Fits a conic to a set of points using linear regression. More...

#include <vector>
#include <iostream>
#include <vnl/vnl_matrix_fixed.h>
#include <vgl/vgl_point_2d.h>
#include <vgl/vgl_homg_point_2d.h>
#include <vgl/vgl_conic.h>
#include <vgl/algo/vgl_norm_trans_2d.h>

Classes
class	vgl_conic_2d_regression< T >

Macros
#define	VGL_CONIC_2D_REGRESSION_INSTANTIATE(T) extern "please include vgl/algo/vgl_conic_2d_regression.hxx first"

Detailed Description

Fits a conic to a set of points using linear regression.

Author: J.L. Mundy

Date: June 17, 2005

Since conic fitting is rather ill-conditioned it is necessary to normalize the point coordinates so that they have equal standard deviations with respect to the center (mean) of the pointset. The points are transformed to have zero mean. The resulting conic is transformed back to the original frame for computing the Sampson approximation to fitting error.

The regression uses the Bookstein algorithm which constrains the conic norm, $ |ax^2 + bxy + cy^2 + dx + ey +f|^2 $ , so that $ a^2 + 0.5*b^2 + c^2 = 2 $ With this normalization, the resulting fit is invariant up to a similarity transform of the pointset. The solution is formulated as an eigenvalue problem as follows:

The scatter matrix S is decomposed as

$ S   =  \begin{array}{cc}     S_{11} & S_{12}
                        \\     S_{21} & S_{22}     \end{array}$
note \f$S_{21} = S_{12}^t\f$

The conic coefficients are \f$v_1 = \{a,b,c\}^t\f$ and \f$v_2 = \{d,e,f\}^t\f$
The Bookstein constraint is expressed by the diagonal matrix
         D = Diag{1, 0.5, 1, 0, 0, 0}
The Lagrangian to be minimized is
$ L =  v_1^t S_{11} v_1  + 2 v_2^t S_{21} v_1  + v_2^t S_{22} v_2
       - \lambda(v_1^t D v_1 -2) $.

Minimizing with respect to v2 gives
\f$ dL/dv_2 = 2 S_{21} v_1 + 2 S_{22} v_2 = 0 \f$

So,  \f$ v_2 = -S_{22}^-1 S_{21} v_1 \f$.

Substituting for v2 in L,

$ L = v_1^t ( S_{11} - S_{12} * S_{22}^-1 * S_{21}) v_1 - \lambda(v_1^t D v_1 -2) = v_1^t ( (S_{11} - S_{12} * S_{22}^-1 * S_{21}) - \lambda D ) v_1 - 2 \lambda $,

$dL/dv_1 = ( (S_{11} - S_{12} * S_{22}^-1 * S_{21}) - \lambda D ) v_1 = 0$ .

So, $\lambda v_1 = D^-1 (S_{11} - S_{12} * S_{22}^-1 * S_{21}) v_1$ .

This eigenvalue problem is solved using singular value decomposition.

  Modifications
   none

Definition in file vgl_conic_2d_regression.h.

Macro Definition Documentation

◆ VGL_CONIC_2D_REGRESSION_INSTANTIATE

#define VGL_CONIC_2D_REGRESSION_INSTANTIATE ( T ) extern "please include vgl/algo/vgl_conic_2d_regression.hxx first"

Definition at line 149 of file vgl_conic_2d_regression.h.

Classes

Macros

Detailed Description

Macro Definition Documentation

◆ VGL_CONIC_2D_REGRESSION_INSTANTIATE