#include <MeasureOfStraightLines.h>

Public Member Functions
	MeasureOfStraightLines ()
	~MeasureOfStraightLines ()
void	selfDisplay (std::ostream &out) const
bool	isValid () const
double	computeMeasure (const std::vector< double > &a, const std::vector< double > &b)
double	computeCentroidA (const std::vector< double > &a, const std::vector< double > &b)
double	computeCentroidB (const std::vector< double > &a, const std::vector< double > &b)
void	setEpsilon (const double aValue)

Private Member Functions
	MeasureOfStraightLines (const MeasureOfStraightLines &other)
MeasureOfStraightLines &	operator= (const MeasureOfStraightLines &other)
double	computeMeasureEdge (double a0, double b0, double a1, double b1)
double	computeCentroidEdge_a (double a0, double b0, double a1, double b1)
double	computeCentroidEdge_b (double a0, double b0, double a1, double b1)
double	__computeCentroidTriApprox_b (double a0, double b0, double a1, double b1)
double	__computeCentroidEdgeApprox_b (double a0, double b0, double a1, double b1)
double	__computeCentroidSquare_b (double x1, double y1, double x2, double y2)
int	sign (const double a)

Private Attributes
double	myEpsilon

Detailed Description

The aim of this class is to compute the measure in the Lebesgues sense of the set of straight lines associated to domains defined as polygons in the (a,b)-parameter space. This parameter space maps the line $ax-y+b=0$ to the point $(a,b)$.

Description of class 'MeasureOfStraightLines'

Aim:

   * @inproceedings{COEURJOLLY:2009:HAL-00432711:1,
   *   title = { {M}easure of {S}traight {L}ines and its {A}pplications in {D}igital {G}eometry},
   *   author = {{C}oeurjolly, {D}avid and {S}ivignon, {I}sabelle},
   *   booktitle = {13th {I}nternational {W}orkshop on {C}ombinatorial {I}mage {A}nalysis 13th {I}nternational {W}orkshop on {C}ombinatorial {I}mage {A}nalysis },
   *   publisher = {{R}esearch {P}ublishing {S}ervices },
   *   pages = {1-12 },
   *   address = {{C}ancun {M}exique },
   *   audience = {internationale },
   *   year = {2009},
   *   URL = {http://hal.archives-ouvertes.fr/hal-00432711/PDF/mesure.pdf},
   * }
   *

See also:: testMeasure.cpp

Todo:: Create a vector<Point> interface

Definition at line 79 of file MeasureOfStraightLines.h.

Constructor & Destructor Documentation

DGtal::MeasureOfStraightLines::MeasureOfStraightLines ( )

inline

Constructor.

Default value

Definition at line 50 of file MeasureOfStraightLines.ih.

References myEpsilon.

{
  myEpsilon = 0.0005;
}

DGtal::MeasureOfStraightLines::~MeasureOfStraightLines ( )

inline

Destructor.

Definition at line 61 of file MeasureOfStraightLines.ih.

{

}

DGtal::MeasureOfStraightLines::MeasureOfStraightLines ( const MeasureOfStraightLines & other )

private

Copy constructor.

Parameters:

other the object to clone. Forbidden by default.

Member Function Documentation

double DGtal::MeasureOfStraightLines::__computeCentroidEdgeApprox_b	(	double	a0,
		double	b0,
		double	a1,
		double	b1
	)

inlineprivate

Approximate the centroid on 'b' on the trapezioid (a0,0)-(a0,b0)-(a1,b1)-(a1,0) (internal function)

Parameters:

a0	abscissa first point.
b0	ordinate of the first point.
a1	abscissa of the second point.
b1	ordinate of the second point.

Approximate the centroid on 'b' on the trapezioid (a0,0)-(a0,b0)-(a1,b1)-(a1,0) (internal function)

A0 -> A1

Definition at line 245 of file MeasureOfStraightLines.ih.

{
  double measure,y2;
  unsigned int nb_step;
  if ( a1 < a0 )
    {
      double tmp = a0;
      a0 = a1;
      a1 = tmp;
      tmp = b0;
      b0 = b1;
      b1 = tmp;
    }
  measure = 0;
  nb_step = ( unsigned int ) floor ( fabs ( a1-a0 ) /myEpsilon );
  double slope = ( b1-b0 ) / ( a1-a0 );
  double decal = b1 - ( b1-b0 ) / ( a1-a0 ) *a1;
  if ( slope == 0 )
    return __computeCentroidSquare_b ( a0, 0, a1, b0 );
  for ( unsigned int i=0; i < nb_step ; i++ )
    {
      y2 = ( b1-b0 ) / ( a1-a0 ) * ( a0+ ( i+1 ) *myEpsilon ) + decal;
      if ( y2>0 )
        measure += __computeCentroidSquare_b ( a0+i*myEpsilon, 0, a0+ ( i+1 ) *myEpsilon, y2 );
      else
        measure += __computeCentroidSquare_b ( a0+i*myEpsilon, y2, a0+ ( i+1 ) *myEpsilon, 0 );
    }
  return measure;
}

double DGtal::MeasureOfStraightLines::__computeCentroidSquare_b	(	double	x1,
		double	y1,
		double	x2,
		double	y2
	)

inlineprivate

Compute the centroid on 'b' on the rectangular domain with vertices (x1,,y1) - (x2,y2)

Precondition:: y1<y2

Parameters:

x1	abscissa first point.
y1	ordinate of the first point.
x2	abscissa of the second point.
y2	ordinate of the second point.

Compute the centroid on 'b' on the rectangular domain with vertices (x1,,y1) - (x2,y2) PRECONDITION: y1<y2

Definition at line 218 of file MeasureOfStraightLines.ih.

{
  double val;
  val = ( ( -1.0* ( sqrt ( 1.0 + pow ( x1,2 ) ) *x2 ) + x1*sqrt ( 1 + pow ( x2,2 ) ) ) * ( pow ( y1,2 ) - pow ( y2,2 ) ) ) / ( 2.0*sqrt ( ( 1.0 + pow ( x1,2 ) ) * ( 1.0 + pow ( x2,2 ) ) ) );
  ASSERT ( val>=0 );
  return val;
}

double DGtal::MeasureOfStraightLines::__computeCentroidTriApprox_b	(	double	a0,
		double	b0,
		double	a1,
		double	b1
	)

inlineprivate

Approximate the centroid on 'b' on the triangle (0,0)-(a0,b0)-(a1,b1) (internal function)

Parameters:

a0	abscissa first point.
b0	ordinate of the first point.
a1	abscissa of the second point.
b1	ordinate of the second point.

Approximate the centroid on 'b' on the triangle (0,0)-(a0,b0)-(a1,b1) (internal function)

Definition at line 285 of file MeasureOfStraightLines.ih.

{
  int signe = sign ( a0*b1 - a1*b0 );
  double measure = 0;
  double m_A0A1=0, m_0A0=0, m_0A1=0;
  if ( ( b0 != 0 ) && ( a0 != 0 ) )
    m_0A0 = __computeCentroidEdgeApprox_b ( 0,0,a0,b0 );
  if ( ( b1 != 0 )  && ( a1 != 0 ) )
    m_0A1 = __computeCentroidEdgeApprox_b ( 0,0,a1,b1 );
  if ( ( a1 != a0 ) )
    m_A0A1 = __computeCentroidEdgeApprox_b ( a0,b0,a1,b1 );
  ASSERT ( m_0A0>=0 );
  ASSERT ( m_0A1>=0 );
  ASSERT ( m_A0A1>=0 );
  if ( a0 < a1 )
    {
      double det = ( a1 * ( b0 - b1 ) - b1* ( a0 - a1 ) );
      if ( det >0 )
        measure = m_0A0 + m_A0A1 - m_0A1;
      else
        measure = m_0A1 - m_0A0 - m_A0A1;
    }
  else
    {
      double det = ( a0 * ( b1 - b0 ) - b0* ( a1 - a0 ) );
      if ( det >0 )
        measure = m_0A1 + m_A0A1 - m_0A0;
      else
        measure = m_0A0 - m_0A1 - m_A0A1;
    }
  ASSERT ( measure>=0 );
  return signe*measure;
}

double DGtal::MeasureOfStraightLines::computeCentroidA	(	const std::vector< double > &	a,
		const std::vector< double > &	b
	)

inline

Compute the abscissa of the centroid of the polygon {(a_i,b_i)} in the (a,b)-parameter space with respect to the measure of lines.

REQUIREMENTS:

The polygon is given counter-clockwise
a_i > 0

Parameters:

a	the a-value of polygon vertices
b	the b-value of polygon vertices

Returns:: the measure value (positive value)

Compute the abscissa of the centroid of the polygon {(a_i,b_i)} in the (a,b)-parameter space

Parameters:

a	the a-value of polygon vertices
b	the b-value of polygon vertices

Returns:: the measure value (positive value)

Definition at line 164 of file MeasureOfStraightLines.ih.

{
  double measure = 0;
  double C_a = 0;
  ASSERT(a.size() == b.size());
  
  for (unsigned int i=0 ; i < a.size() ; i++)
    {
      measure += computeMeasureEdge( a[i], b[i], 
                                     a[ ( i+1 ) % a.size()],b[ ( i+1 ) %a.size()] );
      C_a += computeCentroidEdge_a ( a[i], b[i], 
                                     a[ ( i+1 ) % a.size()],b[ ( i+1 ) %a.size()] );
    } 
  return C_a/measure;
}

double DGtal::MeasureOfStraightLines::computeCentroidB	(	const std::vector< double > &	a,
		const std::vector< double > &	b
	)

inline

Compute the ordinate of the centroid of the polygon {(a_i,b_i)} in the (a,b)-parameter space with respect to the measure of lines. Note that there is a numerical approximation is performed.

REQUIREMENTS:

The polygon is given counter-clockwise
a_i > 0

Parameters:

a	the a-value of polygon vertices
b	the b-value of polygon vertices

Returns:: the measure value (positive value)

Compute the abscissa of the centroid of the polygon {(a_i,b_i)} in the (a,b)-parameter space

Parameters:

a	the a-value of polygon vertices
b	the b-value of polygon vertices

Returns:: the measure value (positive value)

Definition at line 191 of file MeasureOfStraightLines.ih.

{
  double measure = 0;
  double C_b = 0;
  ASSERT(a.size() == b.size());
  
  for (unsigned int i=0 ; i < a.size() ; i++)
    {
      measure += computeMeasureEdge( a[i], b[i], 
                                     a[ ( i+1 ) % a.size()],b[ ( i+1 ) %a.size()] );
      C_b += computeCentroidEdge_b ( a[i], b[i], 
                                     a[ ( i+1 ) % a.size()],b[ ( i+1 ) %a.size()] );
    } 
  return C_b/measure;
}

double DGtal::MeasureOfStraightLines::computeCentroidEdge_a	(	double	a0,
		double	b0,
		double	a1,
		double	b1
	)

inlineprivate

Compute the abscissa of the centroid associated to an edge (a0,b0)-(a1,b1) It returns the measure of the triangle defined by the origin and the edge.

Parameters:

a0	abscissa first point.
b0	ordinate of the first point.
a1	abscissa of the second point.
b1	ordinate of the second point.

Returns:: the measure

Compute the centroid on 'a' on the triangle (0,0)-(a0,b0)-(a1,b1)

Definition at line 356 of file MeasureOfStraightLines.ih.

{
  //double measure=0;
  double delta= ( a0*b1 - a1*b0 );
  if ( ( b0 == 0 ) && ( b1 == 0 ) )
    return 0;
  if ( ( a0 == 0 ) && ( a1 == 0 ) )
    return 0;
  if ( a0 == 0 )
    return delta* ( a1 - boost::math::asinh ( a1 ) ) / ( a1*a1 );
  if ( a1 == 0 )
    return delta* ( a0 - boost::math::asinh ( a0 ) ) / ( a0*a0 );
  if ( a0 == a1 )
    return delta* ( ( -sqrt ( 1 + pow ( a1,-2 ) ) - 1.0/ ( a1*sqrt ( 1 + pow ( a1,2 ) ) ) + a1/sqrt ( 1 + pow ( a1,2 ) ) +
                      ( 2*boost::math::asinh ( a1 ) )  /pow ( a1,2 ) ) /2. );
  return delta* ( ( - ( a1*boost::math::asinh ( a0 ) ) + a0*boost::math::asinh ( a1 ) ) / ( a0* ( a0 - a1 ) *a1 ) );
}

double DGtal::MeasureOfStraightLines::computeCentroidEdge_b	(	double	a0,
		double	b0,
		double	a1,
		double	b1
	)

inlineprivate

Compute the ordinate of the centroid associated to an edge (a0,b0)-(a1,b1) It returns the measure of the triangle defined by the origin and the edge.

Parameters:

a0	abscissa first point.
b0	ordinate of the first point.
a1	abscissa of the second point.
b1	ordinate of the second point.

Returns:: the measure

Compute the centroid on 'b' on the triangle (0,0)-(a0,b0)-(a1,b1)

Definition at line 327 of file MeasureOfStraightLines.ih.

{
  //double measure=0;
  double delta= ( a0*b1 - a1*b0 );
  if ( ( b0 == 0 ) && ( b1 == 0 ) )
    return 0;
  if ( ( a0 == 0 ) && ( a1 == 0 ) )
    return 0;
  if ( a0 == 0 )
    return delta* ( a1* ( ( -2 + sqrt ( 1 + a1*a1 ) ) *b0 + 2*b1 ) + ( b0 - 2*b1 ) *boost::math::asinh ( a1 ) ) / ( 2*a1*a1*a1 );
  if ( a1 == 0 )
    return delta* ( a0* ( 2*b0 + ( -2 + sqrt ( 1 + a0*a0 ) ) *b1 ) + ( -2*b0 + b1 ) *boost::math::asinh ( a0 ) ) / ( 2*a0*a0*a0 );
  if ( a0 == a1 )
    return delta* ( ( - ( ( a1* ( b0 + a1* ( -a1 + sqrt ( 1 + pow ( a1,-2 ) ) *sqrt ( 1 + pow ( a1,2 ) ) ) *
                                  b1 ) ) /sqrt ( 1.0 + pow ( a1,2 ) ) ) + ( b0 + b1 ) *boost::math::asinh ( a1 ) ) /
                    ( 2.*pow ( a1,3 ) ) );
  return __computeCentroidTriApprox_b ( a0,b0,a1,b1 );
}

double DGtal::MeasureOfStraightLines::computeMeasure	(	const std::vector< double > &	a,
		const std::vector< double > &	b
	)

inline

Compute the measure of the polygon {(a_i,b_i)} in the (a,b)-parameter space

REQUIREMENTS:

The polygon is given counter-clockwise
a_i > 0

Parameters:

a	the a-value of polygon vertices
b	the b-value of polygon vertices

Returns:: the measure value (positive value)

Compute the measure of the polygon {(a_i,b_i)} in the (a,b)-parameter space

Parameters:

a	the a-value of polygon vertices
b	the b-value of polygon vertices

Returns:: the measure value (positive value)

Definition at line 140 of file MeasureOfStraightLines.ih.

{
  double measure = 0;
  ASSERT(a.size() == b.size());
  
  for (unsigned int i=0 ; i < a.size() ; i++)
    {
      measure += computeMeasureEdge ( a[i], b[i], 
                                      a[ ( i+1 ) % a.size()],b[ ( i+1 ) %a.size()] );
    } 
  return measure;
}

double DGtal::MeasureOfStraightLines::computeMeasureEdge	(	double	a0,
		double	b0,
		double	a1,
		double	b1
	)

inlineprivate

Compute the measure associated to an edge (a0,b0)-(a1,b1) It returns the measure of the triangle defined by the origin and the edge.

Parameters:

a0	abscissa first point.
b0	ordinate of the first point.
a1	abscissa of the second point.
b1	ordinate of the second point.

Returns:: the measure

Compute the measure associated to the Edge (a0,b0) – (a1,b1) in the (a,b)-space

Definition at line 91 of file MeasureOfStraightLines.ih.

{
  //double measure=0;
  //Aligned with the 'b' axis
  if ( ( b0 == 0 ) && ( b1 == 0 ) )
    return 0;
  //Aligned with the 'b' axis
  if ( ( a0 == 0 ) && ( a1 == 0 ) )
    return 0;
  double delta = ( a0*b1 - a1*b0 );
  if ( a0 == 0 )
    return delta * ( 1.0/ ( 1.0 + sqrt ( 1 + pow ( a1,2 ) ) ) );
  if ( a1 == 0 )
    return delta * ( ( -1.0 + sqrt ( 1.0 + pow ( a0,2 ) ) ) /pow ( a0,2 ) );
  if ( a0 != a1 )
    {
      return  delta * ( ( a0 - a1 + sqrt ( 1 + pow ( a0,2 ) ) *a1 - a0*sqrt ( 1 + pow ( a1,2 ) ) ) / ( pow ( a0,2 ) *a1 - a0*pow ( a1,2 ) ) );
    }
  else
    return  delta * a1/ ( a0* ( 1 + pow ( a1,2 ) + sqrt ( 1 + pow ( a1,2 ) ) ) );
}

bool DGtal::MeasureOfStraightLines::isValid ( ) const

inline

Checks the validity/consistency of the object.

Returns:: 'true' if the object is valid, 'false' otherwise.

Definition at line 82 of file MeasureOfStraightLines.ih.

{
  return true;
}

MeasureOfStraightLines& DGtal::MeasureOfStraightLines::operator= ( const MeasureOfStraightLines & other )

private

Assignment.

Parameters:

other the object to copy.

Returns:: a reference on 'this'. Forbidden by default.

void DGtal::MeasureOfStraightLines::selfDisplay ( std::ostream & out ) const

inline

Writes/Displays the object on an output stream.

Parameters:

out	the output stream where the object is written.

Definition at line 71 of file MeasureOfStraightLines.ih.

{
  out << "[MeasureOfStraightLines]";
}

void DGtal::MeasureOfStraightLines::setEpsilon ( const double aValue )

inline

Set the internal Epsilon threshold for the numerical approximation.

Parameters:

aValue the new epsilon value

Set the epsilon threshold in the numerical approximation.

Parameters:

aValue the new epsilon value

Definition at line 124 of file MeasureOfStraightLines.ih.

{
  myEpsilon = aValue;
}

int DGtal::MeasureOfStraightLines::sign ( const double a )

inlineprivate

Parameters:

a a value

Returns:: the sign of a number (1 or -1)

Definition at line 230 of file MeasureOfStraightLines.ih.

{
  if ( a>=0 )
    return 1;
  else
    return -1;
}

Field Documentation

double DGtal::MeasureOfStraightLines::myEpsilon

private

Definition at line 171 of file MeasureOfStraightLines.h.

Referenced by MeasureOfStraightLines().

The documentation for this class was generated from the following files:

Public Member Functions

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Field Documentation