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Covariance Ellipsoids

Description: Algorithms for 2d and 3d covariance ellipsoids, typically used in slam.

Keywords: ecl statistics

Tutorial Level: INTERMEDIATE

Overview

Implemented by the template class CovarianceEllipsoid - this takes two template parameters. The first is the float type used (usually float/double) and the second, the dimension of the ellipsoid to be determined.

Specialisations are enabled for 2 and 3 dimensional types only as we haven't had a need to develop a general implementation.

Typedefs

Typedefs exist for the template class in the eigen style:

   1 using ecl::CovarianceEllipsoid2f;
   2 using ecl::CovarianceEllipsoid2d;
   3 using ecl::CovarianceEllipsoid3f;
   4 using ecl::CovarianceEllipsoid3d;

Usage

You typically feed the covariance ellipsoid with a positive definite symmetric matrix (eigen matrix). Take care entering the positive definite symmetric matrix - the class does not yet do checking to make sure it is positive definite symmetric.

You can either input the matrix via the constructor - in which case it will automatically calculate the ellipsoid properties, or pass it in later via the compute() method:

   1 Matrix2d M;
   2 M << 3.0, 1.0, 1.0, 5.0; // must be a positive definite, symmetric matrix
   3 
   4 CovarianceEllipsoid2d ellipse(M);
   5 
   6 // OR
   7 
   8 CovarianceEllipsoid2d ellipse;
   9 ellipse.compute(M);

   1 Matrix2d M;
   2 M << 3.0, 1.0, 1.0, 5.0;
   3 CovarianceEllipsoid2d ellipse(M);
   4 
   5 // ellipse minor and major axis lengths
   6 const Vector2d& lengths = ellipse.lengths(); 
   7 double eigen_value_0 = lengths[0]*lengths[0];
   8 
   9 // angle between x-axis and major axis of the ellipse
  10 double angle = ellipse.rotation(); 
  11 
  12 // values of the intercepts of the ellipse with x and y axes
  13 const Vector2d& intercepts = ellipse.intercepts(); 
  14 
  15 // the ellipse axes/covariance eigenvectors as column vectors in a matrix
  16 const Matrix2d& axes = ellipse.axes(); 

The 3d version is similar, but will only calculate intercepts and axes respectively. Note that the axes are sorted (to ensure a right-handed co-ordinate system) and normalised.

   1 Matrix3d P;
   2 double sigmaX(0.1);
   3 double sigmaY(0.5);
   4 double sigmaT(0.3);
   5 P << sigmaX*sigmaX, 0, 0, 0, sigmaY*sigmaY, 0, 0, 0, sigmaT*sigmaT;
   6 CovarianceEllipsoid3d ellipse(P);
   7 
   8 const Vector3d& lengths = ellipse.lengths();
   9 double eigen_value_0 = lengths[0]*lengths[0];
  10 const Matrix3d& axes = ellipse.axes();


2020-01-25 12:37