\ref{sec:ShapeDetectionLevelSetFilter}.
  //
  //  Note that a relatively larger propagation scaling value was required to
  //  segment the white matter. This is due to two factors: the lower
  //  contrast at the border of the white matter and the complex shape of the
  //  structure. Unfortunately the optimal value of these scaling parameters
  //  can only be determined by experimentation. In a real application we
  //  could imagine an interactive mechanism by which a user supervises the
  //  contour evolution and adjusts these parameters accordingly.
  //
  //  \begin{figure} \center
  //  \includegraphics[width=0.24\textwidth]{GeodesicActiveContourImageFilterOutput5}
  //  \includegraphics[width=0.24\textwidth]{GeodesicActiveContourImageFilterOutput6}
  //  \includegraphics[width=0.24\textwidth]{GeodesicActiveContourImageFilterOutput7}
  //  \includegraphics[width=0.24\textwidth]{GeodesicActiveContourImageFilterOutput8}
  //  \itkcaption[GeodesicActiveContourImageFilter segmentations]{Images
  //  generated by the segmentation process based on the
  //  GeodesicActiveContourImageFilter. From left to right: segmentation of
  //  the left ventricle, segmentation of the right ventricle, segmentation of
  //  the white matter, attempt of segmentation of the gray matter.}
  //  \label{fig:GeodesicActiveContourImageFilterOutput2}
  //  \end{figure}
  //
  //  Software Guide : EndLatex
 
  return EXIT_SUCCESS;
}