deletions | additions       diff --git a/Methodolofy Measurement Definitions.tex b/Methodolofy Measurement Definitions.tex  index 0ff8de4..465c4e4 100644  --- a/Methodolofy Measurement Definitions.tex  +++ b/Methodolofy Measurement Definitions.tex 
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An interaction quantifies the spatial relationship between moving objects in a scene, as is depicted in Figure~\ref{fig:conflict-video}. At the most fundamental level, an interaction is defined as a pair of moving objects simultaneously present in a scene over a common time interval (also referred to as a user pair). We further define an instantaneous observation (i.e. in a given video frame) within this time interval as an interaction instant.  This interaction definition is generic, if not naive, as the quality depends largely on how the scene is constructed. For example, the significance of an interaction between two vehicles separated from each other physically (e.g. via a median or alarge  building) may not be comparable to an interaction between two vehicles merely separated by a painted line lane markings  becausethe probability that one  of the vehicles comes into contact with the other vehicle is reduced in implication that  the case of cross  the median. lane marking intentionally or inadvertantly very easily.  This may interfere with collision prediction attempts, particularly if scenes are not consistently selected and geometry is not controlled. One solution is to perform a triage of user pairs based on physical access and proximity. A network topology coupled with a driving distance horizon is proposed. This is not a perfect solution, however, as physical access isn't may not  necessarily be  a binary option. In our median example, discrete choice. For the example of the median,  it is still physically possible, although much  less likely, for vehicles a vehicle  to cross-over cross over  into an opposing lane and cause a collision, although this is something that could be modelled. oncoming traffic.  \subsubsection{Motion Prediction}  \label{motion-prediction}  While vehicle trajectories offer a rich set of observed behavioural data, they do not provide much collision data; this is by design of the proactive road safety approach: predicting collisions should be performed without observing them directly. In order to study collisions, they need to be extrapolated from traffic events with potential for collision. This potential is modelled by predicting future possible  positions between each pair  of vehicles road users  at every instant in time and examining i) situations of particular probability of collision (i.e. threshold) or ii) evolution of the probability of collision over a time series. time.  Several motion prediction models are proposed for study \cite{Mohamed_2013}: study:  \begin{itemize}  \item \textbf{Constant velocity} is the classic motion prediction model, wherein vehicles are projected along straight paths at a constant speed and heading using the velocity vector at that moment in time. This model is the simplest but also makes the most assumptions: only one movement is predicted at every instant, both users do not enter evasive action in the event of a collision course, and the natural (non-reacting) motion of a moving object is a straight path (not always true). These assumptions may be adequate for specific applications of the methodology, e.g. highways \cite{St_Aubin_2013}. The current implementation is based off of \cite{Laureshyn_2010}. 
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where $f$ is the number of frames per second of the video.  \item \textbf{Motion patterns} are a family of models which use machine learning to calculate future position likelihoods from past behaviour \cite{saunier07probabilistic,morris08survey}. \cite{saunier07probabilistic,morris08survey,Sivaraman_2013}.  This type of model is the most promising as motion prediction is probabilistic in nature and inherently models naturalistic behaviour. However, they may not be able to model erratic behaviour such as roadway departures. Motion patterns are also complex to implement and expensive to process. The type of motion pattern being studied for implementation is a discretized motion pattern \cite{St_Aubin_2014}. \end{itemize}  As illustrated in Figure~\ref{fig:prob-collision-space}, motion prediction is performed for each user pair over each instant $t_0$ for a number of time steps of size $\Delta t$ between $t_0$ and some chosen timehorizon. Each motion prediction may generate for two road users a series or a matrix of collision points with a sum of probabilities inferior or equal to 1. This %This  is significantly larger and more difficult to handle than the  trajectory data, and currently cannot be performed in real time.