1. Tracking Multiple Cells By Correspondence Resolution In A Sequential Bayesian Framework Nilanjan Ray Gang Dong Scott T. Acton C.L. Brown Department of Electrical and Computer Engineering University of Virginia Charlottesville, Virginia, US

2. Presentation Overview  The problem  The framework to solve the problem  Track mapping  Simultaneous Tracking and Detection

3. The Problem Can we estimate these paths (the mapping f t ) in a sequential Bayesian framework? Frame t Frame t-1 Frame t-2 dtdt d t-1 d t-2 ftft f t-1 d t : Set of detected cells on frame t

4. Sequential Bayesian Framework We are interested in estimating x t given information (z 1,z 2,…,z t )  z 1:t Applying Bayes rule: (1) Measurement z t is conditionally independent on the current state x t : (2) Current state is conditionally independent on immediate past state:. Incorporating (1) and (2): Assumptions: Note 1: dimension of x t may be a (non-random) variable over t Note 2: dimension of z t may be a (non-random) variable over t

5. Sequential Bayesian Framework… Sequential MAP estimation Marginal probability distribution for p(x t |z 1:t ): Likelihood Motion prior

6. from (A) by Hastings’ MCMC Algorithm: (1)Randomly choose a sample from (2)Generate a sample (3)Generate and compute (4)Set if u>r, else set Sequential Markov Chain Monte Carlo (MCMC) Computation If we approximate the posterior density at (t-1) by a set of samples then the posterior density at t becomes We can generate samples (A)

7. Sequential Track Map Estimation and Detection Refinement such that the restricted mapping (function): is one-to-one. We define track mapping (function) as: Apply Sequential MCMC to: We also define detection refinement mapping as:

8. Sampling Via Reverse Track Map Apply sequential MCMC sampling Let’s consider a reverse track map: such that One can uniquely construct f t-1 from g t-1 and vice-versa, so: which implies

9. A Generic Sequential MCMC Algorithm For ease of sampling we assume the density Factors as:

10. Sampling for Detection Refinement Map Assume detection refinement depends only on current track map Our choice of detection refinement density for a cell tracking problem We also assume measurement depends only on detection refinement map Our choice of measurement density MH ratio for sampling of detection refinement map:

11. Sampling for Track Map Where, h(.) is the motion model, a choice might be:

12. Detection and Track Likelihood Detection likelihood for the “ligocyte” video: Track likelihood for the “ligocyte” video:

13. Tracking Video Dong, please insert a few good example videos

14. Experimental Results

15. Summary A single framework –no ad hoc combination of detection and tracking –Variable number of targets automatically taken care of: no ad hoc computation –Detection and tracking becomes cooperative, performance of each may improve –No explicit effort to compute “track-to- measurement” association

16. Future Plan  Instead of starting with a initial crude detection of cells, we like to dynamically detect cells as tracking proceeds  Mathematical Implication: the dimension of the set d t is a random variable  This stochastic dynamic behavior can be modeled in the Bayes’ rule by point process formalism