From a5fc1628e625009ca656e087cd334b33993780d6 Mon Sep 17 00:00:00 2001 From: Markus Bullmann Date: Wed, 14 Feb 2018 14:07:27 +0100 Subject: [PATCH] Usage --- tex/chapters/usage.tex | 41 ++++++++++++++++++++++++++++++++++++++++- 1 file changed, 40 insertions(+), 1 deletion(-) diff --git a/tex/chapters/usage.tex b/tex/chapters/usage.tex index edce483..24b8e3b 100644 --- a/tex/chapters/usage.tex +++ b/tex/chapters/usage.tex @@ -7,4 +7,43 @@ % Anschließend Filterung per Box Filter über das Histogram % - Wenn möglich parallel (SIMD, GPU) % - separiert in jeder dim einzeln -% Maximum aus Filter ergebnis nehmen \ No newline at end of file +% Maximum aus Filter ergebnis nehmen + +The objective of our method is to allow reliable recover the most probable state from a time-sequential Monte Carlo sensor fusion system. +Assuming a sample based representation, our method allows to estimate the density of the unknown distribution of the state space in a narrow time frame. +Such systems are often used to obtain an estimation of the most probable state in near real time. +As the density estimation poses only a single step in the whole process, its computation needs to be as fast as possible. +% not taking to much time from the frame + +%Consider a set of two-dimensional samples, presumably generated from e.g. particle filter system. +Assuming that the generated samples are often stored in a sequential list, the first step is to create a grid representation. +In order to efficiently compute the grid and to allocate the required memory the extrema of the samples need to be known in advance. +These limits might be given by the application, for example, the position of a pedestrian within a building is limited by the physical dimensions of the building. +Such knowledge should be integrated into the system to avoid a linear search over the sample set, naturally reducing the computation time. + +The second parameter to be defined by the application is the size of the grid, which can be set directly or defined in terms of bin sizes. +As the number of grid points directly affects both computation time and accuracy, a suitable grid should be as coarse as possible but at the same time narrow enough to produce an estimate sufficiently fast with an acceptable approximation error. + +Given the extreme values of the samples and the number of grid points $G$, the computation of the grid has a linear complexity of \landau{N} where $N$ is the number of samples. +If the extreme values are unknown, an additional $\landau{N}$ search is required. +The grid is stored as an linear array in memory, thus its space complexity is $\landau{G}$. + +Next, the binned data is filtered with a Gaussian using the box filter approximation. +The box filter width is derived from the standard deviation of the approximated Gaussian, which is in turn equal to the bandwidth of the KDE. +However, the bandwidth $h$ needs to be scaled according to the grid size. +This is necessary as $h$ is defined in the input space of the KDE, i.e. in relation to the sample data. +In contrast, the bandwidth of a BKDE is defined in the context of the binned data, which differs from the unbinned data due to the discretisation of the samples. +For this reason, $h$ needs to be divided by the bin size to account the discrepancy between the different sampling spaces. + +Given the scaled bandwidth the required box filter width can be computed. % as in \eqref{label} +Due to its best runtime performance the recursive box filter implementation is used. +If multivariate data is processed, the algorithm is easily extended due to its separability. +Each filter pass is computed in $\landau{G}$ operations, however an additional memory buffer is required. + +While the integer-sized box filter requires fewest operations, it causes a larger approximation error due to rounding errors. +Depending on the required accuracy the extended box filter algorithm can further improve the estimation results, with only a small additional overhead. +Due to its simple indexing scheme, the recursive box filter can easily be computed in parallel using SIMD operations or parallel computation cores. + +Finally, the most likely state can be obtained from the filtered data, i.e. from the estimated discrete density, by searching filtered data for its maximum value. + +