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

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

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

Cell fluorescence data

Spatial Poisson point process Point process over S Intensity function ( ) Point process over S Intensity function ( ) Density: f ( )= ( )/ = z S (z)dz Density: f ( )= ( )/ = z S (z)dz Realized locations Z N ={z 1,...,z N } Realized locations Z N ={z 1,...,z N } Likelihood Likelihood

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

Dynamic Spatial DPM (DSDPM) DPM at each time point DPM at each time point Time evolution of mixture model parameters induces dynamic model for time-varying intensity function Time evolution of mixture model parameters induces dynamic model for time-varying intensity function Dynamic spatial point process Intensity function Z t-1 ZtZt Z t+1 Parameters of DPMs t-1 ( ) t ( ) t+1 ( ) t-1 t t+1

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

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

Particle filter implementation At time t 2 For each particle i=1,...,N For each particle i=1,...,N Evolve m t (i) according to the Generalized Polya Urn Evolve m t (i) according to the Generalized Polya Urn i,t and s i,t Update i,t and s i,t via near constant velocity model Split process Split process Sample c t (i) q(c t |m t (i), t|t-1 (i),Z t ) Sample c t (i) q(c t |m t (i), t|t-1 (i),Z t ) Sample t (i) q( t | t|t-1 (i),c t (i), Z t ) Sample t (i) q( t | t|t-1 (i),c t (i), Z t ) Compute importance weights Compute importance weights Resampling if needed Resampling if needed

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

Trajectory of cells

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

Reference Doucet, A., De Freitas, J. and Gordon, N. (eds.): Sequential Monte Carlo Methods in Practice. New York: Springer, (2001) 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. 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, 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, B. Vo, and W. K. Ma. The Gaussian mixture Probability Hypothesis Density filter. IEEE Transactions on Signal Processing, B. Vo, and W. K. Ma. The Gaussian mixture Probability Hypothesis Density filter. IEEE Transactions on Signal Processing, Chunlin Ji, 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. Chunlin Ji, 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.

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