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
Presentation Overview The problem The framework to solve the problem Track mapping Simultaneous Tracking and Detection
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
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
Sequential Bayesian Framework… Sequential MAP estimation Marginal probability distribution for p(x t |z 1:t ): Likelihood Motion prior
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)
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:
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
A Generic Sequential MCMC Algorithm For ease of sampling we assume the density Factors as:
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:
Sampling for Track Map Where, h(.) is the motion model, a choice might be:
Detection and Track Likelihood Detection likelihood for the “ligocyte” video: Track likelihood for the “ligocyte” video:
Tracking Video Dong, please insert a few good example videos
Experimental Results
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
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