Fixed FE 1

tex/chapters/introduction.tex

@@ -3,10 +3,10 @@
 
 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. 
+Therefore a sample or particle is a representation of one possible system state, \eg{} 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, 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 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{Fetzer2016OMC}. 
+This value is then calculated by means of simple parametric point estimators, \eg{} the weighted-average over all samples, the sample with the highest weight or by assuming other parametric statistics like normal distributions \cite{Fetzer2016OMC}. 
 %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
 
@@ -19,14 +19,14 @@ Additionally, in most practical scenarios the sample size and therefore the reso
 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. 
 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 maxima of the density estimation and thus avoids the aforementioned drawbacks.
-However, non-parametric estimators tend to consume a large amount of computational time, which renders them unpractical for real time scenarios.
+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 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 estimation problem as a filtering operation.
-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.
+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 for Gaussian filters: the box filter.
 By the central limit theorem, multiple recursion of a box filter yields an approximative Gaussian filter \cite{kovesi2010fast}.