Salut,

Pour ma première contribution (c++), je vous propose une régression circulaire sur un ensemble de points (x; y).

Voilà la déclaration de la classe Matrix (matrix.h) :

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#ifndef __MATRIX__
#define __MATRIX__
 
#include <vector>
 
struct Matrix
{
	/// @brief Solve the system AX=Y where A and Y are known and X is unknown.
	/// @param [in] A A n*n matrix where n = A.size() / 2 = Y.size().
	/// @param [in] X The matrix to find. X is resized if needed.
	/// @param [in] Y A 1*n matrix where n = A.size() / 2 = Y.size().
	/// @throw std::runtime_error If an error occured.
	static void gaussjordan(std::vector<double> const & A, std::vector<double> & X, std::vector<double> const & Y);
 
	/// @brief Process to the regression of the points (xlist; ylist) by a circle.
	/// @param [in] xlist List of the x coordinates of the points that should fit a circle.
	/// @param [in] ylist List of the y coordinates of the points that should fit a circle.
	/// @param [in] xcenter The x coordinate of the center of the circle computed from the points (xlist; ylist).
	/// @param [in] ycenter The y coordinate of the center of the circle computed from the points (xlist; ylist).
	/// @param [in] radius The radius of the circle computed from the points (xlist; ylist).
	/// @param [in] rmse The root mean square error of the regression computed as the root squared sum of the difference between the computed radius and the distances between the computed center and the points (xlist; ylist)
	/// @throw std::runtime_error If an error occured.
	static void circularRegression(std::vector<double> const & xlist, std::vector<double> const & ylist, double & xcenter, double & ycenter, double & radius, double & rmse);
}; // Matrix
 
#endif
et voilà l'implémentation de la classe Matrix (matrix.cpp) :

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#include <limits>
#include <stdexcept>
#include <utility>
#include <map>
#include <cmath>
#include "matrix.h"
 
void Matrix::gaussjordan(std::vector<double> const & A, std::vector<double> & X, std::vector<double> const & Y)
{
  // get the size of the matrix and check system input
  unsigned int const n = Y.size();
  unsigned int const m = n + 1;
  if(A.size() != n * n)
    {
      throw std::runtime_error("Bad size with A and Y arguments");
    }
 
  // resize the ouptut 
  if(X.size() != n)
    {
      X.resize(n);
    }
 
  std::map<unsigned int, unsigned int> swapped;
  for(unsigned int row = 0; row < n; ++row)
    {
      swapped[row] = row;
    }
 
  // here is the system to solve
  std::vector<double> system(n * n + n);
 
  // array of pointers on each row of the system
  std::vector<double *> rows(n);
 
  // setup the system
  std::vector<double>::const_iterator a = A.begin();
  std::vector<double>::const_iterator y = Y.begin();
  std::vector<double>::iterator s = system.begin();
  for(unsigned int row = 0; row < n; ++row)
    {
      // keep the address of the first item of the current row
      rows[row] = &(*s);
 
      // set A
      for(unsigned int col = 0; col < n; ++col, ++a, ++s)
	{
	  *s = *a;	  
	}
 
      // set Y
      *s++ = *y++;
    }
 
  // find X to solve  the system AX=Y
  for(unsigned int ipivot = 0; ipivot < n; ++ipivot)
    {
      double pivot = rows[ipivot][ipivot];
 
      // if the current pivot is null, search for another line with a valid pivot (for the column ipivot)
      if(fabs(pivot) <= std::numeric_limits<double>::epsilon())
	{
	  // browse next row, searching for a valid pivot at column ipivot
	  for(unsigned int row = ipivot + 1; row < n; ++row)
	    {
	      // if found, swap the rows pointer (not the memory)
	      pivot = rows[row][ipivot];
	      if(fabs(pivot) > std::numeric_limits<double>::epsilon())
		{
		  std::swap(rows[ipivot], rows[row]);
		  std::swap(swapped[ipivot], swapped[row]);
		  break;
		}	      
	    }	  
	}
 
      // if a valid pivot was not found, then the matrix can not be inversed
      if(fabs(pivot) <= std::numeric_limits<double>::epsilon())
	{
	  throw std::runtime_error("The system can not be solved");
	}
 
      // normalize the row of the pivot so as the pivot is set to 1.0
      for(unsigned int col = 0; col < m; ++col)
	{
	  rows[ipivot][col] /= pivot;
	}
 
      // for all row, except the one of the pivot, set the item of the column "ipivot" ot 0
      for(unsigned int row = 0; row < n; ++row)
	{
	  if(row == ipivot)
	    {
	      continue;
	    }
 
	  double const factor = rows[row][ipivot];
	  for(unsigned int col = 0; col < m; ++col)
	    {
	      rows[row][col] -= factor * rows[ipivot][col];
	    }
	}
    }
 
  // now the system is solved, X is the last column of the system
  for(unsigned int row = 0; row < n; ++row)
    {      
      X[row] = system[swapped[row] * m + m - 1];
    }
}
 
void Matrix::circularRegression(std::vector<double> const & xlist, std::vector<double> const & ylist, double & xcenter, double & ycenter, double & radius, double & rmse)
{
  xcenter = 0.0;
  ycenter = 0.0;
  radius = 0.0;
  rmse = 0.0;
 
  // compute the A (3x3 matrix) and B (3x1 matrix)
  std::vector<double> A(9);
  std::vector<double> B(3);
  std::fill(A.begin(), A.end(), 0);
  std::fill(B.begin(), B.end(), 0);
  for(unsigned int index = 0; index < xlist.size(); ++index)
    {
      double const & x = xlist[index];
      double const & y = ylist[index];
      double const xx = x * x;
      double const yy = y * y;
      double const xy = x * y;
      double const xxx = xx * x;
      double const xxy = xx * y;
      double const yyy = yy * y;
      double const xyy = x * yy;
      A[0] += 2 * xx;
      A[1] += 2 * xy;
      A[2] += x;
      A[3] += 2 * xy;
      A[4] += 2 * yy;
      A[5] += y;
      A[6] += 2 * x;
      A[7] += 2 * y;
      A[8] += 1;
      B[0] += xxx + xyy; 
      B[1] += xxy + yyy;
      B[2] += xx + yy;
    }
 
  // solve the system AX=B using a the gauss-jordan method
  std::vector<double> X(3);
  gaussjordan(A, X, B);
 
  // circle details extraction
  xcenter = X[0];
  ycenter = X[1];
  radius = sqrt(X[2] + xcenter * xcenter + ycenter * ycenter);
 
  // compute the root mean square
  for(unsigned int index = 0; index < xlist.size(); ++index)
    {
      double const & x = xlist[index];
      double const & y = ylist[index];
      double const distance = sqrt((x - xcenter) * (x - xcenter) + (y - ycenter) * (y - ycenter));
      rmse += (radius - distance) * (radius - distance);
    }
 
  rmse /= (double)xlist.size();
  rmse = sqrt(rmse);
}
Le tout s'utilise très facilement (en c++ je rappelle) via le code :

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// variables
std::vector<double> xlist; // ... The list of the x coordinates of the points that match a circle.
std::vector<double> ylist; // ... The list of the y coordinates of the points that match a circle.
double xcenter = 0.0; // To store the x coordinate of the center of the circle computed by the function.
double ycenter = 0.0; // To store the y coordinate of the center of the circle computed by the function.
double radius = 0.0; // To store the radius of the circle computed by the function.
double rmse = 0.0; // To store the root mean square error resulted from the regression.
 
try
{
   // regression of the points by a circle
   Matrix::circularRegression(xlist, ylist, xcenter, ycenter, radius, rmse);
}
catch(std::exception const & error)
{
   std::cout << error.what() << std::endl;
}
catch(...)
{
   std::cout << "An undefined error has occured" << std::endl;
}
Et un main (main.cpp) pour tester tout ça :

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#include <iostream>
#include <iomanip>
#include <cmath>
#include <stdio.h>
#include <stdlib.h>
#include "matrix.h"
 
struct Circle
{
  double xcenter;
  double ycenter;
  double radius;
}; // Circle
 
int main()
{
  // for std::stw
  unsigned int const setwv = 5;
 
  // definition of the circle
  Circle const expected = {20.5, -3.7, 8.0};
  Circle computed = {0.0, 0.0, 0.0};
 
  // lists of the points on the circle used to make the regression
  std::vector<double> xlist;
  std::vector<double> ylist;
 
  // display the informations about the expected circle 
  std::cout << "  ---------------------------------------------------------------------";
  std::cout << std::endl;
  std::cout << "||" << std::right << std::setw(2 * setwv) << "Expected circle : " << std::endl;
  std::cout << "||" << std::right << std::setw(4 * setwv) <<  "x center = " << expected.xcenter << std::endl;
  std::cout << "||" << std::right << std::setw(4 * setwv) <<  "y center = " << expected.ycenter << std::endl;
  std::cout << "||" << std::right << std::setw(4 * setwv) <<  "radius = " << expected.radius << std::endl;
 
  // display the titles of the informations about the points on the circle
  std::cout << "  ---------------------------------------------------------------------";
  std::cout << std::endl;
  std::cout << "||"
	    << std::right << std::setw(2 * setwv) << "x original" << "|" 
	    << std::right << std::setw(2 * setwv) << "x noised" << "|"  
	    << std::right << std::setw(2 * setwv) << "noise"
	    << std::right << std::setw(5) << "||"
	    << std::right << std::setw(2 * setwv) << "y original" << "|"
	    << std::right << std::setw(2 * setwv) << "y noised" << "|" 
	    << std::right << std::setw(2 * setwv) << "noise" << "||";
  std::cout << std::endl;
  std::cout << "  ---------------------------------------------------------------------";
  std::cout << std::endl;
 
  // compute the points used to make the regression
  for(unsigned int angle = 0; angle < 360; angle += 12)
    {
      double const radian = 2 * 3.1415 * (double)angle / 360.0;
      double const x = expected.xcenter + expected.radius * cos(radian);
      double const y = expected.ycenter + expected.radius * sin(radian);
      double const xnoise = 2.0 * (double)(rand() % 200 - 100) / 100.0;
      double const ynoise = 2.0 * (double)(rand() % 200 - 100) / 100.0;
      xlist.push_back(x + xnoise);
      ylist.push_back(y + ynoise);
 
      // values
      std::cout << "||"
		<< std::right << std::setw(2 * setwv) << x<< "|" 
		<< std::right << std::setw(2 * setwv) << xlist.back() << "|" 
		<< std::right << std::setw(2 * setwv) << xnoise
		<< std::right << std::setw(5) << "||"
		<< std::right << std::setw(2 * setwv) << y << "|" 
		<< std::right << std::setw(2 * setwv) << ylist.back() << "|" 
		<< std::right << std::setw(2 * setwv) << ynoise << "||";
      std::cout << std::endl;
    }
 
  double rmse = 0.0;
 
  try
    {
      // regression of the points by a circle
      Matrix::circularRegression(xlist, ylist, computed.xcenter, computed.ycenter, computed.radius, rmse);
    }
  catch(std::exception const & error)
    {
      std::cout << error.what() << std::endl;
      return 0;
    }
  catch(...)
    {
      std::cout << "An undefined error has occured" << std::endl;
      return 0;
    }
 
  // display the informations about the computed circle 
  std::cout << "  ---------------------------------------------------------------------";
  std::cout << std::endl;
  std::cout << "||" << std::right << std::setw(2 * setwv) << "Computed circle : " << std::endl;
  std::cout << "||" << std::right << std::setw(4 * setwv) <<  "x center = " << computed.xcenter << std::endl;
  std::cout << "||" << std::right << std::setw(4 * setwv) <<  "y center = " << computed.ycenter << std::endl;
  std::cout << "||" << std::right << std::setw(4 * setwv) <<  "radius = " << computed.radius << std::endl;
  std::cout << "||" << std::right << std::setw(4 * setwv) <<  "rmse = " << rmse << std::endl;
  std::cout << "  ---------------------------------------------------------------------";
  std::cout << std::endl;
 
  return 0;
}
Le code ne requiert aucune autre dépendance que la Standart Template Library (native sur tous les compilateurs), pas de boost, opencv ou autres. Vous compilez sous gcc par la commande suivante :

g++ main.cpp matrix.cpp -o test

La sortie est du style :

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  ---------------------------------------------------------------------
||Expected circle :
||         x center = 20.5
||         y center = -3.7
||           radius = 8
  ---------------------------------------------------------------------
||x original|  x noised|     noise   ||y original|  y noised|     noise||
  ---------------------------------------------------------------------
||      28.5|     27.32|     -1.18   ||      -3.7|     -4.36|     -0.66||
||   28.3252|   29.0052|      0.68   ||  -2.03675|  -2.03675|         0||
||   27.8084|   29.1884|      1.38   || -0.446197| 0.0338029|      0.48||
||   26.9722|   26.5322|     -0.44   ||   1.00216|   2.16216|      1.16||
||   25.8532|   27.0932|      1.24   ||   2.24503|   1.52503|     -0.72||
||   24.5002|   24.6002|       0.1   ||   3.22808|   4.12808|       0.9||
||   22.9724|   22.5924|     -0.38   ||   3.90836|   2.44836|     -1.46||
||   21.3366|   22.5566|      1.22   ||   4.25614|   4.07614|     -0.18||
||   19.6642|   21.5642|       1.9   ||   4.25622|   5.09622|      0.84||
||   18.0283|   16.5683|     -1.46   ||   3.90859|   2.62859|     -1.28||
||   16.5004|   18.3204|      1.82   ||   3.22845|   1.30845|     -1.92||
||   15.1474|   15.1874|      0.04   ||   2.24552|   3.30552|      1.06||
||   14.0282|   13.8682|     -0.16   ||   1.00276|   2.64276|      1.64||
||   13.1919|   11.6119|     -1.58   ||  -0.44552|  -0.12552|      0.32||
||    12.675|    13.035|      0.36   ||  -2.03603|  -2.13603|      -0.1||
||      12.5|     11.44|     -1.06   ||  -3.69926|  -3.17926|      0.52||
||   12.6747|   14.0947|      1.42   ||  -5.36252|  -4.60252|      0.76||
||   13.1913|   12.5713|     -0.62   ||  -6.95313|  -6.71313|      0.24||
||   14.0273|   13.3673|     -0.66   ||  -8.40156|  -8.42156|     -0.02||
||   15.1463|   13.8463|      -1.3   ||  -9.64453|  -9.76453|     -0.12||
||   16.4991|   16.5591|      0.06   ||  -10.6277|  -12.4077|     -1.78||
||   18.0269|   18.4669|      0.44   ||  -11.3081|  -10.6481|      0.66||
||   19.6627|   19.1227|     -0.54   ||  -11.6561|  -12.3761|     -0.72||
||   21.3351|   22.1551|      0.82   ||  -11.6563|  -11.4363|      0.22||
||    22.971|    22.031|     -0.94   ||  -11.3088|  -11.9488|     -0.64||
||   24.4989|   25.4389|      0.94   ||  -10.6288|  -11.7488|     -1.12||
||   25.8521|   25.0921|     -0.76   ||  -9.64602|  -8.50602|      1.14||
||   26.9714|   25.7114|     -1.26   ||  -8.40336|  -9.22336|     -0.82||
||   27.8078|   28.2678|      0.46   ||  -6.95516|  -6.13516|      0.82||
||   28.3249|   28.9049|      0.58   ||   -5.3647|   -3.8047|      1.56||
  ---------------------------------------------------------------------
||Computed circle :
||         x center = 20.3688
||         y center = -3.76401
||           radius = 8.25942
||             rmse = 1.01834
  ---------------------------------------------------------------------
Bon je veux pas faire de la pub pour mon blog, mais je veux pas non plus ré-expliquer le fonctionnement du code. Le lien vers des explications détaillées est le suivant :

http://floriansella.free.fr/index.ph...y=ti&billid=92

Histoire de pas faire un truc creux, la régression d'un ensemble de points par un cercle (xc; yc; R) fonctionne comme la régression d'un ensemble de points par une droite. C'est une régression au sens des moindres carrés et on parvient à un système matriciel qu'il suffit de résoudre par la méthode basique de Gauss-Jordan (pivot de Gauss).

Flo.