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Inference of Non-Overlapping Camera Network Topology by Measuring Statistical Dependence

Inference of Non-Overlapping Camera Network Topology by Measuring Statistical Dependence. Date ： 2009.01.21. Motivation . With the price of camera devices getting cheaper, the wide-area surveillance system is becoming a trend for the future.

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Inference of Non-Overlapping Camera Network Topology by Measuring Statistical Dependence

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  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 • 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 T

  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 ((xi, yπ(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 ((xi, yπ(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 xi may not have corresponding yπ(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

  21. Results

  22. Results

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