1. Inference of Non-Overlapping Camera Network Topology by Measuring Statistical Dependence Date ： 2009.01.21

2. Motivation With the price of camera devices getting cheaper, the wide-area surveillance system is becoming a trend for the future. For the purpose of achieving the wide-area surveillance, there is a big problem we need to take care : non-overlapping FOVs.

3. Non-overlapping FOVs More practical in the real word But we begin to bump into many problems… - Correspondence between cameras, but the difference view angles of camera may enhance the difficulty - Hard to do the calibration, so we may not have the information of relative positions and orientations between cameras

4. Correspondence btw Cameras Means we have to indentify the same object in different cameras. Usually by space-time and appearance feature. But actually in wide-area surveillance, because cameras may be widely separated and objects may occupy only a few pixels, so is difficult to solve.

5. View from Another Angle To link the objects across no-overlapping FOVs, we need to know the connectivity of movement between FOVs. Turn into the problem of finding the topology of the camera network.

6. What’s our goal now? Want to determine the network structure relating cameras, and the typical transitions between cameras, based on noisy observations of moving objects in the cameras. Departure and arrival locations in each camera view are nodes in the network. An arc between a departure node and an arrival node denotes connectivity (like transition)

7. First Consider a simple case…

8. What feature can we use? Object occupy little pixels  Appearance may fails Space relationship btw cameras unknown  Space may fails Aha! Time may be a good feature to use!

9. The model T Departure and arrival locations in each camera view are nodes in the network. An arc between a departure node and an arrival node denotes connectivity  Transition Time Distribution

10. Problem Formulation Now, given departure and arrival time observations X and Y, they are connected by the transition time T. T

11. Main Hypothesis If camera are connected, the arrival time Y might be easily predict from departure time X.  There is a regularity between X and Y  Given the correspondence, the transition time distribution is highly structured  Dependence between X and Y is strong

12. Problem Formulation How do we measure the dependency ? By Mutual Information ! write in terms of entropy…

13. Problem Formulation Since from the graphical model, we have Y = T(X)  Y = X + T (assumed X indept. with T) Therefore relate h(Y|X) to the entropy of T h(Y|X) = h(X+T|X) = h(T|X) = h(T) T

14. Problem Formulation We get I(X;Y) = h(Y) – h(T)  maximizing statistical dependence is the same as minimizing the entropy of the distribution of transformation T Distribution of T is decided by the matching π between X and Y ! (π is a permutation for correspondence btw X and Y)

15. Problem Formulation So what we want to do now is trying to find the matching ((x i, y π(i) )) whose transition time distribution have lowest entropy  maximum dependence But how to compute the entropy?  by Parzen density estimater

16. Problem Formulation Okay, but how to find the matching ((x i, y π(i) ))?  It’s a NP hard problem… Look for approximation algorithms  Markov Chain Monte Carlo (MCMC) Briefly, MCMC is a way to draw samples from the posterior distribution of matchings given the data.

17. Markov Chain Monte Carlo We use the most general MCMC algorithm  Metropolis-Hastings Sampler

18. Markov Chain Monte Carlo The key to the efficiency of an MCMC algorithm is the choice of proposal distribution q(.). Here we use 3 types of proposals for sampling matches: 1. Add 2. Delete 3. Swap The new sample is accepted with probability proportional to the relative likelihood of the new sample vs. the current one. The likelihood of a correspondence is proportional to the log probability of the corresponding transformations.

19. Missing Matches But actually not all the observations in X and Y will be matched, but contain missing matches. (some x i may not have corresponding y π(i) ) Consider missing data as outliers, and model the distribution of transformations as a mixture of the true and outlier distributions. Usually use a uniform outlier distribution

20. Results