\section{Introduction}

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, e.g. the position of a pedestrian within a building. 
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.  
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 "best guess". 
This value is then calculated by means of simple parametric point estimators, e.g. the weighted-average over all samples, the sample with the highest weight or by assuming other parametric statistics like normal distributions \cite{}. 
%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

While such methods 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.  
As a result, those techniques are not able to provide an accurate statement about the most probable state, rather causing misleading or false outcomes. 
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.
\commentByMarkus{Vlt. noch drauf eingehen, dass avg. eben in die Mitte geht?}  
Additionally, in most practical scenarios the sample size and therefore the resolution is limited, causing the variance of the sample based estimate to be high \cite{Verma2003}.

It is obvious, that a computation of the full posterior could solve the above, but finding such an analytical solution is an intractable problem, what is the reason for applying a sample representation in the first place. 
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, it is easy to find the "real" most probable state and thus to avoid the aforementioned drawbacks.
However, non-parametric estimators tend to consume a large amount of computational time, which renders them unpractical for real time scenarios.
Nevertheless, the availability of a fast processing density estimate might improve the accuracy of today's sensor fusion systems without sacrificing their real time capability.

\commentByToni{Der nachfolgende Satz ist ziemlich wichtig. Find ich aktuell noch nicht gut. Allgemein sollte ihr jetzt noch ca eine viertel Seite ein wenig die Methode grob beschrieben werden.
The basic idea ...
We formalize this ...
Our experiments support our ..
}

In this paper, a novel approximation approach for rapid computation of the KDE is presented.
%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. 

 




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