Dynamic Spatial Mixture Dynamic Spatial Mixture Modelling and its Application in Modelling and its Application in

Cell TrackingCell Tracking- Work in Progress -- Work in Progress -

Chunlin Ji & Mike WestChunlin Ji & Mike WestDepartment of Statistical Department of Statistical

Sciences, Sciences, Duke UniversityDuke University

SMC Mid-Program WorkshopSMC Mid-Program WorkshopFebruary 19, 2009February 19, 2009

OutlineOutline Spatial Inhomogeneous Point ProcessSpatial Inhomogeneous Point Process

Dynamic Spatial Mixture ModellingDynamic Spatial Mixture Modelling

Particle Filter ImplementationParticle Filter Implementation

Cell Fluorescence Imaging TrackingCell Fluorescence Imaging Tracking

Conclusion and Future WorksConclusion and Future Works

IntroductionIntroduction

Dynamic spatial inhomogeneous Dynamic spatial inhomogeneous point processpoint process

Potential application areasPotential application areas Multi-target tracking, particularly for Multi-target tracking, particularly for

extended targetextended target Cell fluorescence imaging trackingCell fluorescence imaging tracking

Existing methodsExisting methods Probability hypothesis density (PHD) Probability hypothesis density (PHD)

filter filter (Vo and Ma, 2006; Clark et al., 2007)(Vo and Ma, 2006; Clark et al., 2007)

Poisson models for extended target Poisson models for extended target tracking tracking (Gilholm et al., 2005)(Gilholm et al., 2005)

Cell fluorescence dataCell fluorescence data

Spatial Poisson point Spatial Poisson point processprocess

Point process over SPoint process over S Intensity Intensity function function (())

Density: Density: ff(()=)=(()/)/ = = zz S S (z)dz (z)dz

Realized locations ZRealized locations ZNN={z={z11,...,z,...,zNN}}

LikelihoodLikelihood

Spatial Dirichlet process Spatial Dirichlet process mixture (DPM) model mixture (DPM) model (Ji et al. (Ji et al.

2009)2009)

Flexible model for spatially varying Flexible model for spatially varying ff(())

Bivariate Gaussian mixture ff(())

Hierarchical DP prior over Hierarchical DP prior over parametersparameters

Dynamic Spatial DPM Dynamic Spatial DPM (DSDPM)(DSDPM)

DPM at each time pointDPM at each time point

Time evolution of mixture model Time evolution of mixture model parameters induces dynamic model for parameters induces dynamic model for time-varying intensity functiontime-varying intensity function

Dynamic spatial point process

Intensity function

Zt-1 Zt Zt+1

Parameters of DPMs

t-

1()t() t+1(

)

t-1

t

t+1

How points “move” in How points “move” in DSDPMDSDPM

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 cellsDynamic model for cells

Component locations: Component locations: i,ti,t={={i,ti,t, s, si,ti,t}, }, i,ti,t ~ position of cells ~ position of cells

ssi,ti,t ~ parameters describing shape/appearance ~ parameters describing shape/appearance of cellsof cells

“Near constant” velocity model for i,t i,t and s and si,ti,t

Split process to simulate cell division:Split process to simulate cell division: - e.g. if s- e.g. if si,ti,t says cell is “large”, then cell splits says cell is “large”, then cell splits

Particle filter Particle filter implementationimplementation

At time At time tt 2 2 For each particle For each particle ii=1,...,=1,...,NN

Evolve Evolve mmtt((ii)) according to the Generalized Polya according to the Generalized Polya

UrnUrn Update i,t i,t and s and si,t i,t via near constant velocity

model Split processSplit process Sample cSample ctt

((ii)) qq(c(ctt||mmtt((ii)),,t|t-t|t-11

((ii)),Z,Ztt)) Sample Sample tt

((ii)) qq((tt||t|t-t|t-11((ii)),c,ctt

((ii)), Z, Ztt)) Compute importance weightsCompute importance weights

Resampling if neededResampling if needed

Tracking resultTracking 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 cellsTrajectory of cells

Further workFurther work

Data association and track Data association and track managementmanagement

Dynamic lineage analysisDynamic lineage analysis Observation generation methodsObservation generation methods

Result of image segmentation Result of image segmentation Original image--fluorescence imageOriginal image--fluorescence image Feature points, e.g. Harris Feature Feature points, e.g. Harris Feature

PointsPoints Performance evaluation of MTTPerformance evaluation of MTT

ReferenceReference Doucet, A., De Freitas, J. and Gordon, N. (eds.): Sequential Monte Doucet, A., De Freitas, J. and Gordon, N. (eds.): Sequential Monte

Carlo Methods in Practice. New York: Springer, (2001)Carlo Methods in Practice. New York: Springer, (2001)

F. Caron, M. Davy, and A. Doucet. Generalized poly urn for time-F. Caron, M. Davy, and A. Doucet. Generalized poly urn for time-varying dirichlet process mixtures. Proceedings of the International varying dirichlet process mixtures. Proceedings of the International Conference on Uncertainty in Artificial Intelligence(UAI),2007.Conference on Uncertainty in Artificial Intelligence(UAI),2007.

K. Gilholm, S.J. Godsill, S. Maskell, and D. Salmond. Poisson models K. Gilholm, S.J. Godsill, S. Maskell, and D. Salmond. Poisson models for extended target and group tracking. In Proc. SPIE: Signal and for extended target and group tracking. In Proc. SPIE: Signal and Data Processing of Small Targets, 2005.Data Processing of Small Targets, 2005.

B. Vo, and W. K. Ma. The Gaussian mixture Probability Hypothesis B. Vo, and W. K. Ma. The Gaussian mixture Probability Hypothesis Density filter. IEEE Transactions on Signal Processing, 2006.Density filter. IEEE Transactions on Signal Processing, 2006.

Chunlin Ji, Daniel Merl, Thomas Kepler and Mike West. "Spatial Chunlin Ji, Daniel Merl, Thomas Kepler and Mike West. "Spatial Mixture Modelling for Partially Observed Point Processes: Mixture Modelling for Partially Observed Point Processes: Application to Cell Intensity Mapping in Immunology." Bayesian Application to Cell Intensity Mapping in Immunology." Bayesian Analysis, invited revision. Analysis, invited revision.

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