FFT2D.h

Go to the documentation of this file.

// -*- Mode: C++; tab-width: 2; -*-
// vi: set ts=2:
//

#ifndef BALL_MATHS_TFFT2D_H
#define BALL_MATHS_TFFT2D_H

#ifndef BALL_COMMON_EXCEPTION_H
# include <BALL/COMMON/exception.h>
#endif

#ifndef BALL_DATATYPE_REGULARDATA2D_H
# include <BALL/DATATYPE/regularData2D.h>
#endif

#ifndef BALL_MATHS_VECTOR2_H
# include <BALL/MATHS/vector2.h>
#endif

#include <math.h>
#include <complex>
#include <fftw3.h>

#include <BALL/MATHS/fftwCommon.h>


namespace BALL
{
  template <typename ComplexTraits>
  class TFFT2D 
    : public TRegularData2D<std::complex<typename ComplexTraits::ComplexPrecision> >
  {
    public:
    
      typedef std::complex<typename ComplexTraits::ComplexPrecision> Complex;
      typedef TRegularData2D<std::complex<typename ComplexTraits::ComplexPrecision> > ComplexVector;
      typedef typename TRegularData2D<std::complex<typename ComplexTraits::ComplexPrecision> >::IndexType IndexType;

      BALL_CREATE(TFFT2D)

      
 
      
      TFFT2D();

      TFFT2D(const TFFT2D &data);

       // ldn is not any longer the binary logarithm but the absolute number of grid points
      TFFT2D(Size nX, Size nY, double stepPhysX=1., double stepPhysY=1., Vector2 origin=Vector2(0.,0.), bool inFourierSpace=false);

      virtual ~TFFT2D();
      


      const TFFT2D& operator = (const TFFT2D& fft_2d);
      
      virtual void clear();
      
      virtual void destroy();



      bool operator == (const TFFT2D& fft_2d) const;
      
      // @name Accessors
      
      void doFFT();

      void doiFFT();

      bool translate(const Vector2& trans_origin);

      bool setPhysStepWidth(double new_width_x, double new_width_y);

      double getPhysStepWidthX() const;

      double getPhysStepWidthY() const;

      double getFourierStepWidthX() const;

      double getFourierStepWidthY() const;

      double getPhysSpaceMinX() const;

      double getPhysSpaceMinY() const;

      double getPhysSpaceMaxX() const;

      double getPhysSpaceMaxY() const;

      double getFourierSpaceMinX() const;

      double getFourierSpaceMinY() const;

      double getFourierSpaceMaxX() const;

      double getFourierSpaceMaxY() const;
        
      Size getMaxXIndex() const;

      Size getMaxYIndex() const;
        
      Size getNumberOfInverseTransforms() const;

      Vector2 getGridCoordinates(Position position) const;

      Complex getData(const Vector2& pos) const;

      Complex getInterpolatedValue(const Vector2& pos) const;

      void setData(const Vector2& pos, Complex val);

      Complex& operator[](const Vector2& pos);

      const Complex& operator[](const Vector2& pos) const;

      const Complex& operator [] (const IndexType& index) const { return TRegularData2D<Complex>::operator [](index); }

      Complex& operator [] (const IndexType& index) {  return TRegularData2D<Complex>::operator [](index); }
      
      Complex& operator[](const Position& pos)
      {
        return TRegularData2D<Complex>::operator [] (pos);
      }

      const Complex& operator[](const Position& pos) const
      {
        return TRegularData2D<Complex>::operator [] (pos);
      }
      
      // 
      void setNumberOfFFTTransforms(Size num)
      {
        numPhysToFourier_ = num;
      }
      
      // 
      void setNumberOfiFFTTransforms(Size num)
      {
        numFourierToPhys_ = num;
      }
      
      Complex phase(const Vector2& pos) const;
        
      bool isInFourierSpace() const;

    protected:
      Size lengthX_, lengthY_;
      bool inFourierSpace_;
      Size numPhysToFourier_;
      Size numFourierToPhys_;
      Vector2 origin_;
      double stepPhysX_, stepPhysY_;
      double stepFourierX_, stepFourierY_;
      Vector2 minPhys_, maxPhys_;
      Vector2 minFourier_, maxFourier_;
      
      // new version for FFTW3
      typename ComplexTraits::FftwPlan planForward_;
      typename ComplexTraits::FftwPlan planBackward_;

      
      // to control plan calculation with new fftw3
      Size dataLength_;
      Complex *dataAdress_;
      bool planCalculated_;
      
  };
  
  typedef TFFT2D<BALL_FFTW_DEFAULT_TRAITS> FFT2D;
  
  template <typename ComplexTraits>
  const TRegularData2D<typename TFFT2D<ComplexTraits>::Complex>& operator<< 
      (TRegularData2D<typename TFFT2D<ComplexTraits>::Complex>& to, const TFFT2D<ComplexTraits>& from);
  
  template <typename ComplexTraits>
  const RegularData2D& operator << (RegularData2D& to, const TFFT2D<ComplexTraits>& from);
  
  template <typename ComplexTraits>
  TFFT2D<ComplexTraits>::TFFT2D()
    : TRegularData2D<Complex>(),
      dataLength_(0),
      dataAdress_(0),
      planCalculated_(false)
  {
  }
  
  template <typename ComplexTraits>
  bool TFFT2D<ComplexTraits>::operator == (const TFFT2D<ComplexTraits>& fft2D) const
  {
    // test whether data_.size() == fft2D.data_.size()
    // instead of testing 2 lengths. Better for vector handling.
    
    if (lengthX_ == fft2D.lengthX_ &&
        lengthY_ == fft2D.lengthY_ &&
        origin_ == fft2D.origin_ &&
        stepPhysX_ == fft2D.stepPhysX_ &&
        stepPhysY_ == fft2D.stepPhysY_ &&
        stepFourierX_ == fft2D.stepFourierX_ &&
        stepFourierY_ == fft2D.stepFourierY_ &&
        minPhys_ == fft2D.minPhys_ &&
        maxPhys_ == fft2D.maxPhys_ &&
        minFourier_ == fft2D.minFourier_ &&
        maxFourier_ == fft2D.maxFourier_ &&
        numPhysToFourier_ == fft2D.numPhysToFourier_ &&
        numFourierToPhys_ == fft2D.numFourierToPhys_)
    {
      Vector2 min  = inFourierSpace_ ?  minFourier_  :   minPhys_;
      Vector2 max  = inFourierSpace_ ?  maxFourier_  :   maxPhys_;
      double stepX = inFourierSpace_ ? stepFourierX_ : stepPhysX_;
      double stepY = inFourierSpace_ ? stepFourierY_ : stepPhysY_;  
      
      for (double posX=min.x; posX<=max.x; posX+=stepX)
      {
        for (double posY=min.y; posY<=max.y; posY+=stepY)
        {
          if (getData(Vector2(posX,posY)) != fft2D.getData(Vector2(posX,posY)))
          {
            return false;
          }
        }
      }
      
      return true;
    }
  
    return false;
  }
  
  template <typename ComplexTraits>
  bool TFFT2D<ComplexTraits>::translate(const Vector2& trans_origin)
  {
    Position internalOriginX = (Position) Maths::rint(trans_origin.x*stepPhysX_);
    Position internalOriginY = (Position) Maths::rint(trans_origin.y*stepPhysY_);
    
    if ((internalOriginX <= lengthX_) && (internalOriginY <= lengthY_))
    {
      origin_.x = trans_origin.x;
      origin_.y = trans_origin.y;
      
      minPhys_ = Vector2(-origin_.x,-origin_.y);
      maxPhys_ = Vector2(((lengthX_-1)*stepPhysX_)-origin_.x,((lengthY_-1)*stepPhysY_)-origin_.y);
      minFourier_ = Vector2(-(lengthX_/2.-1)*stepFourierX_,-(lengthY_/2.-1)*stepFourierY_);
      maxFourier_ = Vector2((lengthX_/2.)*stepFourierX_,(lengthY_/2.)*stepFourierY_);
     
      return true;
    }
    else
    {
      return false;
    }
  }

  template <typename ComplexTraits>
  bool TFFT2D<ComplexTraits>::setPhysStepWidth(double new_width_x, double new_width_y)
  {
    if ((new_width_x <= 0) || (new_width_y <= 0))
    {
      return false;
    }
    else
    {
      stepPhysX_ = new_width_x;
      stepPhysY_ = new_width_y;
      stepFourierX_ = 2.*M_PI/(stepPhysX_*lengthX_);
      stepFourierY_ = 2.*M_PI/(stepPhysY_*lengthY_);

      minPhys_ = Vector2(-origin_.x,-origin_.y);
      maxPhys_ = Vector2(((lengthX_-1)*stepPhysX_)-origin_.x,((lengthY_-1)*stepPhysY_)-origin_.y);
      minFourier_ = Vector2(-(lengthX_/2.-1)*stepFourierX_,-(lengthY_/2.-1)*stepFourierY_);
      maxFourier_ = Vector2((lengthX_/2.)*stepFourierX_,(lengthY_/2.)*stepFourierY_);
  
      return true;
    }
  }
  
  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getPhysStepWidthX() const
  {
    return stepPhysX_;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getPhysStepWidthY() const
  {
    return stepPhysY_;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getFourierStepWidthX() const
  {
    return stepFourierX_;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getFourierStepWidthY() const
  {
    return stepFourierY_;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getPhysSpaceMinX() const
  {
    return minPhys_.x;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getPhysSpaceMinY() const
  {
    return minPhys_.y;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getPhysSpaceMaxX() const
  {
    return maxPhys_.x;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getPhysSpaceMaxY() const
  {
    return maxPhys_.y;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getFourierSpaceMinX() const
  {
    return minFourier_.x;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getFourierSpaceMinY() const
  {
    return minFourier_.y;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getFourierSpaceMaxX() const
  {
    return maxFourier_.x;
  }

  template <typename ComplexTraits>
  double TFFT2D<ComplexTraits>::getFourierSpaceMaxY() const
  {
    return maxFourier_.y;
  }
  
  template <typename ComplexTraits>
  Size TFFT2D<ComplexTraits>::getMaxXIndex() const
  {
    return (lengthX_ - 1);
  }
  
  template <typename ComplexTraits>
  Size TFFT2D<ComplexTraits>::getMaxYIndex() const
  {
    return (lengthY_ - 1);
  }
  
  template <typename ComplexTraits>
  Size TFFT2D<ComplexTraits>::getNumberOfInverseTransforms() const
  {
    return numFourierToPhys_;
  }
  
  // new for compatibility with FFT3D
  template <typename ComplexTraits>
  Vector2 TFFT2D<ComplexTraits>::getGridCoordinates(Position position) const
  {
    if (!inFourierSpace_)
    {
      if (position >= ComplexVector::size())
      {
        throw Exception::OutOfGrid(__FILE__, __LINE__);
      }
    
      Vector2 r;
      Position  x, y;


      // AR: ??????
      y = position % lengthY_;
      x = position / lengthY_;

      r.set(-origin_.x + (float)x * stepPhysX_,
            -origin_.y + (float)y * stepPhysY_);

      return r;
    }
    else
    {
      if (position >= ComplexVector::size())
      {
        throw Exception::OutOfGrid(__FILE__, __LINE__);
      }
    
      Vector2 r;
      Index x, y;
  
      // AR: ??????
      y = position % lengthY_;
      x = position / lengthY_;

      if (x>=lengthX_/2.)
      {
        x-=lengthX_;
      }
      
      if (y>=lengthY_/2.)
      {
        y-=lengthY_;
      }

      r.set((float)x * stepFourierX_,
            (float)y * stepFourierY_);

      return r;
    }
  }
  
  
  
  template <typename ComplexTraits>
  typename TFFT2D<ComplexTraits>::Complex TFFT2D<ComplexTraits>::getData(const Vector2& pos) const
  {
    Complex result;
    double normalization=1.;

    if (!inFourierSpace_)
    {
      result = (*this)[pos];
      normalization=1./((float)pow((float)(lengthX_*lengthY_),(int)numFourierToPhys_));
    }
    else
    {
      result = (*this)[pos] * phase(pos);
      normalization=1./(2.*M_PI)*(stepPhysX_*stepPhysY_)/((float)pow((float)(lengthX_*lengthY_),(int)numFourierToPhys_));
    }

    result *= normalization;
    
    return result;
  }

  template <typename ComplexTraits>
  typename TFFT2D<ComplexTraits>::Complex TFFT2D<ComplexTraits>::getInterpolatedValue(const Vector2& pos) const
  {
    Complex result;
    
    Vector2 min  = inFourierSpace_ ? minFourier_   :   minPhys_;
    Vector2 max  = inFourierSpace_ ? maxFourier_   :   maxPhys_;
    double stepX = inFourierSpace_ ? stepFourierX_ : stepPhysX_;
    double stepY = inFourierSpace_ ? stepFourierY_ : stepPhysY_;
    
    if (    (pos.x < min.x) || (pos.y < min.y)
         || (pos.x > max.x) || (pos.y > max.y)  )
    {
      throw Exception::OutOfGrid(__FILE__, __LINE__);
    }

    Vector2 h(pos.x - min.x, pos.y - min.y);
    double modX = fmod((double)h.x,stepX);
    double modY = fmod((double)h.y,stepY);

    if (modX==0 && modY ==0) // we are on the grid
    {
      return getData(pos);
    }

    double beforeX = floor(h.x/stepX)*stepX+ min.x;
    double beforeY = floor(h.y/stepY)*stepY+ min.y;
    double afterX  =  ceil(h.x/stepX)*stepX+ min.x;
    double afterY  =  ceil(h.y/stepY)*stepY+ min.y;
      
    double tx = (pos.x - beforeX)/stepX;
    double ty = (pos.y - beforeY)/stepY;

    result  = getData(Vector2(beforeX,beforeY))*(typename ComplexTraits::ComplexPrecision)((1.-tx)*(1.-ty));
    result += getData(Vector2(afterX, beforeY))*(typename ComplexTraits::ComplexPrecision)(    tx *(1.-ty));
    result += getData(Vector2(beforeX,afterY ))*(typename ComplexTraits::ComplexPrecision)((1.-tx)*    ty );
    result += getData(Vector2(afterX, afterY ))*(typename ComplexTraits::ComplexPrecision)(    tx *    ty );

    return result;
  }

  template <typename ComplexTraits>
  void TFFT2D<ComplexTraits>::setData(const Vector2& pos, Complex val)
  {
    Complex dummy;
  
    if (!inFourierSpace_)
    {
      dummy = Complex(val.real()*((float)pow((float)(lengthX_*lengthY_),(int)numFourierToPhys_)),
                        val.imag()*((float)pow((float)(lengthX_*lengthY_),(int)numFourierToPhys_)));
  
      (*this)[pos]=dummy;
    }
    else
    {
      val*=phase(pos)*(typename ComplexTraits::ComplexPrecision)((2*M_PI/(stepPhysX_*stepPhysY_)))
                     *(typename ComplexTraits::ComplexPrecision)pow((typename ComplexTraits::ComplexPrecision)(lengthX_*lengthY_),(int)numFourierToPhys_);
      
      dummy = val;
    
      (*this)[pos]=dummy;
    }
  }

  template <typename ComplexTraits>
  typename TFFT2D<ComplexTraits>::Complex& TFFT2D<ComplexTraits>::operator[](const Vector2& pos)
  {
    Index internalPos;

    if (!inFourierSpace_)
    {
      Index i, j;
      
      i = (Index) Maths::rint((pos.x+origin_.x)/stepPhysX_);
      j = (Index) Maths::rint((pos.y+origin_.y)/stepPhysY_);

      internalPos = j + i*lengthY_;
    }
    else
    {
      Index i, j;

      i = (Index) Maths::rint(pos.x/stepFourierX_);
      j = (Index) Maths::rint(pos.y/stepFourierY_);

      if (i<0)
      {
        i+=lengthX_;
      }

      if (j<0)
      {
        j+=lengthY_;
      }

      internalPos = (j + i*lengthY_);
    }

    if ((internalPos < 0) || (internalPos>=(Index) (lengthX_*lengthY_)))
    {
      throw Exception::OutOfGrid(__FILE__, __LINE__);
    }
    
    return operator [] (internalPos);
  }

  template <typename ComplexTraits>
  const typename TFFT2D<ComplexTraits>::Complex& TFFT2D<ComplexTraits>::operator[](const Vector2& pos) const
  {
    Index internalPos;

    if (!inFourierSpace_)
    {
      Index i, j;
      
      i = (Index) Maths::rint((pos.x+origin_.x)/stepPhysX_);
      j = (Index) Maths::rint((pos.y+origin_.y)/stepPhysY_);

      internalPos = j + i*lengthY_;
    }
    else
    {
      Index i, j;

      i = (Index) Maths::rint(pos.x/stepFourierX_);
      j = (Index) Maths::rint(pos.y/stepFourierY_);

      if (i<0)
      {
        i+=lengthX_;
      }

      if (j<0)
      {
        j+=lengthY_;
      }

      internalPos = (j + i*lengthY_);
    }

    if ((internalPos < 0) || (internalPos>=(Index) (lengthX_*lengthY_)))
    {
      throw Exception::OutOfGrid(__FILE__, __LINE__);
    }
    
    return operator [] (internalPos);
  }
  
  template <typename ComplexTraits>
  typename TFFT2D<ComplexTraits>::Complex TFFT2D<ComplexTraits>::phase(const Vector2& pos) const
  {
    double phase = 2.*M_PI*(  Maths::rint(pos.x/stepFourierX_)*Maths::rint(origin_.x/stepPhysX_)
                              /lengthX_
                            + Maths::rint(pos.y/stepFourierY_)*Maths::rint(origin_.y/stepPhysY_)
                              /lengthY_ );

    Complex result = Complex(cos(phase), sin(phase));
            
    return result;
  }
  
  template <typename ComplexTraits>
  bool TFFT2D<ComplexTraits>::isInFourierSpace() const
  {
    return inFourierSpace_;
  }
  
  template <typename ComplexTraits>
  const TRegularData2D<typename TFFT2D<ComplexTraits>::Complex>& operator << 
    (TRegularData2D<typename TFFT2D<ComplexTraits>::Complex>& to, const TFFT2D<ComplexTraits>& from)
  {
    // first decide if the FFT3D data is in Fourier space.
    if (!from.isInFourierSpace())
    {
      // create a new grid
      Size lengthX = from.getMaxXIndex()+1;
      Size lengthY = from.getMaxYIndex()+1;
      
      TRegularData2D<typename TFFT2D<ComplexTraits>::Complex> newGrid(TRegularData2D<typename TFFT2D<ComplexTraits>::Complex>::IndexType(lengthX, lengthY),
                                      Vector2(from.getPhysSpaceMinX(), from.getPhysSpaceMinY()),
                                      Vector2(from.getPhysSpaceMaxX(), from.getPhysSpaceMaxY()));

      // and fill it
      double normalization=1./(pow((float)(lengthX*lengthY),from.getNumberOfInverseTransforms()));
      typename TFFT2D<ComplexTraits>::Complex dataIn;
      typename TFFT2D<ComplexTraits>::Complex dataOut;
      
      for (Position i = 0; i < from.size(); i++)
      {
        Position x, y;

        y =  i % lengthY;
        x =  i / lengthY;

        dataIn  = from[i];
        dataOut = dataIn;
        
        newGrid[x + y*lengthY] = dataOut*(typename ComplexTraits::ComplexPrecision)normalization;
      }

      to = newGrid;

      return to;
    }
    else
    {
      // we are in Fourier space
      
      // create a new grid
      Size lengthX = from.getMaxXIndex()+1;
      Size lengthY = from.getMaxYIndex()+1;
      //float stepPhysX = from.getPhysStepWidthX();
      //float stepPhysY = from.getPhysStepWidthY();
      float stepFourierX = from.getFourierStepWidthX();
      float stepFourierY = from.getFourierStepWidthY();


      
      TRegularData2D<typename TFFT2D<ComplexTraits>::Complex> newGrid(TRegularData2D<typename TFFT2D<ComplexTraits>::Complex>::IndexType(lengthX, lengthY),
                                      Vector2(from.getFourierSpaceMinX(), 
                                              from.getFourierSpaceMinY()),
                                      Vector2(from.getFourierSpaceMaxX(),
                                              from.getFourierSpaceMaxY()));

      // and fill it
      // AR: old double normalization=1./(sqrt(2.*M_PI))*(stepPhysX*stepPhysY*stepPhysZ)/(pow((float)(lengthX*lengthY*lengthZ),from.getNumberOfInverseTransforms()));
      double normalization=1./(2.*M_PI)/(pow((float)(lengthX*lengthY),from.getNumberOfInverseTransforms()));
      
      
      Index x, y;
      Vector2 r;
      typename TFFT2D<ComplexTraits>::Complex dataIn;
      typename TFFT2D<ComplexTraits>::Complex dataOut;
  
      for (Position i = 0; i < from.size(); i++)
      {
        y =  i % lengthY;
        x =  i / lengthY;

        if (x>lengthX/2.)
        {
          x-=lengthX;
        }

        if (y>lengthY/2.)
        {
          y-=lengthY;
        }

        r.set((float)x * stepFourierX,
              (float)y * stepFourierY);

        dataIn = from[i];
        dataOut = dataIn;
        
        newGrid[x + y*lengthY] = dataOut*(typename ComplexTraits::ComplexPrecision)normalization*from.phase(r);
      }

      to = newGrid;

      return to;
    }
  }
  
  template <typename ComplexTraits>
  const RegularData2D& operator << (RegularData2D& to, const TFFT2D<ComplexTraits>& from)
  {
    // first decide if the FFT2D data is in Fourier space.
    if (!from.isInFourierSpace())
    {
      // create a new grid
      Size lengthX = from.getMaxXIndex()+1;
      Size lengthY = from.getMaxYIndex()+1;
      
      RegularData2D newGrid(RegularData2D::IndexType(lengthX, lengthY),
                            Vector2(from.getPhysSpaceMinX(), 
                                    from.getPhysSpaceMinY()),
                            Vector2(from.getPhysSpaceMaxX(),
                                    from.getPhysSpaceMaxY()));

      // and fill it
      double normalization = 1./(pow((float)(lengthX*lengthY),from.getNumberOfInverseTransforms()));
      typename TFFT2D<ComplexTraits>::Complex dataIn;
      typename TFFT2D<ComplexTraits>::Complex dataOut;

      typename TFFT2D<ComplexTraits>::IndexType current_index;
      typename RegularData2D::IndexType regdat_index;
      for (current_index.x = 0; current_index.x < lengthX; current_index.x++)
      {
        for (current_index.y = 0; current_index.y < lengthY; current_index.y++)
        {
          regdat_index.x = current_index.x;
          regdat_index.y = current_index.y;

          dataIn  = from[current_index];
          dataOut = dataIn;
        
          newGrid[regdat_index] = dataOut.real()*normalization;
        }
      }

      to = newGrid;

      return to;
    }
    else
    {
      // we are in Fourier space
      
      // create a new grid
      Size lengthX = from.getMaxXIndex()+1;
      Size lengthY = from.getMaxYIndex()+1;
      //float stepPhysX = from.getPhysStepWidthX();
      //float stepPhysY = from.getPhysStepWidthY();
      float stepFourierX = from.getFourierStepWidthX();
      float stepFourierY = from.getFourierStepWidthY();
  
      RegularData2D newGrid(RegularData2D::IndexType(lengthX, lengthY), Vector2(from.getFourierSpaceMinX(), from.getFourierSpaceMinY()), Vector2(from.getFourierSpaceMaxX(), from.getFourierSpaceMaxY()));

      // and fill it
      // AR: old version double normalization=1./(sqrt(2.*M_PI))*(stepPhysX*stepPhysY*stepPhysZ)/(pow((float)(lengthX*lengthY*lengthZ),from.getNumberOfInverseTransforms()));
      double normalization=1./(2.*M_PI)/(pow((float)(lengthX*lengthY),from.getNumberOfInverseTransforms()));
      
      Index x, y;
      signed int xp, yp;
      Vector2 r;
      typename TFFT2D<ComplexTraits>::Complex dataIn;
      typename TFFT2D<ComplexTraits>::Complex dataOut;
  
      RegularData2D::IndexType current_index;
      for (Position i = 0; i < from.size(); i++)
      {
        y =  i % lengthY;
        x =  i / lengthY;
        
        xp = x;
        yp = y;

        if (xp>=lengthX/2.)
        {
          xp-=(int)lengthX;
        }
        if (yp>=lengthY/2.)
        {
          yp-=(int)lengthY;
        }

        if (x>=lengthX/2.)
        {
          x-=(int)(lengthX/2.);
        }
        else
        {
          x+=(int)(lengthX/2.);
        }

        if (y>=lengthY/2.)
        {
          y-=(int)(lengthY/2.);
        }
        else
        {
          y+=(int)(lengthY/2.);
        }


        r.set((float)xp * stepFourierX,
              (float)yp * stepFourierY);

        dataIn = from[i];
        dataOut = dataIn;

        current_index.x = x;
        current_index.y = y;

        newGrid[current_index] = (dataOut*(typename ComplexTraits::ComplexPrecision)normalization*from.phase(r)).real();
      }

      to = newGrid;

      return to;
    }
  }
  
  
  
}

#endif // BALL_MATHS_TFFT2D_H

---