Usage

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
+% 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.
+
+