diff --git a/tex/chapters/introduction.tex b/tex/chapters/introduction.tex index 9440699..4058981 100644 --- a/tex/chapters/introduction.tex +++ b/tex/chapters/introduction.tex @@ -1,54 +1,41 @@ \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 noise measurements and a state transition function provides the system's dynamics. +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 calculating a single value, also known as sample statistic, to serve as a "best guess". -This values is often calculated by means of simple parametric point estimators, e.g. using weighted-average of all samples or that one sample with the highest overall weight \cite{}. +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 -%multimodalities... +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. +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 .. +} -%interested in the most proper state within the state space of the dynamic system -%echte antwort computationel complex deswegen %weighted-average -> problem multimodal; sample mit höhsten wert -> springt viel rum -%-> Density -> KDE - -%Egal auf welchem Weg das sample set entstanden ist, am ende muss ein verwertbarer wert rauskommen. irgendein - -After calculating - - - -In real world scenarios - - -%find the state that describs our probleme the best -% - -% ... in many real world scenarios an estimate of the problem state is required e.g. the position of a pedestrian within a building... -%this is often done by calculating the weighted-average of all samples or - -%however multimodalities. - -% in the optimal case - -bessere entscheidung kde raus machen, als einfach nur - -to receive this information - -based upon a set of descrete samples - -%for this purpose parameteric estimators like ... are often used in real time scenarios because of their low complexity and short computatinal time. - -% however, -non parameteric estimators like kde +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. -\cite{Deinzer01-CIV} + + + % KDE wellknown nonparametic estimation method % Flexibility is paid with slow speed % Finding optimal bandwidth diff --git a/tex/egbib.bib b/tex/egbib.bib index f35ada7..938d8f5 100644 --- a/tex/egbib.bib +++ b/tex/egbib.bib @@ -2880,3 +2880,14 @@ year = {2003} year={2017, submitted}, } +@inproceedings{Verma2003, +author = {Verma, Vandi and Thrun, Sebastian and Simmons, Reid}, +doi = {10.1.1.68.4380}, +booktitle={Proc. of the International Joint Conference on Artificial Intelligence (IJCAI)}, +pages = {976--984}, +title = {{Variable resolution particle filter}}, +year = {2003} +} + + +