29 Palms Vehicle Detection

1. 29 Palms Vehicle Detection (what we wanted to do)

2. 1 0 2 5 3 7 4 9 5 10 1 10 1 0.1 2 16 2 5.2 Update 3 19 3 6.9 Pattern 4 ? 4 9 5 23 5 9.8 Vehicle Tracking Vehicles trigger network events Pattern ESTIMATES Events are matched to a dynamic pattern to determine vehicle parameters Vehicle speed = 17 MPH Entry time = 21:09:04 Regression Analysis Event

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

4. 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.

5. Linear formulation • For each vehicle, there are two parameters: • t0 – the time the vehicle passes through point p = 0 • v – the speed of the vehicle (positive for increasing p) • Each node contributes one equation, if we combine all nodes: • t1 = t0 + (1/v)p1 • t2 = t0 + (1/v)p2… etc. • This can be cast as the equation Ax = b with:

6. Least squares solution This overdetermined system can be solved using the least squares solution x = (ATA)-1ATb Matrix inversion is only a 2x2! Akin to fitting a line through a series of points – here our slope is (1/v) and our t0 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.

7. Position update Once we calculate v, go back and make a new guess at each pi ti = (1/v)pinew + t0 pinew = (ti-t0)v Update according to some non-catastrophic weighted rule like: Better Results As time progresses Make Initial guess For pi’s Detect Vehicle (fix pi’s) Update Positions (fix t0, v)

8. 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?)