Dynamic Spatial Mixture Modelling and its Application in Cell Tracking - Work in Progress -

1. Dynamic Spatial Mixture Modelling and its Application in Cell Tracking- Work in Progress - ChunlinJi & Mike West Department of Statistical Sciences, Duke University SMC Mid-Program Workshop February 19, 2009

2. Outline • Spatial Inhomogeneous Point Process • Dynamic Spatial Mixture Modelling • Particle Filter Implementation • Cell Fluorescence Imaging Tracking • Conclusion and Future Works

3. Introduction • Dynamic spatial inhomogeneous point process • Potential application areas • Multi-target tracking, particularly for extended target • Cell fluorescence imaging tracking • Existing methods • Probability hypothesis density (PHD) filter (Vo and Ma, 2006; Clark et al., 2007) • Poisson models for extended target tracking (Gilholm et al., 2005)

4. Cell fluorescence data

5. Spatial Poisson point process • Point process over S Intensity function () • Density: f()=()/ = z S(z)dz • Realized locations ZN={z1,...,zN} • Likelihood

6. Spatial Dirichlet process mixture (DPM) model (Ji et al. 2009) • Flexible model for spatially varying f() • Bivariate Gaussian mixture f() • Hierarchical DP prior over parameters

7. Dynamic Spatial DPM (DSDPM) • DPM at each time point • Time evolution of mixture model parameters induces dynamic model for time-varying intensity function Zt-1 Zt Zt+1 Dynamic spatial point process t-1() t() t+1() Intensity function t-1 t t+1 Parameters of DPMs

8. How points “move” in DSDPM Generalized Polya Urn scheme (Caron et al., 2007) (1) t-1 (2) t|t-1 (3) t|t-1 (4) t

9. Dynamic model for cells • Component locations: i,t={i,t, si,t}, • i,t ~ position of cells • si,t ~ parameters describing shape/appearance of cells • “Near constant” velocity model for i,t and si,t • Split process to simulate cell division: - e.g. if si,t says cell is “large”, then cell splits

10. Particle filter implementation At time t 2 • For each particle i=1,...,N • Evolve mt(i) according to the Generalized Polya Urn • Update i,t and si,t via near constant velocity model • Split process • Sample ct(i)q(ct|mt(i),t|t-1(i),Zt) • Sample t(i)q(t|t|t-1(i),ct(i), Zt) • Compute importance weights • Resampling if needed

11. Tracking result • Cells represented by blue color are segmented from the original movie • Green dots are the estimation of center positions of cells from the PF.

12. Trajectory of cells

13. Further work • Data association and track management • Dynamic lineage analysis • Observation generation methods • Result of image segmentation • Original image--fluorescence image • Feature points, e.g. Harris Feature Points • Performance evaluation of MTT

14. Reference • Doucet, A., De Freitas, J. and Gordon, N. (eds.): Sequential Monte Carlo Methods in Practice. New York: Springer, (2001) • F. Caron, M. Davy, and A. Doucet. Generalized poly urn for time-varying dirichlet process mixtures. Proceedings of the International Conference on Uncertainty in Artificial Intelligence(UAI),2007. • K. Gilholm, S.J. Godsill, S. Maskell, and D. Salmond. Poisson models for extended target and group tracking. In Proc. SPIE: Signal and Data Processing of Small Targets, 2005. • B. Vo, and W. K. Ma. The Gaussian mixture Probability Hypothesis Density filter. IEEE Transactions on Signal Processing, 2006. • ChunlinJi, Daniel Merl, Thomas Kepler and Mike West. "Spatial Mixture Modelling for Partially Observed Point Processes: Application to Cell Intensity Mapping in Immunology." Bayesian Analysis, invited revision.

15. Thank You