54 lines
5.1 KiB
TeX
54 lines
5.1 KiB
TeX
\section{Introduction}
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Sensor fusion approaches are often based upon probabilistic descriptions like particle filters, using samples to represent the distribution of a dynamical system.
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To update the system recursively in time, probabilistic sensor models process the noisy measurements and a state transition function provides the system's dynamics.
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Therefore a sample or particle is a representation of one possible system state, e.g. the position of a pedestrian within a building.
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In most real world scenarios one is then interested in finding the most probable state within the state space, to provide the \qq{best estimate} of the underlying problem.
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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}.
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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{}.
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%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)
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%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
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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.
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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.
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As a result, those techniques are not able to provide an accurate statement about the most probable state, rather causing misleading or false outcomes.
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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.
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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}.
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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.
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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).
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With this, it is easy to find the \qq{real} most probable state and thus to avoid the aforementioned drawbacks.
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However, non-parametric estimators tend to consume a large amount of computational time, which renders them unpractical for real time scenarios.
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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.
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%Therefore, this paper presents a novel approximation approach for rapid computation of the KDE.
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%In this paper, a well known approximation of the Gaussian filter is used to speed up the computation of the KDE.
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In this paper, a novel approximation approach for rapid computation of the KDE is presented.
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The basic idea is to interpret the estimation problem as a filtering operation.
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We show that computing the KDE with a Gaussian kernel on pre-binned data is equal to applying a Gaussian filter on the binned data.
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This allows us to use a well known approximation scheme for Gaussian filters: the box filter.
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By the central limit theorem, multiple recursion of a box filter yields an approximative Gaussian filter \cite{kovesi2010fast}.
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This process converges quite fast to a reasonable close approximation of the ideal Gaussian.
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In addition, a box filter can be computed extremely fast by a computer, due to its intrinsic simplicity.
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While the idea to use several box filter passes to approximate a Gaussian has been around for a long, the application to obtain a fast KDE is new.
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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 \landau{N} and a negligible overhead, even for small sample sets.
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In addition, it requires only a few elementary operations and is highly parallelizable.
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%\todo{Mehrdimensionen mit aufnehmen. das das abgedeckt ist! }
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%linear complexity and easy parall
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%ist immer gleich schnell.
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%andere rießen daten, wir weniger daten.
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%low complexity, only requires a few elementar operations
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%produces nearly no overhead.
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% time sequential, fixed computation time, pre binned data!!
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% KDE wellknown nonparametic estimation method
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% Flexibility is paid with slow speed
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% Finding optimal bandwidth
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% Expensive computation
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