bspline_data.hpp

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/*
Copyright (c) 2006, Michael Kazhdan and Matthew Bolitho
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namespace pcl
{
  namespace poisson
  {

    // BSplineData //
    // Support[i]:
    //          Odd:  i +/- 0.5 * ( 1 + Degree )
    //                  i - 0.5 * ( 1 + Degree ) < 0
    // <=>              i < 0.5 * ( 1 + Degree )
    //                  i + 0.5 * ( 1 + Degree ) > 0
    // <=>              i > - 0.5 * ( 1 + Degree )
    //                  i + 0.5 * ( 1 + Degree ) > r
    // <=>      i > r - 0.5 * ( 1 + Degree )
    //                  i - 0.5 * ( 1 + Degree ) < r
    // <=>      i < r + 0.5 * ( 1 + Degree )
    //          Even: i + 0.5 +/- 0.5 * ( 1 + Degree )
    //                  i - 0.5 * Degree < 0
    // <=>              i < 0.5 * Degree
    //                  i + 1 + 0.5 * Degree > 0
    // <=>              i > -1 - 0.5 * Degree
    //                  i + 1 + 0.5 * Degree > r
    // <=>              i > r - 1 - 0.5 * Degree
    //                  i - 0.5 * Degree < r
    // <=>              i < r + 0.5 * Degree
    template< int Degree > inline bool LeftOverlap( unsigned int depth , int offset )
    {
      offset <<= 1;
      if( Degree & 1 ) return (offset < 1+Degree) && (offset > -1-Degree );
      else             return (offset <   Degree) && (offset > -2-Degree );
    }
    template< int Degree > inline bool RightOverlap( unsigned int depth , int offset )
    {
      offset <<= 1;
      int r = 1<<(depth+1);
      if( Degree & 1 ) return (offset > 2-1-Degree) && (offset < 2+1+Degree );
      else             return (offset > 2-2-Degree) && (offset < 2+  Degree );
    }
    template< int Degree > inline int ReflectLeft( unsigned int depth , int offset )
    {
      if( Degree&1 ) return   -offset;
      else           return -1-offset;
    }
    template< int Degree > inline int ReflectRight( unsigned int depth , int offset )
    {
      int r = 1<<(depth+1);
      if( Degree&1 ) return r  -offset;
      else           return r-1-offset;
    }

    template< int Degree , class Real >
    BSplineData<Degree,Real>::BSplineData( void )
    {
      vvDotTable = dvDotTable = ddDotTable = NULL;
      valueTables = dValueTables = NULL;
      functionCount = sampleCount = 0;
    }

    template< int Degree , class Real >
    BSplineData< Degree , Real >::~BSplineData(void)
    {
      if( functionCount )
      {
        if( vvDotTable ) delete[] vvDotTable;
        if( dvDotTable ) delete[] dvDotTable;
        if( ddDotTable ) delete[] ddDotTable;

        if(  valueTables ) delete[]  valueTables;
        if( dValueTables ) delete[] dValueTables;
      }
      vvDotTable = dvDotTable = ddDotTable = NULL;
      valueTables = dValueTables=NULL;
      functionCount = 0;
    }

    template<int Degree,class Real>
    void BSplineData<Degree,Real>::set( int maxDepth , bool useDotRatios , bool reflectBoundary )
    {
      this->useDotRatios    = useDotRatios;
      this->reflectBoundary = reflectBoundary;

      depth = maxDepth;
      // [Warning] This assumes that the functions spacing is dual
      functionCount = BinaryNode< double >::CumulativeCenterCount( depth );
      sampleCount   = BinaryNode< double >::CenterCount( depth ) + BinaryNode< double >::CornerCount( depth );
      baseFunctions = new PPolynomial<Degree>[functionCount];
      baseBSplines = new BSplineComponents[functionCount];

      baseFunction = PPolynomial< Degree >::BSpline();
      for( int i=0 ; i<=Degree ; i++ ) baseBSpline[i] = Polynomial< Degree >::BSplineComponent( i ).shift( double(-(Degree+1)/2) + i - 0.5 );
      dBaseFunction = baseFunction.derivative();
      StartingPolynomial< Degree > sPolys[Degree+3];

      for( int i=0 ; i<Degree+3 ; i++ )
      {
        sPolys[i].start = double(-(Degree+1)/2) + i - 1.5;
        sPolys[i].p *= 0;
        if(         i<=Degree   )  sPolys[i].p += baseBSpline[i  ].shift( -1 );
        if( i>=1 && i<=Degree+1 )  sPolys[i].p += baseBSpline[i-1];
        for( int j=0 ; j<i ; j++ ) sPolys[i].p -= sPolys[j].p;
      }
      leftBaseFunction.set( sPolys , Degree+3 );
      for( int i=0 ; i<Degree+3 ; i++ )
      {
        sPolys[i].start = double(-(Degree+1)/2) + i - 0.5;
        sPolys[i].p *= 0;
        if(         i<=Degree   )  sPolys[i].p += baseBSpline[i  ];
        if( i>=1 && i<=Degree+1 )  sPolys[i].p += baseBSpline[i-1].shift( 1 );
        for( int j=0 ; j<i ; j++ ) sPolys[i].p -= sPolys[j].p;
      }
      rightBaseFunction.set( sPolys , Degree+3 );
      dLeftBaseFunction  =  leftBaseFunction.derivative();
      dRightBaseFunction = rightBaseFunction.derivative();
      leftBSpline = rightBSpline = baseBSpline;
      leftBSpline [1] +=  leftBSpline[2].shift( -1 ) ,  leftBSpline[0] *= 0;
      rightBSpline[1] += rightBSpline[0].shift(  1 ) , rightBSpline[2] *= 0;
      double c , w;
      for( int i=0 ; i<functionCount ; i++ )
      {
        BinaryNode< double >::CenterAndWidth( i , c , w );
        baseFunctions[i] = baseFunction.scale(w).shift(c);
        baseBSplines[i] = baseBSpline.scale(w).shift(c);
        if( reflectBoundary )
        {
          int d , off , r;
          BinaryNode< double >::DepthAndOffset( i , d , off );
          r = 1<<d;
          if     ( off==0   ) baseFunctions[i] =  leftBaseFunction.scale(w).shift(c);
          else if( off==r-1 ) baseFunctions[i] = rightBaseFunction.scale(w).shift(c);
          if     ( off==0   ) baseBSplines[i] =  leftBSpline.scale(w).shift(c);
          else if( off==r-1 ) baseBSplines[i] = rightBSpline.scale(w).shift(c);
        }
      }
    }
    template<int Degree,class Real>
    void BSplineData<Degree,Real>::setDotTables( int flags )
    {
      clearDotTables( flags );
      int size = ( functionCount*functionCount + functionCount )>>1;
      int fullSize = functionCount*functionCount;
      if( flags & VV_DOT_FLAG )
      {
        vvDotTable = new Real[size];
        memset( vvDotTable , 0 , sizeof(Real)*size );
      }
      if( flags & DV_DOT_FLAG )
      {
        dvDotTable = new Real[fullSize];
        memset( dvDotTable , 0 , sizeof(Real)*fullSize );
      }
      if( flags & DD_DOT_FLAG )
      {
        ddDotTable = new Real[size];
        memset( ddDotTable , 0 , sizeof(Real)*size );
      }
      double vvIntegrals[Degree+1][Degree+1];
      double vdIntegrals[Degree+1][Degree  ];
      double dvIntegrals[Degree  ][Degree+1];
      double ddIntegrals[Degree  ][Degree  ];
      int vvSums[Degree+1][Degree+1];
      int vdSums[Degree+1][Degree  ];
      int dvSums[Degree  ][Degree+1];
      int ddSums[Degree  ][Degree  ];
      SetBSplineElementIntegrals< Degree   , Degree   >( vvIntegrals );
      SetBSplineElementIntegrals< Degree   , Degree-1 >( vdIntegrals );
      SetBSplineElementIntegrals< Degree-1 , Degree   >( dvIntegrals );
      SetBSplineElementIntegrals< Degree-1 , Degree-1 >( ddIntegrals );

      for( int d1=0 ; d1<=depth ; d1++ )
        for( int off1=0 ; off1<(1<<d1) ; off1++ )
        {
          int ii = BinaryNode< Real >::CenterIndex( d1 , off1 );
          BSplineElements< Degree > b1( 1<<d1 , off1 , reflectBoundary ? BSplineElements<Degree>::NEUMANN   : BSplineElements< Degree>::NONE );
          BSplineElements< Degree-1 > db1;
          b1.differentiate( db1 );

          int start1 , end1;

          start1 = -1;
          for( int i=0 ; i<int(b1.size()) ; i++ ) for( int j=0 ; j<=Degree ; j++ )
          {
            if( b1[i][j] && start1==-1 ) start1 = i;
            if( b1[i][j] ) end1 = i+1;
          }
          for( int d2=d1 ; d2<=depth ; d2++ )
          {
            for( int off2=0 ; off2<(1<<d2) ; off2++ )
            {
              int start2 = off2-Degree;
              int end2   = off2+Degree+1;
              if( start2>=end1 || start1>=end2 ) continue;
              start2 = std::max< int >( start1 , start2 );
              end2   = std::min< int >(   end1 ,   end2 );
              if( d1==d2 && off2<off1 ) continue;
              int jj = BinaryNode< Real >::CenterIndex( d2 , off2 );
              BSplineElements< Degree > b2( 1<<d2 , off2 , reflectBoundary ? BSplineElements<Degree>::NEUMANN   : BSplineElements< Degree>::NONE );
              BSplineElements< Degree-1 > db2;
              b2.differentiate( db2 );

              int idx = SymmetricIndex( ii , jj );
              int idx1 = Index( ii , jj ) , idx2 = Index( jj , ii );

              memset( vvSums , 0 , sizeof( int ) * ( Degree+1 ) * ( Degree+1 ) );
              memset( vdSums , 0 , sizeof( int ) * ( Degree+1 ) * ( Degree   ) );
              memset( dvSums , 0 , sizeof( int ) * ( Degree   ) * ( Degree+1 ) );
              memset( ddSums , 0 , sizeof( int ) * ( Degree   ) * ( Degree   ) );
              for( int i=start2 ; i<end2 ; i++ )
              {
                for( int j=0 ; j<=Degree ; j++ ) for( int k=0 ; k<=Degree ; k++ ) vvSums[j][k] +=  b1[i][j] *  b2[i][k];
                for( int j=0 ; j<=Degree ; j++ ) for( int k=0 ; k< Degree ; k++ ) vdSums[j][k] +=  b1[i][j] * db2[i][k];
                for( int j=0 ; j< Degree ; j++ ) for( int k=0 ; k<=Degree ; k++ ) dvSums[j][k] += db1[i][j] *  b2[i][k];
                for( int j=0 ; j< Degree ; j++ ) for( int k=0 ; k< Degree ; k++ ) ddSums[j][k] += db1[i][j] * db2[i][k];
              }
              double vvDot = 0 , dvDot = 0 , vdDot = 0 , ddDot = 0;
              for( int j=0 ; j<=Degree ; j++ ) for( int k=0 ; k<=Degree ; k++ ) vvDot += vvIntegrals[j][k] * vvSums[j][k];
              for( int j=0 ; j<=Degree ; j++ ) for( int k=0 ; k< Degree ; k++ ) vdDot += vdIntegrals[j][k] * vdSums[j][k];
              for( int j=0 ; j< Degree ; j++ ) for( int k=0 ; k<=Degree ; k++ ) dvDot += dvIntegrals[j][k] * dvSums[j][k];
              for( int j=0 ; j< Degree ; j++ ) for( int k=0 ; k< Degree ; k++ ) ddDot += ddIntegrals[j][k] * ddSums[j][k];
              vvDot /= (1<<d2);
              ddDot *= (1<<d2);
              vvDot /= ( b1.denominator * b2.denominator );
              dvDot /= ( b1.denominator * b2.denominator );
              vdDot /= ( b1.denominator * b2.denominator );
              ddDot /= ( b1.denominator * b2.denominator );
              if( fabs(vvDot)<1e-15 ) continue;
              if( flags & VV_DOT_FLAG ) vvDotTable [idx] = Real( vvDot );
              if( useDotRatios )
              {
                if( flags & DV_DOT_FLAG ) dvDotTable[idx1] = Real( dvDot / vvDot );
                if( flags & DV_DOT_FLAG ) dvDotTable[idx2] = Real( vdDot / vvDot );
                if( flags & DD_DOT_FLAG ) ddDotTable[idx ] = Real( ddDot / vvDot );
              }
              else
              {
                if( flags & DV_DOT_FLAG ) dvDotTable[idx1] = Real( dvDot );
                if( flags & DV_DOT_FLAG ) dvDotTable[idx2] = Real( dvDot );
                if( flags & DD_DOT_FLAG ) ddDotTable[idx ] = Real( ddDot );
              }
            }
            BSplineElements< Degree > b;
            b = b1;
            b.upSample( b1 );
            b1.differentiate( db1 );
            start1 = -1;
            for( int i=0 ; i<int(b1.size()) ; i++ ) for( int j=0 ; j<=Degree ; j++ )
            {
              if( b1[i][j] && start1==-1 ) start1 = i;
              if( b1[i][j] ) end1 = i+1;
            }
          }
        }
    }
    template<int Degree,class Real>
    void BSplineData<Degree,Real>::clearDotTables( int flags )
    {
      if( (flags & VV_DOT_FLAG) && vvDotTable ) delete[] vvDotTable , vvDotTable = NULL;
      if( (flags & DV_DOT_FLAG) && dvDotTable ) delete[] dvDotTable , dvDotTable = NULL;
      if( (flags & DD_DOT_FLAG) && ddDotTable ) delete[] ddDotTable , ddDotTable = NULL;
    }
    template< int Degree , class Real >
    void BSplineData< Degree , Real >::setSampleSpan( int idx , int& start , int& end , double smooth ) const
    {
      int d , off , res;
      BinaryNode< double >::DepthAndOffset( idx , d , off );
      res = 1<<d;
      double _start = ( off + 0.5 - 0.5*(Degree+1) ) / res - smooth;
      double _end   = ( off + 0.5 + 0.5*(Degree+1) ) / res + smooth;
      //   (start)/(sampleCount-1) >_start && (start-1)/(sampleCount-1)<=_start
      // => start > _start * (sampleCount-1 ) && start <= _start*(sampleCount-1) + 1
      // => _start * (sampleCount-1) + 1 >= start > _start * (sampleCount-1)
      start = int( floor( _start * (sampleCount-1) + 1 ) );
      if( start<0 ) start = 0;
      //   (end)/(sampleCount-1)<_end && (end+1)/(sampleCount-1)>=_end
      // => end < _end * (sampleCount-1 ) && end >= _end*(sampleCount-1) - 1
      // => _end * (sampleCount-1) > end >= _end * (sampleCount-1) - 1
      end = int( ceil( _end * (sampleCount-1) - 1 ) );
      if( end>=sampleCount ) end = sampleCount-1;
    }
    template<int Degree,class Real>
    void BSplineData<Degree,Real>::setValueTables( int flags , double smooth )
    {
      clearValueTables();
      if( flags &   VALUE_FLAG )  valueTables = new Real[functionCount*sampleCount];
      if( flags & D_VALUE_FLAG ) dValueTables = new Real[functionCount*sampleCount];
      PPolynomial<Degree+1> function;
      PPolynomial<Degree>  dFunction;
      for( int i=0 ; i<functionCount ; i++ )
      {
        if(smooth>0)
        {
          function  = baseFunctions[i].MovingAverage(smooth);
          dFunction = baseFunctions[i].derivative().MovingAverage(smooth);
        }
        else
        {
          function  = baseFunctions[i];
          dFunction = baseFunctions[i].derivative();
        }
        for( int j=0 ; j<sampleCount ; j++ )
        {
          double x=double(j)/(sampleCount-1);
          if(flags &   VALUE_FLAG){ valueTables[j*functionCount+i]=Real( function(x));}
          if(flags & D_VALUE_FLAG){dValueTables[j*functionCount+i]=Real(dFunction(x));}
        }
      }
    }
    template<int Degree,class Real>
    void BSplineData<Degree,Real>::setValueTables(int flags,double valueSmooth,double normalSmooth){
      clearValueTables();
      if(flags &   VALUE_FLAG){ valueTables=new Real[functionCount*sampleCount];}
      if(flags & D_VALUE_FLAG){dValueTables=new Real[functionCount*sampleCount];}
      PPolynomial<Degree+1> function;
      PPolynomial<Degree>  dFunction;
      for(int i=0;i<functionCount;i++){
        if(valueSmooth>0)       { function=baseFunctions[i].MovingAverage(valueSmooth);}
        else                            { function=baseFunctions[i];}
        if(normalSmooth>0)      {dFunction=baseFunctions[i].derivative().MovingAverage(normalSmooth);}
        else                            {dFunction=baseFunctions[i].derivative();}

        for(int j=0;j<sampleCount;j++){
          double x=double(j)/(sampleCount-1);
          if(flags &   VALUE_FLAG){ valueTables[j*functionCount+i]=Real( function(x));}
          if(flags & D_VALUE_FLAG){dValueTables[j*functionCount+i]=Real(dFunction(x));}
        }
      }
    }


    template<int Degree,class Real>
    void BSplineData<Degree,Real>::clearValueTables(void){
      if( valueTables){delete[]  valueTables;}
      if(dValueTables){delete[] dValueTables;}
      valueTables=dValueTables=NULL;
    }

    template<int Degree,class Real>
    inline int BSplineData<Degree,Real>::Index( int i1 , int i2 ) const { return i1*functionCount+i2; }
    template<int Degree,class Real>
    inline int BSplineData<Degree,Real>::SymmetricIndex( int i1 , int i2 )
    {
      if( i1>i2 ) return ((i1*i1+i1)>>1)+i2;
      else        return ((i2*i2+i2)>>1)+i1;
    }
    template<int Degree,class Real>
    inline int BSplineData<Degree,Real>::SymmetricIndex( int i1 , int i2 , int& index )
    {
      if( i1<i2 )
      {
        index = ((i2*i2+i2)>>1)+i1;
        return 1;
      }
      else
      {
        index = ((i1*i1+i1)>>1)+i2;
        return 0;
      }
    }


    // BSplineElementData //
    template< int Degree >
    BSplineElements< Degree >::BSplineElements( int res , int offset , int boundary )
    {
      denominator = 1;
      this->resize( res , BSplineElementCoefficients<Degree>() );

      for( int i=0 ; i<=Degree ; i++ )
      {
        int idx = -_off + offset + i;
        if( idx>=0 && idx<res ) (*this)[idx][i] = 1;
      }
      if( boundary!=0 )
      {
        _addLeft( offset-2*res , boundary ) , _addRight( offset+2*res , boundary );
        if( Degree&1 ) _addLeft( offset-res , boundary ) , _addRight(  offset+res     , boundary );
        else           _addLeft( -offset-1  , boundary ) , _addRight( -offset-1+2*res , boundary );
      }
    }
    template< int Degree >
    void BSplineElements< Degree >::_addLeft( int offset , int boundary )
    {
      int res = int( this->size() );
      bool set = false;
      for( int i=0 ; i<=Degree ; i++ )
      {
        int idx = -_off + offset + i;
        if( idx>=0 && idx<res ) (*this)[idx][i] += boundary , set = true;
      }
      if( set ) _addLeft( offset-2*res , boundary );
    }
    template< int Degree >
    void BSplineElements< Degree >::_addRight( int offset , int boundary )
    {
      int res = int( this->size() );
      bool set = false;
      for( int i=0 ; i<=Degree ; i++ )
      {
        int idx = -_off + offset + i;
        if( idx>=0 && idx<res ) (*this)[idx][i] += boundary , set = true;
      }
      if( set ) _addRight( offset+2*res , boundary );
    }
    template< int Degree >
    void BSplineElements< Degree >::upSample( BSplineElements< Degree >& high ) const
    {
      fprintf( stderr , "[ERROR] B-spline up-sampling not supported for degree %d\n" , Degree );
      exit( 0 );
    }
    template<>
    void BSplineElements< 1 >::upSample( BSplineElements< 1 >& high ) const
    {
      high.resize( size()*2 );
      high.assign( high.size() , BSplineElementCoefficients<1>() );
      for( int i=0 ; i<int(size()) ; i++ )
      {
        high[2*i+0][0] += 1 * (*this)[i][0];
        high[2*i+0][1] += 0 * (*this)[i][0];
        high[2*i+1][0] += 2 * (*this)[i][0];
        high[2*i+1][1] += 1 * (*this)[i][0];

        high[2*i+0][0] += 1 * (*this)[i][1];
        high[2*i+0][1] += 2 * (*this)[i][1];
        high[2*i+1][0] += 0 * (*this)[i][1];
        high[2*i+1][1] += 1 * (*this)[i][1];
      }
      high.denominator = denominator * 2;
    }
    template<>
    void BSplineElements< 2 >::upSample( BSplineElements< 2 >& high ) const
    {
      //    /----\
      //   /      \
      //  /        \  = 1  /--\       +3    /--\     +3      /--\   +1        /--\
      // /          \     /    \           /    \           /    \           /    \
      // |----------|     |----------|   |----------|   |----------|   |----------|

      high.resize( size()*2 );
      high.assign( high.size() , BSplineElementCoefficients<2>() );
      for( int i=0 ; i<int(size()) ; i++ )
      {
        high[2*i+0][0] += 1 * (*this)[i][0];
        high[2*i+0][1] += 0 * (*this)[i][0];
        high[2*i+0][2] += 0 * (*this)[i][0];
        high[2*i+1][0] += 3 * (*this)[i][0];
        high[2*i+1][1] += 1 * (*this)[i][0];
        high[2*i+1][2] += 0 * (*this)[i][0];

        high[2*i+0][0] += 3 * (*this)[i][1];
        high[2*i+0][1] += 3 * (*this)[i][1];
        high[2*i+0][2] += 1 * (*this)[i][1];
        high[2*i+1][0] += 1 * (*this)[i][1];
        high[2*i+1][1] += 3 * (*this)[i][1];
        high[2*i+1][2] += 3 * (*this)[i][1];

        high[2*i+0][0] += 0 * (*this)[i][2];
        high[2*i+0][1] += 1 * (*this)[i][2];
        high[2*i+0][2] += 3 * (*this)[i][2];
        high[2*i+1][0] += 0 * (*this)[i][2];
        high[2*i+1][1] += 0 * (*this)[i][2];
        high[2*i+1][2] += 1 * (*this)[i][2];
      }
      high.denominator = denominator * 4;
    }

    template< int Degree >
    void BSplineElements< Degree >::differentiate( BSplineElements< Degree-1 >& d ) const
    {
      d.resize( this->size() );
      d.assign( d.size()  , BSplineElementCoefficients< Degree-1 >() );
      for( int i=0 ; i<int(this->size()) ; i++ ) for( int j=0 ; j<=Degree ; j++ )
      {
        if( j-1>=0 )   d[i][j-1] -= (*this)[i][j];
        if( j<Degree ) d[i][j  ] += (*this)[i][j];
      }
      d.denominator = denominator;
    }
    // If we were really good, we would implement this integral table to store
    // rational values to improve precision...
    template< int Degree1 , int Degree2 >
    void SetBSplineElementIntegrals( double integrals[Degree1+1][Degree2+1] )
    {
      for( int i=0 ; i<=Degree1 ; i++ )
      {
        Polynomial< Degree1 > p1 = Polynomial< Degree1 >::BSplineComponent( i );
        for( int j=0 ; j<=Degree2 ; j++ )
        {
          Polynomial< Degree2 > p2 = Polynomial< Degree2 >::BSplineComponent( j );
          integrals[i][j] = ( p1 * p2 ).integral( 0 , 1 );
        }
      }
    }


  }
}