Gaussian Filtering Using a Spherical-Radial Double Exponential Cubature

Abstract

Gaussian filters use quadrature rules or cubature rules to recursively solve Gaussian-weighted integrals. Classical and contemporary methods use stable rules with a minimal number of cubature points to achieve the highest accuracy. Gaussian quadrature is widely believed to be optimal due to its polynomial degree of exactness and higher degree cubature methods often require complex optimization to solve moment equations. In this paper, Gaussian-weighted integrals and Gaussian filtering are approached using a double exponential (DE) transformation and the trapezoidal rule. The DE rule is principled in high rates of convergence for certain integrands and the DE transform ensures that the trapezoidal rule maximizes its performance. A novel spherical-radial cubature rule is derived for Gaussian-weighted integrals where it is shown to be perfectly stable and highly efficient. A new Gaussian filter is then built on top of this cubature rule. The filter is shown to be stable with bounded estimation error. The effect of varying the number of cubature points on filter stability and convergence is also examined. The advantages of the DE method over comparable Gaussian filters and their cubature methods are outlined. These advantages are realized in two numerical examples: a challenging non-polynomial integral and a benchmark filtering problem. The results show that simple and fundamental cubature methods can lead to great improvements in performance when applied correctly.