Inference of Non-Overlapping Inference of Non-Overlapping

Camera Network Topology by Camera Network Topology by Measuring Statistical Measuring Statistical

DependenceDependence

Date ： 2009.01.21

MotivationWith 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.

Non-overlapping FOVsMore practical in the real wordBut 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

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.

View from Another AngleTo 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.

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)

First Consider a simple case…

What feature can we use?Object occupy little pixels Appearance may

failsSpace relationship btw cameras unknown

Space may failsAha! Time may be a good feature to use!

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

Problem FormulationNow, given departure and arrival time

observations X and Y, they are connected by the transition time T.

T

Main HypothesisIf 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

Problem FormulationHow do we measure the dependency ?By Mutual Information !

write in terms of entropy…

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

Problem FormulationWe 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)

Problem FormulationSo 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

Problem FormulationOkay, 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.

Markov Chain Monte CarloWe use the most general MCMC algorithm

Metropolis-Hastings Sampler

Markov Chain Monte CarloThe 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.

Missing MatchesBut 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

Results

Results

Results