29 Palms Vehicle Detection (what we wanted to do)

Vehicle Tracking ? Event Pattern Regression Analysis ESTIMATES Vehicle speed = 17 MPH Entry time = 21:09: Update Pattern Vehicles trigger network events Events are matched to a dynamic pattern to determine vehicle parameters

One-dimensional problem Each node i has a name n i and a position p i along the road The nodes don’t know their positions exactly, but we do need some estimate p

Problem restrictions Assumption 1: There is only one vehicle in the network at a time This lets us separate events – deal with them independently So… an event is characterized by the time that the vehicle reaches each node in the network Assumption 2: The vehicle moves with a constant velocity v. This means that the relationship between time and position is linear, p = vt.

Linear formulation For each vehicle, there are two parameters: 1.t 0 – the time the vehicle passes through point p = 0 2.v – the speed of the vehicle (positive for increasing p) Each node contributes one equation, if we combine all nodes: t 1 = t 0 + (1/v)p 1 t 2 = t 0 + (1/v)p 2 … etc. This can be cast as the equation Ax = b with:

Least squares solution This overdetermined system can be solved using the least squares solution x = (A T A) -1 A T b Matrix inversion is only a 2x2! Akin to fitting a line through a series of points – here our slope is (1/v) and our t 0 is the y-intercept We can use as many equations (at least two) as we want – each node can poll its neighbours and use as much information as desired, allowing for missed readings Messages sent are: “Hi, I saw the vehicle at time t and I think that I’m at position p”. We don’t even care who you are.

Position update Once we calculate v, go back and make a new guess at each p i t i = (1/v)p i new + t 0 p i new = (t i -t 0 )v Update according to some non-catastrophic weighted rule like: Detect Vehicle (fix p i ’s) Update Positions (fix t 0, v) Make Initial guess For p i ’s Better Results As time progresses

Extensions? (Are our restrictions problematic for a more realistic scenario?) One vehicle at a time + constant velocity requirements can be satisfied providing we consider a small enough region of the network at a time The 1-D to 2-D extension is more difficult… matched against a single pattern, would require more patterns (perhaps one for vertical and one for horizontal travel at each node?)