Nonlinear Multidimensional Bayesian Estimation with Fourier Densities

Abstract

Abstract-Efficiently implementing nonlinear Bayesian estimators is still an unsolved problem, especially for the multidimensional case. A trade-off between estimation quality and demand on computational resources has to be found. Using multidimensional Fourier series as representation for probability density functions, so called Fourier densities, is proposed. To ensure non-negativity, the approximation is performed indirectly via Ψ-densities, of which the absolute square represent the Fourier density. It is shown that Ψ-densities can be determined using the efficient fast Fourier transform algorithm and their coefficients have an ordering with respect to the Hellinger metric. Furthermore, the multidimensional Bayesian estimator based on Fourier densities is derived in closed form. That allows an efficient realization of the Bayesian estimator where the demands on computational resources are adjustable.

Fourier density approximation for belief propagation in wireless sensor networks

“…[5] and [6] derived the basic operations using Fourier density approximation. Here some important equations related to BP are briefly described.…”

“…Recently, [5] and [6] ensured the non-negativity of Fourier series by approximating the square root of the density instead of the density itself. The usage of Fourier series in nonlinear Bayesian filtering is also derived in [5] and [6]. Using Fourier density approximation, the belief can be represented sufficiently by only a small number of Fourier coefficients.…”

“…Fourier series were first employed to estimate probability densities in [4]. Recently, [5] and [6] ensured the non-negativity of Fourier series by approximating the square root of the density instead of the density itself. The usage of Fourier series in nonlinear Bayesian filtering is also derived in [5] and [6].…”

Density trees for efficient nonlinear state estimation

“…The partitioning from one layer to the next is similar to n-dimensional Octrees [6] or B-trees [7]. The container structure allows for the use of different density representation forms in the tree nodes, from very simple forms, such as Dirac impulses or uniform distributions, to very elaborate and complex ones, such as Gaussian mixture densities [8] or Fourier densities [9], even mixed in the same tree. This offers a compromise between the number of tree nodes and the number of parameters of individual node functions.…”

Direct fusion of Dirac mixture densities using an efficient approximation in joint state space

“…Unfortunately, the resulting Gaussian densities are not capable of representing arbitrary density functions that may appear in estimation problems of arbitrary nonlinear systems. Other density representations include Gaussian mixture densities [7], gridbased approaches [8], simple moments of probability density functions [9], exponential densities [3], fourier series [10], [11], the representation by means of sample sets [12], or Dirac mixture densities [13], which are capable of representing more general densities.…”