worked on related work

tex/chapters/relatedwork.tex

@@ -2,23 +2,64 @@
 \label{sec:relatedWork}
 
 We consider indoor localization to be a time-sequential, non-linear and non-Gaussian state estimation problem. 
-Such problems are often solved by using Bayesian filters, which update the state estimation recursively
+Such problems are often solved using Bayesian filters, which update a state estimation recursively
 with every new incoming measurement. 
 A powerful method to obtain numerical results for this approach are particle filters.
 
-In context of indoor localisation, particle filter approximate a probability distribution describing the pedestrian's possible whereabouts by using a set of weighted random samples (particles).
+In context of indoor localization, particle filter approximate a probability distribution describing the pedestrian's possible whereabouts by using a set of weighted random samples (particles).
 Here, new particles are drawn according to some importance distribution, often represented by the state transition, which models the dynamics of the system.
+\todo{statt dynamics of the system vlt: the pedestrian's movement?}
 Those particles are then weighted by the state evaluation given different sensor measurements.
-A resampling step is deployed to prevent that only a small number of particles have a signifcant weight \cite{chen2003bayesian}.
-Most localisation approaches differ mainly in how the transition and evaluation steps are implemented and the available sensors are incorporated \cite{Fetzer-16, Ebner-16, Hilsenbeck2014}.
+A resampling step is deployed to prevent that only a small number of particles have a significant weight \cite{chen2003bayesian}.
+Most localization approaches differ mainly in how the transition and evaluation steps are implemented and the available sensors are incorporated \cite{Fetzer-16, Ebner-16, Hilsenbeck2014}.
+\todo{hier ist irgendwie ein harter cut zu dem nächsten satz}
 Additionally, within this paper we present a method, which is designed to run solely on a commercial smartphone.
 
-In its most basic form, the state transition is given by.. einfach distanz und heading.. intersection with walls usw.
+%In its most basic form, the state transition is given by.. einfach distanz und heading.. intersection with walls usw.
+
+%\todo{nochmal mit frank klären was wir jetzt GENAU machen.}
+
+The system's dynamics describe a pedestrian's potential movement within the building.
+This can be formulated as the question \emph{``Given the pedestrian's current position (and heading) are known,
+where could he be after a certain amount of time?''}.
+Obviously, the answer to this question depends on the pedestrian's walking behavior, any nearby architecture
+and thus the building's floorplan.
+
+Assuming the pedestrian to walk almost straight towards his current heading with a known, constant walking speed,
+the most basic form of state transition simply rejects all movements, where the line-of-sight
+between current position and potential destination is blocked by an obstacle. \cite{TODO, wir-2014? nurminen?}
+
+Despite its simplicity, this approach suffers from several drawbacks. The intersection-test can be costly,
+depending on the number of used particles and the complexity of the building. Furthermore, it is limited mainly
+to 2D transitions within the plane. Smooth 3D transitions, like walking stairs, would require much more complex
+intersection tests \cite{todo}.
+
+To overcome both limitations, the building's floorplan can be used to derive a graph-based structure,
+like voronoi diagrams or fixed-distance grids, 
+moving all costly intersection tests into a one-time offline phase \cite{wir, voronoi}.
+Hereafter, graph-based random walks along the created data-structure can be used as a fast transition approximation.
+When generating nodes and edges along stairs and elevators, this also allows for smooth transitions in 3D space \cite{todo, wir}.
+Furthermore, the nodes can be used to store additional information, like their distance towards a pedestrian's desired destination.
+Such information can be included during the transitions step, \eg{} increasing the likelihood of all potential movements that approach
+this destination \cite{wirFusion16}.
+
+However, the graph-based approach also imposes some potential issues. When using a gridded graph, the spacing between adjacent
+nodes directly represents the transition's accuracy. Likewise, the amount of required memory to represent the floorplan 
+scales about quadratically with this spacing. Even though nodes/edges are only created for actually walkable areas (like a sparse cube),
+large buildings require millions of nodes and might not fit into memory at once.
+Furthermore, (large) outdoor regions between adjacent buildings require unnecessarily large amounts
+of memory to be modeled \cite{todo}. While voronoi diagrams have the ability to mitigate this issue to some degree,
+they usually suffer from reduced accuracy for large open spaces, as many implementations only use the edges to estimate potential movements. 
+\todo{das mit dem voronoi issue, wo viele kleine facetten in offenen bereichen verwendet wurden}.
+
+We therefore present a novel technique based on continuous walks along a navigation mesh.
+Like the graph, the mesh, consisting of triangles sharing adjacent edges,
+is created once during an offline phase, based on the buildings 3D floorplan.
+Using large triangles reduces the memory footprint dramatically (a few megabytes for large buildings)
+while still increasing the quality (triangle-edges directly adhere to architectural-edges) and allows
+for truly continuous transitions along the surface spanned by all triangles. 
 
-\todo{nochmal mit frank klären was wir jetzt GENAU machen.}
 
-These disadvantages can be avoided by using spatial models
-like indoor graphs. Besonders geometric spatial models sind beliebt
 
 \todo{kurz auf voronoi eingehen mit neueren papern und dann auf grid basierte eingehen. schreiben das wir in previous work auch solche benutzt haben, aber das problem ist halt der gigantische speicheraufwand. deshalb haben wir uns für triangle based entscheiden, die erstellung ist einfacher, die verfahren sind aus der spieletheorie bekannt und erfolgreich im einatz. natürlich ist das ganze ein wenig rechenaufwendiger, da nun bla und blub gemacht werden muss, jedoch ist das laufen realisischer und nicht auf 45 grad winkel begrenzt. es wird also eine höhere genaugikeit erwartet, bei stark reduzierten speicher und zugrifssbedarf auf das netz.}