SO3vonMisesFisherKernel

von Mises Fisher kernel on the orientation space

The von Mises Fisher kernel \(\psi_{\kappa}\in L^2(\mathcal{SO}(3))\) is a nonnegative function depending on a parameter \(\kappa>0\) and is defined by its Chebyshev series

\[ \psi_{\kappa}(t) = \sum\limits_{n=0}^{\infty} \frac{\mathcal{I}_n(\kappa}-\mathcal{I}_{n+1}(\kappa)} {\mathcal{I}_0(\kappa)-\mathcal{I}_1(\kappa)} \, \mathcal U_{2n}(t)\]

or directly by

\[ \psi_{\kappa}(\cos\frac{\omega®}2) = \frac1{\mathcal{I}_0(\kappa)-\mathcal{I}_1(\kappa)} \, \mathrm{e}^{\kappa \cos\omega®}\]

while \(\mathcal I_n,\,n\in\mathbb N_0\) denotes the the modified Bessel functions of first kind

\[ \mathcal I_n (\kappa) = \frac1{\pi} \int_0^{\pi} \mathrm e^{\kappa \, \cos \omega} \, \cos n\omega \, \mathrm d\omega \].

Syntax

psi = SO3vonMisesFisherKernel(100)
psi = SO3vonMisesFisherKernel('halfwidth',5*degree)

Input

Output