\section{Introduction}
\label{sec:intro}

Sensor fusion approaches are often based upon probabilistic descriptions like particle filters, using samples to represent the distribution of a dynamical system. 
To update the system recursively in time, probabilistic sensor models process the noisy measurements and a state transition function provides the system's dynamics. 
Therefore a sample or particle is a representation of one possible system state, \eg{} the position of a pedestrian within a building.
The set of all particles represents the posterior of the system.
In most real world scenarios one is then interested in finding the most probable state within the state space, to provide the best estimate of the underlying problem, generally speaking, solving the state estimation problem.
In the discrete manner of a sample representation this is often done by providing a single value, also known as sample statistic, to serve as a \qq{best guess}.  

This value is often calculated by means of simple parametric point estimators, \eg{} the weighted-average over all samples or, the sample with the highest weight \cite{Fetzer2016OMC}.
While such point estimators are computational fast and suitable most of the time, it is not uncommon that they fail to recover the state in more complex scenarios.
Especially time-sequential, non-linear and non-Gaussian state spaces, depending upon a high number of different sensor types, frequently suffer from a multimodal representation of the posterior distribution.
For example, in a localization scenario where a bimodal distribution represents the current posterior, a reliable position estimation is more likely to be at one of the modes, instead of somewhere in-between, like provided by a simple weighted-average estimation.
Additionally, in most practical scenarios the sample size, and hence the resolution is limited, causing the variance of the sample based estimate to be high \cite{Verma2003}.

%da muss es doch noch andere methoden geben... verflixt und zugenäht... aber grundsätzlich ist ein weighted average doch ein point estimator? (https://www.statlect.com/fundamentals-of-statistics/point-estimation)
%Für related work brauchen wir hier definitiv quellen. einige berechnen ja auch https://en.wikipedia.org/wiki/Sample_mean_and_covariance oder nehmen eine gewisse verteilung für die sample menge and und berechnen dort die parameter

It is obvious, that a computation of the full posterior could solve the above, but finding such an analytical solution is an intractable problem, which is the reason for applying a sample representation in the first place.
A feasible alternative is to estimate the parameters of a specific parametric model based on the sample set, assuming that the unknown distribution is approximately a parametric distribution, \eg{} like the normal distribution.
Given the estimated parameters the most probable state can be obtained from the parameterized density function.
%In the case of multi-modalities several parametric distributions can be combined into a mixture distribution.
However, parametric models fail when the assumption does not fit the underlying model.
For many real world problems assuming a parametric distribution is too limiting as the posterior is changing over time.
As a result, those techniques are not able to provide an accurate statement about the most probable state, rather causing misleading or false outcomes. 

Another promising way is to recover the probability density function from the sample set itself, by using a non-parametric estimator like a kernel density estimation (KDE). 
With this, the \qq{real} most probable state is given by the maximum of the density estimation and thus avoids the aforementioned drawbacks.
However, non-parametric estimators tend to consume a large amount of computation time, which renders them unpractical for real time scenarios.
Nevertheless, the availability of a fast processing non-parametric density estimate might improve the accuracy of today's sensor fusion systems without sacrificing their real time capability.

%Therefore, this paper presents a novel approximation approach for rapid computation of the KDE. 
%In this paper, a well known approximation of the Gaussian filter is used to speed up the computation of the KDE. 
In this paper, a novel approximation approach for rapid computation of the KDE is presented.
The basic idea is to interpret the density estimation problem as a filtering operation.
We show that computing the KDE with a Gaussian kernel on binned data is equal to applying a Gaussian filter on the binned data.
This allows us to use a well known approximation scheme based on multiple recursions of a box filter, which yields an approximative Gaussian filter given by the central limit theorem \cite{kovesi2010fast}.


This process converges quite fast to a reasonable close approximation of the ideal Gaussian.
In addition, a box filter can be computed extremely fast by a computer, due to its intrinsic simplicity.
While the idea to use several box filter passes to approximate a Gaussian has been around for a long time, the application to obtain a fast KDE is new.
Especially in time critical and time sequential sensor fusion scenarios, the here presented approach  outperforms other state of the art solutions, due to a fully linear complexity and a negligible overhead, even for small sample sets.
In addition, it requires only a few elementary operations and is highly parallelizable.


%\todo{Mehrdimensionen mit aufnehmen. das das abgedeckt ist! }

%linear complexity and easy parall
%ist immer gleich schnell. 
%andere rießen daten, wir weniger daten. 
%low complexity, only requires a few elementar operations 
%produces nearly no overhead. 


% time sequential, fixed computation time, pre binned data!!

% KDE wellknown nonparametic estimation method
% Flexibility is paid with slow speed
% Finding optimal bandwidth
% Expensive computation
