Hi all.
I'm trying to understand the unscented transformation with respect to the unscendted kalman filter. I think I have understood the basic idea, but I have some troubles with the definition of the sigma points.
I quote from "Unscented Filtering and Nonlinear Estimation" by Julier and Uhlmann: "A set of sigma points are chosen so that their mean and covariance are x_bar and Sigma_x . The nonlinear function is applied to each point, in turn, to yield a cloud of transformed points. The statistics of the transformed points can then be calculated to form an estimate of the nonlinearly transformed mean and covariance"
They then give an example of a symmetric set of sigma points that lie on the sqrt(N)'th covariance contour. The sigma points are then given by rows of the square root matrix of N*Sigma_x for example calculated from the Cholesky decomposition.
My questions are: What do they mean by the sqrt(N)'th contour? Why do they choose the sqrt(N)'th covariance contour?
Do they mean the set of points where the mahalanobis distance of x equals sqrt(N)? In other words where (x - x_bar)^T * Sigma_x¨^-1 * (x - x_bar) = sqrt(n)? If this is the case, why is this set of points given by the square root matrix?
Thanks in advance.