# Find the location of points of the 2D projection on 3D image

Hi, I used this code to get the projection of my 3D image on the axial plane. I did some image processing on the 2D projection and I got some points on the 2D image. Now I want to find the coordinates of those points on corresponding 3D image, Does anyone know how I can do it? I really appreciate it if you can help me with that!

// Software Guide : BeginLatex
//
// \index{Iterators!and image slices}
//
// The \doxygen{ImageSliceIteratorWithIndex} class is an extension of
// \doxygen{ImageLinearIteratorWithIndex} from iteration along lines to
// iteration along both lines \emph{and planes} in an image.
// A \emph{slice} is a 2D
// plane spanned by two vectors pointing along orthogonal coordinate axes.  The
// slice orientation of the slice iterator is defined by specifying its two
// spanning axes.
//
// \begin{itemize}
// \index{itk::Image\-Slice\-Iterator\-With\-Index!SetFirstDirection()}
// \item \textbf{\code{SetFirstDirection()}}
// Specifies the first coordinate axis
// direction of the slice plane.
//
// \index{itk::Image\-Slice\-Iterator\-With\-Index!SetSecondDirection()}
// \item \textbf{\code{SetSecondDirection()}}
// Specifies the second coordinate axis
// direction of the slice plane.
// \end{itemize}
//
// Several new methods control movement from slice to slice.
//
// \begin{itemize}
//
// \index{itk::Image\-Slice\-Iterator\-With\-Index!NextSlice()}
// \item \textbf{\code{NextSlice()}} Moves the iterator to the beginning pixel
// location of the next slice in the image.  The origin of the next slice is
// calculated by incrementing the current origin index along the fastest
// increasing dimension of the image subspace which excludes the first and
// second dimensions of the iterator.
//
// \index{itk::Image\-Slice\-Iterator\-With\-Index!PreviousSlice()}
// \item \textbf{\code{PreviousSlice()}} Moves the iterator to the \emph{last
// valid pixel location} in the previous slice.  The origin of the previous
// slice is calculated by decrementing the current origin index along the
// fastest increasing dimension of the image subspace which excludes the first
// and second dimensions of the iterator.
//
// \index{itk::Image\-Slice\-Iterator\-With\-Index!IsAtReverseEndOfSlice()}
// \item \textbf{\code{IsAtReverseEndOfSlice()}} Returns true if the iterator
// points to \emph{one position before} the beginning pixel of the current
// slice.
//
// \index{itk::Image\-Slice\-Iterator\-With\-Index!IsAtEndOfSlice()}
// \item \textbf{\code{IsAtEndOfSlice()}} Returns true if the iterator points
// to \emph{one position past} the last valid pixel of the current slice.
//
// \end{itemize}
//
// The slice iterator moves line by line using \code{NextLine()} and
// \code{PreviousLine()}.  The line direction is parallel to the \emph{second}
// Section~\ref{sec:itkImageLinearIteratorWithIndex}).
//
// \index{itk::Image\-Slice\-Iterator\-With\-Index!example of using|(}
// The next code example calculates the maximum intensity projection along one
// of the coordinate axes of an image volume.  The algorithm is straightforward
// using ImageSliceIteratorWithIndex because we can coordinate
// movement through a slice of the 3D input image with movement through the 2D
// planar output.
//
// Here is how the algorithm works.  For each 2D slice of the input, iterate
// through all the pixels line by line. Copy a pixel value to the corresponding
// position in the 2D output image if it is larger than the value already
// contained there.  When all slices have been processed, the output image is
// the desired maximum intensity projection.
//
// We include a header for the const version of the slice iterator. For writing
// values to the 2D projection image, we use the linear iterator from the
// previous section.  The linear iterator is chosen because it can be set to
// follow the same path in its underlying 2D image that the slice iterator
// follows over each slice of the 3D image.
//
// Software Guide : EndLatex

#include "itkImage.h"
#include "itkMath.h"

// Software Guide : BeginCodeSnippet
#include "itkImageSliceConstIteratorWithIndex.h"
#include "itkImageLinearIteratorWithIndex.h"
// Software Guide : EndCodeSnippet
#include "itkImageFileWriter.h"

int
main(int argc, char * argv[])
{
//   Verify the number of parameters on the command line.
if (argc < 4)
{
std::cerr << "Missing parameters. " << std::endl;
std::cerr << "Usage: " << std::endl;
std::cerr << argv[0] << " inputImageFile outputImageFile projectionDirection"
<< std::endl;
return EXIT_FAILURE;
}

// Software Guide : BeginLatex
//
// The pixel type is defined as \code{unsigned short}.  For this application,
// we need two image types, a 3D image for the input, and a 2D image for the
// intensity projection.
//
// Software Guide : EndLatex

// Software Guide : BeginCodeSnippet
using PixelType = unsigned short;
using ImageType2D = itk::Image<PixelType, 2>;
using ImageType3D = itk::Image<PixelType, 3>;
// Software Guide : EndCodeSnippet

// Software Guide : BeginLatex
//
//  A slice iterator type is defined to walk the input image.
//
// Software Guide : EndLatex

// Software Guide : BeginCodeSnippet
using LinearIteratorType = itk::ImageLinearIteratorWithIndex<ImageType2D>;
using SliceIteratorType = itk::ImageSliceConstIteratorWithIndex<ImageType3D>;
// Software Guide : EndCodeSnippet

using WriterType = itk::ImageFileWriter<ImageType2D>;

ImageType3D::ConstPointer inputImage;
try
{
}
catch (const itk::ExceptionObject & err)
{
std::cerr << "ExceptionObject caught !" << std::endl;
std::cerr << "err" << std::endl;
return EXIT_FAILURE;
}

// Software Guide : BeginLatex
//
// The projection direction is read from the command line. The projection image
// will be the size of the 2D plane orthogonal to the projection direction.
// Its spanning vectors are the two remaining coordinate axes in the volume.
// These axes are recorded in the \code{direction} array.
//
// Software Guide : EndLatex

// Software Guide : BeginCodeSnippet
auto projectionDirection = static_cast<unsigned int>(::std::stoi(argv[3]));

unsigned int i, j;
unsigned int direction[2];
for (i = 0, j = 0; i < 3; ++i)
{
if (i != projectionDirection)
{
direction[j] = i;
j++;
}
}
// Software Guide : EndCodeSnippet

// Software Guide : BeginLatex
//
// The \code{direction} array is now used to define the projection image size
// based on the input image size.  The output image is created so that its
// common dimension(s) with the input image are the same size.  For example,
// if we project along the $x$ axis of the input, the size and origin of the
// $y$ axes of the input and output will match.  This makes the code slightly
// more complicated, but prevents a counter-intuitive rotation of the output.
//
// Software Guide : EndLatex

// Software Guide : BeginCodeSnippet
ImageType2D::RegionType            region;
ImageType2D::RegionType::SizeType  size;
ImageType2D::RegionType::IndexType index;

ImageType3D::RegionType requestedRegion = inputImage->GetRequestedRegion();

index[direction[0]] = requestedRegion.GetIndex()[direction[0]];
index[1 - direction[0]] = requestedRegion.GetIndex()[direction[1]];
size[direction[0]] = requestedRegion.GetSize()[direction[0]];
size[1 - direction[0]] = requestedRegion.GetSize()[direction[1]];

region.SetSize(size);
region.SetIndex(index);

ImageType2D::Pointer outputImage = ImageType2D::New();

outputImage->SetRegions(region);
outputImage->Allocate();
// Software Guide : EndCodeSnippet

// Software Guide : BeginLatex
//
// Next we create the necessary iterators.  The const slice iterator walks
// the 3D input image, and the non-const linear iterator walks the 2D output
// image. The iterators are initialized to walk the same linear path through
// a slice.  Remember that the \emph{second} direction of the slice iterator
// defines the direction that linear iteration walks within a slice.
//
// Software Guide : EndLatex

// Software Guide : BeginCodeSnippet
SliceIteratorType  inputIt(inputImage, inputImage->GetRequestedRegion());
LinearIteratorType outputIt(outputImage, outputImage->GetRequestedRegion());

inputIt.SetFirstDirection(direction[1]);
inputIt.SetSecondDirection(direction[0]);

outputIt.SetDirection(1 - direction[0]);
// Software Guide : EndCodeSnippet

// Software Guide: BeginLatex
//
// Now we are ready to compute the projection.  The first step is to initialize
// all of the projection values to their nonpositive minimum value.  The
// projection values are then updated row by row from the first slice of the
// input.  At the end of the first slice, the input iterator steps to the first
// row in the next slice, while the output iterator, whose underlying image
// consists of only one slice, rewinds to its first row.  The process repeats
// until the last slice of the input is processed.
//
// Software Guide : EndLatex

// Software Guide : BeginCodeSnippet
outputIt.GoToBegin();
while (!outputIt.IsAtEnd())
{
while (!outputIt.IsAtEndOfLine())
{
outputIt.Set(itk::NumericTraits<unsigned short>::NonpositiveMin());
++outputIt;
}
outputIt.NextLine();
}

inputIt.GoToBegin();
outputIt.GoToBegin();

while (!inputIt.IsAtEnd())
{
while (!inputIt.IsAtEndOfSlice())
{
while (!inputIt.IsAtEndOfLine())
{
outputIt.Set(std::max(outputIt.Get(), inputIt.Get()));
++inputIt;
++outputIt;
}
outputIt.NextLine();
inputIt.NextLine();
}
outputIt.GoToBegin();
inputIt.NextSlice();
}
// Software Guide : EndCodeSnippet

WriterType::Pointer writer = WriterType::New();
writer->SetFileName(argv[2]);
writer->SetInput(outputImage);
try
{
writer->Update();
}
catch (const itk::ExceptionObject & err)
{
std::cerr << "ExceptionObject caught !" << std::endl;
std::cerr << "err" << std::endl;
return EXIT_FAILURE;
}

// Software Guide : BeginLatex
//
// Running this example code on the 3D image
// \code{Examples/Data/BrainProtonDensity3Slices.mha} using the $z$-axis as
// the axis of projection gives the image shown in
// Figure~\ref{fig:ImageSliceIteratorWithIndexOutput}.
//
// \begin{figure}
// \centering
// \includegraphics[width=0.4\textwidth]{ImageSliceIteratorWithIndexOutput}
// \itkcaption[Maximum intensity projection using ImageSliceIteratorWithIndex]{The
// maximum intensity projection through three slices of a volume.}
// \protect\label{fig:ImageSliceIteratorWithIndexOutput}
// \end{figure}
//
//
// \index{itk::Image\-Slice\-Iterator\-With\-Index!example of using|)}
//
// Software Guide : EndLatex

return EXIT_SUCCESS;
}


Determining location of 3D points from their 2D projections is under-constrained problem. Unless you record 3D position during projection step, you can’t go back later.