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29 Palms Vehicle Detection

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

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29 Palms Vehicle Detection

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  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?)

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