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/*=========================================================================
*
* Copyright Insight Software Consortium
*
* 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
*
* http://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: {BrainT1SliceBorder20.png}
// INPUTS: {BrainProtonDensitySliceShifted13x17y.png}
// OUTPUTS: {MultiResImageRegistration1Output.png}
// ARGUMENTS: 128
// OUTPUTS: {MultiResImageRegistration1CheckerboardBefore.png}
// OUTPUTS: {MultiResImageRegistration1CheckerboardAfter.png}
// Software Guide : EndCommandLineArgs
// Software Guide : BeginLatex
//
// \index{itk::ImageRegistrationMethod!Multi-Resolution}
// \index{itk::ImageRegistrationMethod!Multi-Modality}
// \index{itk::Multi\-Resolution\-Image\-Registration\-Method}
//
// This example illustrates the use of the
// \doxygen{MultiResolutionImageRegistrationMethod} to solve a simple
// multi-modality registration problem. In addition to the two input images,
// a transform, a metric, an interpolator and an optimizer, the
// multi-resolution framework also requires two image pyramids for creating
// the sequence of downsampled images. To begin the example, we include the
// headers of the registration components we will use.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
#include "itkMultiResolutionImageRegistrationMethod.h"
#include "itkTranslationTransform.h"
#include "itkMattesMutualInformationImageToImageMetric.h"
#include "itkRegularStepGradientDescentOptimizer.h"
#include "itkImage.h"
// Software Guide : EndCodeSnippet
#include "itkImageFileReader.h"
#include "itkImageFileWriter.h"
#include "itkResampleImageFilter.h"
#include "itkCastImageFilter.h"
#include "itkCheckerBoardImageFilter.h"
// Software Guide : BeginLatex
//
// The MultiResolutionImageRegistrationMethod solves a registration
// problem in a coarse to fine manner as illustrated in Figure
// \ref{fig:MultiResRegistrationConcept}. The registration is first performed
// at the coarsest level using the images at the first level of the fixed and
// moving image pyramids. The transform parameters determined by the
// registration are then used to initialize the registration at the next finer
// level using images from the second level of the pyramids. This process is
// repeated as we work up to the finest level of image resolution.
//
// \begin{figure}
// \center
// \includegraphics[width=\textwidth]{MultiResRegistrationConcept}
// \itkcaption[Conceptual representation of Multi-Resolution
// registration]{Conceptual representation of the multi-resolution registration process.}
// \label{fig:MultiResRegistrationConcept}
// \end{figure}
//
// Software Guide : EndLatex
// Software Guide : BeginLatex
//
// In a typical registration scenario, a user will tweak component settings
// or even swap out components between multi-resolution levels. For example,
// when optimizing at a coarse resolution, it may be possible to take more
// aggressive step sizes and have a more relaxed convergence criterion.
// Another possible scheme is to use a simple translation transform for the
// initial coarse registration and upgrade to an affine transform at the
// finer levels.
//
// Tweaking the components between resolution levels can be done using ITK's
// implementation of the \emph{Command/Observer} design pattern. Before
// beginning registration at each resolution level,
// MultiResolutionImageRegistrationMethod invokes an
// IterationEvent. The registration components can be changed by
// implementing a \doxygen{Command} which responds to the
// event. A brief description the interaction between events and commands was
// previously presented in Section \ref{sec:MonitoringImageRegistration}.
//
// We will illustrate this mechanism by changing the parameters of the
// optimizer between each resolution level by way of a simple interface
// command. First, we include the header file of the Command class.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
#include "itkCommand.h"
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// Our new interface command class is called
// \code{RegistrationInterfaceCommand}. It derives from
// Command and is templated over the
// multi-resolution registration type.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
template <typename TRegistration>
class RegistrationInterfaceCommand : public itk::Command
{
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// We then define \code{Self}, \code{Superclass}, \code{Pointer},
// \code{New()} and a constructor in a similar fashion to the
// \code{CommandIterationUpdate} class in Section
// \ref{sec:MonitoringImageRegistration}.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
public:
typedef RegistrationInterfaceCommand Self;
typedef itk::Command Superclass;
typedef itk::SmartPointer<Self> Pointer;
itkNewMacro( Self );
protected:
RegistrationInterfaceCommand() {};
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// For convenience, we declare types useful for converting pointers
// in the \code{Execute()} method.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
public:
typedef TRegistration RegistrationType;
typedef RegistrationType * RegistrationPointer;
typedef itk::RegularStepGradientDescentOptimizer OptimizerType;
typedef OptimizerType * OptimizerPointer;
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// Two arguments are passed to the \code{Execute()} method: the first
// is the pointer to the object which invoked the event and the
// second is the event that was invoked.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
void Execute(itk::Object * object, const itk::EventObject & event)
{
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// First we verify if that the event invoked is of the right type.
// If not, we return without any further action.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
if( !(itk::IterationEvent().CheckEvent( &event )) )
{
return;
}
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// We then convert the input object pointer to a RegistrationPointer.
// Note that no error checking is done here to verify if the
// \code{dynamic\_cast} was successful since we know the actual object
// is a multi-resolution registration method.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
RegistrationPointer registration = static_cast<RegistrationPointer>( object );
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// If this is the first resolution level we set the maximum step length
// (representing the first step size) and the minimum step length (representing
// the convergence criterion) to large values. At each subsequent resolution
// level, we will reduce the minimum step length by a factor of 10 in order to
// allow the optimizer to focus on progressively smaller regions. The maximum
// step length is set up to the current step length. In this way, when the
// optimizer is reinitialized at the beginning of the registration process for
// the next level, the step length will simply start with the last value used
// for the previous level. This will guarantee the continuity of the path
// taken by the optimizer through the parameter space.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
OptimizerPointer optimizer = static_cast< OptimizerPointer >(registration->GetModifiableOptimizer() );
std::cout << "-------------------------------------" << std::endl;
std::cout << "MultiResolution Level : "
<< registration->GetCurrentLevel() << std::endl;
std::cout << std::endl;
if ( registration->GetCurrentLevel() == 0 )
{
optimizer->SetMaximumStepLength( 16.00 );
optimizer->SetMinimumStepLength( 0.005 );
}
else
{
optimizer->SetMaximumStepLength(
optimizer->GetMaximumStepLength() * 0.25 );
optimizer->SetMinimumStepLength(
optimizer->GetMinimumStepLength() * 0.1 );
}
}
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// Another version of the \code{Execute()} method accepting a \code{const}
// input object is also required since this method is defined as pure virtual
// in the base class. This version simply returns without taking any action.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
void Execute(const itk::Object * , const itk::EventObject & )
{ return; }
};
// Software Guide : EndCodeSnippet
// The following section of code implements an observer
// that will monitor the evolution of the registration process.
//
class CommandIterationUpdate : public itk::Command
{
public:
typedef CommandIterationUpdate Self;
typedef itk::Command Superclass;
typedef itk::SmartPointer<Self> Pointer;
itkNewMacro( Self );
protected:
CommandIterationUpdate() {};
public:
typedef itk::RegularStepGradientDescentOptimizer OptimizerType;
typedef const OptimizerType * OptimizerPointer;
void Execute(itk::Object *caller, const itk::EventObject & event) ITK_OVERRIDE
{
Execute( (const itk::Object *)caller, event);
}
void Execute(const itk::Object * object, const itk::EventObject & event) ITK_OVERRIDE
{
OptimizerPointer optimizer = static_cast< OptimizerPointer >( object );
if( !(itk::IterationEvent().CheckEvent( &event )) )
{
return;
}
std::cout << optimizer->GetCurrentIteration() << " ";
std::cout << optimizer->GetValue() << " ";
std::cout << optimizer->GetCurrentPosition() << std::endl;
}
};
int main( int argc, const char *argv[] )
{
if( argc < 4 )
{
std::cerr << "Missing Parameters " << std::endl;
std::cerr << "Usage: " << argv[0];
std::cerr << " fixedImageFile movingImageFile ";
std::cerr << " outputImagefile [backgroundGrayLevel]";
std::cerr << " [checkerBoardBefore] [checkerBoardAfter]";
std::cerr << " [useExplicitPDFderivatives ] " << std::endl;
std::cerr << " [numberOfBins] [numberOfSamples ] " << std::endl;
return EXIT_FAILURE;
}
const unsigned int Dimension = 2;
typedef unsigned short PixelType;
const std::string fixedImageFile = argv[1];
const std::string movingImageFile = argv[2];
const std::string outImagefile = argv[3];
const PixelType backgroundGrayLevel = (argc >4 )? atoi(argv[4]): 100;
const std::string checkerBoardBefore = (argc >5 )? argv[5]: "";
const std::string checkerBoardAfter = (argc >6 )? argv[6]: "";
const bool useExplicitPDFderivatives = (argc >7 )? static_cast<bool>(atoi(argv[7])): false;
const int numberOfBins = (argc >8 )? atoi(argv[8]): 0;
const int numberOfSamples = (argc >9 )? atoi(argv[9]): 0;
typedef itk::Image< PixelType, Dimension > FixedImageType;
typedef itk::Image< PixelType, Dimension > MovingImageType;
// Software Guide : BeginLatex
//
// The fixed and moving image types are defined as in previous
// examples. Due to the recursive nature of the process by which the
// downsampled images are computed by the image pyramids, the output
// images are required to have real pixel types. We declare this internal
// image type to be \code{InternalPixelType}:
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
typedef float InternalPixelType;
typedef itk::Image< InternalPixelType, Dimension > InternalImageType;
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// The types for the registration components are then derived using
// the internal image type.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
typedef itk::TranslationTransform< double, Dimension > TransformType;
typedef itk::RegularStepGradientDescentOptimizer OptimizerType;
typedef itk::LinearInterpolateImageFunction<
InternalImageType,
double > InterpolatorType;
typedef itk::MattesMutualInformationImageToImageMetric<
InternalImageType,
InternalImageType > MetricType;
typedef itk::MultiResolutionImageRegistrationMethod<
InternalImageType,
InternalImageType > RegistrationType;
// Software Guide: EndCodeSnippet
// Software Guide : BeginLatex
//
// In the multi-resolution framework, a
// \doxygen{MultiResolutionPyramidImageFilter} is used to create a pyramid
// of downsampled images. The size of each downsampled image is specified
// by the user in the form of a schedule of shrink factors. A description
// of the filter and the format of the schedules are found in
// Section \ref{sec:ImagePyramids}. For this example, we will simply use
// the default schedules.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
typedef itk::MultiResolutionPyramidImageFilter<
InternalImageType, InternalImageType > FixedImagePyramidType;
typedef itk::MultiResolutionPyramidImageFilter<
InternalImageType, InternalImageType > MovingImagePyramidType;
// Software Guide: EndCodeSnippet
// All the components are instantiated using their \code{New()} method
// and connected to the registration object as in previous example.
//
TransformType::Pointer transform = TransformType::New();
OptimizerType::Pointer optimizer = OptimizerType::New();
InterpolatorType::Pointer interpolator = InterpolatorType::New();
RegistrationType::Pointer registration = RegistrationType::New();
MetricType::Pointer metric = MetricType::New();
FixedImagePyramidType::Pointer fixedImagePyramid =
FixedImagePyramidType::New();
MovingImagePyramidType::Pointer movingImagePyramid =
MovingImagePyramidType::New();
registration->SetOptimizer( optimizer );
registration->SetTransform( transform );
registration->SetInterpolator( interpolator );
registration->SetMetric( metric );
registration->SetFixedImagePyramid( fixedImagePyramid );
registration->SetMovingImagePyramid( movingImagePyramid );
typedef itk::ImageFileReader< FixedImageType > FixedImageReaderType;
typedef itk::ImageFileReader< MovingImageType > MovingImageReaderType;
FixedImageReaderType::Pointer fixedImageReader = FixedImageReaderType::New();
MovingImageReaderType::Pointer movingImageReader = MovingImageReaderType::New();
fixedImageReader->SetFileName( fixedImageFile );
movingImageReader->SetFileName( movingImageFile );
// Software Guide : BeginLatex
//
// The fixed and moving images are read from a file. Before connecting
// these images to the registration we need to cast them to the internal
// image type using \doxygen{CastImageFilters}.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
typedef itk::CastImageFilter<
FixedImageType, InternalImageType > FixedCastFilterType;
typedef itk::CastImageFilter<
MovingImageType, InternalImageType > MovingCastFilterType;
FixedCastFilterType::Pointer fixedCaster = FixedCastFilterType::New();
MovingCastFilterType::Pointer movingCaster = MovingCastFilterType::New();
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// The output of the readers is connected as input to the cast
// filters. The inputs to the registration method are taken from the
// cast filters.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
fixedCaster->SetInput( fixedImageReader->GetOutput() );
movingCaster->SetInput( movingImageReader->GetOutput() );
registration->SetFixedImage( fixedCaster->GetOutput() );
registration->SetMovingImage( movingCaster->GetOutput() );
// Software Guide : EndCodeSnippet
fixedCaster->Update();
registration->SetFixedImageRegion(
fixedCaster->GetOutput()->GetBufferedRegion() );
typedef RegistrationType::ParametersType ParametersType;
ParametersType initialParameters( transform->GetNumberOfParameters() );
initialParameters[0] = 0.0; // Initial offset in mm along X
initialParameters[1] = 0.0; // Initial offset in mm along Y
registration->SetInitialTransformParameters( initialParameters );
metric->SetNumberOfHistogramBins( 128 );
metric->SetNumberOfSpatialSamples( 50000 );
if( argc > 8 )
{
// optionally, override the values with numbers taken from the command line arguments.
metric->SetNumberOfHistogramBins( numberOfBins );
}
if( argc > 9 )
{
// optionally, override the values with numbers taken from the command line arguments.
metric->SetNumberOfSpatialSamples( numberOfSamples );
}
// Software Guide : BeginLatex
//
// Given that the Mattes Mutual Information metric uses a random iterator in
// order to collect the samples from the images, it is usually convenient to
// initialize the seed of the random number generator.
//
// \index{itk::Mattes\-Mutual\-Information\-Image\-To\-Image\-Metric!ReinitializeSeed()}
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
metric->ReinitializeSeed( 76926294 );
// Software Guide : EndCodeSnippet
if( argc > 7 )
{
// Define whether to calculate the metric derivative by explicitly
// computing the derivatives of the joint PDF with respect to the Transform
// parameters, or doing it by progressively accumulating contributions from
// each bin in the joint PDF.
metric->SetUseExplicitPDFDerivatives( useExplicitPDFderivatives );
}
optimizer->SetNumberOfIterations( 200 );
optimizer->SetRelaxationFactor( 0.9 );
optimizer->SetGradientMagnitudeTolerance( 1e-6 );
// Create the Command observer and register it with the optimizer.
//
CommandIterationUpdate::Pointer observer = CommandIterationUpdate::New();
optimizer->AddObserver( itk::IterationEvent(), observer );
// Software Guide : BeginLatex
//
// Once all the registration components are in place we can create
// an instance of our interface command and connect it to the
// registration object using the \code{AddObserver()} method.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
typedef RegistrationInterfaceCommand<RegistrationType> CommandType;
CommandType::Pointer command = CommandType::New();
registration->AddObserver( itk::IterationEvent(), command );
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// We set the number of multi-resolution levels to three and trigger the
// registration process by calling \code{Update()}.
//
// \index{itk::Multi\-Resolution\-Image\-Registration\-Method!SetNumberOfLevels()}
// \index{itk::Multi\-Resolution\-Image\-Registration\-Method!Update()}
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
registration->SetNumberOfLevels( 3 );
try
{
registration->Update();
std::cout << "Optimizer stop condition: "
<< registration->GetOptimizer()->GetStopConditionDescription()
<< std::endl;
}
catch( itk::ExceptionObject & err )
{
std::cout << "ExceptionObject caught !" << std::endl;
std::cout << err << std::endl;
return EXIT_FAILURE;
}
// Software Guide : EndCodeSnippet
ParametersType finalParameters = registration->GetLastTransformParameters();
double TranslationAlongX = finalParameters[0];
double TranslationAlongY = finalParameters[1];
unsigned int numberOfIterations = optimizer->GetCurrentIteration();
double bestValue = optimizer->GetValue();
// Print out results
//
std::cout << "Result = " << std::endl;
std::cout << " Translation X = " << TranslationAlongX << std::endl;
std::cout << " Translation Y = " << TranslationAlongY << std::endl;
std::cout << " Iterations = " << numberOfIterations << std::endl;
std::cout << " Metric value = " << bestValue << std::endl;
// Software Guide : BeginLatex
//
// Let's execute this example using the following images
//
// \begin{itemize}
// \item BrainT1SliceBorder20.png
// \item BrainProtonDensitySliceShifted13x17y.png
// \end{itemize}
//
// The output produced by the execution of the method is
//
// \begin{verbatim}
// 0 -0.419408 [11.0796, 11.5431]
// 1 -0.775143 [18.0515, 25.9442]
// 2 -0.621443 [15.2813, 18.4392]
// 3 -1.00688 [7.81465, 15.567]
// 4 -0.733843 [11.7844, 16.0582]
// 5 -1.17593 [15.2929, 17.9792]
//
// 0 -0.902265 [13.4257, 17.2627]
// 1 -1.21519 [11.6959, 16.2588]
// 2 -1.04207 [12.6029, 16.68]
// 3 -1.21741 [13.4286, 17.2439]
// 4 -1.21605 [12.9899, 17.0041]
// 5 -1.26825 [13.163, 16.8237]
//
// 0 -1.25692 [13.0716, 16.909]
// 1 -1.29465 [12.9896, 17.0033]
// 2 -1.30922 [13.0513, 16.9934]
// 3 -1.30722 [13.0205, 16.9987]
// 4 -1.30978 [12.9897, 17.0039]
//
// Result =
// Translation X = 12.9897
// Translation Y = 17.0039
// Iterations = 6
// Metric value = -1.30921
// \end{verbatim}
//
// These values are a close match to the true misalignment of $(13,17)$
// introduced in the moving image.
//
// Software Guide : EndLatex
typedef itk::ResampleImageFilter<
MovingImageType,
FixedImageType > ResampleFilterType;
TransformType::Pointer finalTransform = TransformType::New();
finalTransform->SetParameters( finalParameters );
finalTransform->SetFixedParameters( transform->GetFixedParameters() );
ResampleFilterType::Pointer resample = ResampleFilterType::New();
resample->SetTransform( finalTransform );
resample->SetInput( movingImageReader->GetOutput() );
FixedImageType::Pointer fixedImage = fixedImageReader->GetOutput();
resample->SetSize( fixedImage->GetLargestPossibleRegion().GetSize() );
resample->SetOutputOrigin( fixedImage->GetOrigin() );
resample->SetOutputSpacing( fixedImage->GetSpacing() );
resample->SetOutputDirection( fixedImage->GetDirection() );
resample->SetDefaultPixelValue( backgroundGrayLevel );
typedef unsigned char OutputPixelType;
typedef itk::Image< OutputPixelType, Dimension > OutputImageType;
typedef itk::CastImageFilter<
FixedImageType,
OutputImageType > CastFilterType;
typedef itk::ImageFileWriter< OutputImageType > WriterType;
WriterType::Pointer writer = WriterType::New();
CastFilterType::Pointer caster = CastFilterType::New();
writer->SetFileName( outImagefile );
caster->SetInput( resample->GetOutput() );
writer->SetInput( caster->GetOutput() );
writer->Update();
//
// Generate checkerboards before and after registration
//
typedef itk::CheckerBoardImageFilter< FixedImageType > CheckerBoardFilterType;
CheckerBoardFilterType::Pointer checker = CheckerBoardFilterType::New();
checker->SetInput1( fixedImage );
checker->SetInput2( resample->GetOutput() );
caster->SetInput( checker->GetOutput() );
writer->SetInput( caster->GetOutput() );
resample->SetDefaultPixelValue( 0 );
// Before registration
TransformType::Pointer identityTransform = TransformType::New();
identityTransform->SetIdentity();
resample->SetTransform( identityTransform );
for (int q=0; q< argc; ++q)
{
std::cout << q << " " << argv[q] << std::endl;
}
if( checkerBoardBefore != std::string("") )
{
writer->SetFileName( checkerBoardBefore );
writer->Update();
}
// After registration
resample->SetTransform( finalTransform );
if( checkerBoardAfter != std::string("") )
{
writer->SetFileName( checkerBoardAfter );
writer->Update();
}
// Software Guide : BeginLatex
//
// \begin{figure}
// \center
// \includegraphics[width=0.32\textwidth]{MultiResImageRegistration1Output}
// \includegraphics[width=0.32\textwidth]{MultiResImageRegistration1CheckerboardBefore}
// \includegraphics[width=0.32\textwidth]{MultiResImageRegistration1CheckerboardAfter}
// \itkcaption[Multi-Resolution registration input images]{Mapped moving image
// (left) and composition of fixed and moving images before (center) and
// after (right) registration.}
// \label{fig:MultiResImageRegistration1Output}
// \end{figure}
//
// The result of resampling the moving image is presented in the left image
// of Figure \ref{fig:MultiResImageRegistration1Output}. The center and
// right images of the figure depict a checkerboard composite of the fixed
// and moving images before and after registration.
//
// Software Guide : EndLatex
// Software Guide : BeginLatex
//
// \begin{figure}
// \center
// \includegraphics[height=0.44\textwidth]{MultiResImageRegistration1TraceTranslations}
// \includegraphics[height=0.44\textwidth]{MultiResImageRegistration1TraceMetric}
// \itkcaption[Multi-Resolution registration output images]{Sequence of
// translations and metric values at each iteration of the optimizer.}
// \label{fig:MultiResImageRegistration1Trace}
// \end{figure}
//
// Figure \ref{fig:MultiResImageRegistration1Trace} (left) shows
// the sequence of translations followed by the optimizer as it searched
// the parameter space. The right side of the same figure shows the
// sequence of metric values computed as the optimizer searched the
// parameter space. From the trace, we can see that with the more
// aggressive optimization parameters we get quite close to the optimal
// value within 4 iterations with the remaining iterations just doing fine
// adjustments. It is interesting to compare these results with the ones
// of the single resolution example in Section
// \ref{sec:MultiModalityRegistrationMattes}, where 24 iterations were
// required as more conservative optimization parameters had to be used.
//
// Software Guide : EndLatex
return EXIT_SUCCESS;
}
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