# Tracking Algorithms. Lecture17: Stochastic Tracking. Joint Probability and Graphical Model. Probabilistic Tracking

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1 Tracking Algorithms (2015S) Lecture17: Stochastic Tracking Bohyung Han CSE, POSTECH Deterministic methods Given input video and current state, tracking result is always same. Local search Least square tracking Mean shift Gradient ascent (or decent) algorithms Probabilistic or stochastic methods Each trial may produce a different tracking result. Statistical search (e.g., by sampling) Kalman filter/extended Kalman filter Particle filter 2 Probabilistic Tracking Target state is determined through a statistical process. For example, the target state is estimated based on density function, sometimes by sampling for target state and corresponding. Joint Probability and Graphical Model Graphical model Probabilistic model for which a graph denotes the conditional independence structure between random variables Enables a simpler representation of joint probability Directed or undirected,,,,,,, 3 4

2 In stochastic process Markov Property The conditional probability distribution of future states of the process depends only upon the present state, not on the sequence of events that preceded it. In the discrete time stochastic process, Markov property can be defined as In graphical model,,, ) Sequential Bayesian Filtering Estimation of an unknown probability density function PDF is estimated recursively over time using Incoming measurement Mathematical process model (dynamic model) Extension of (static) Bayesian estimation Observation changes overtime : state variable : variable : : 5 6 Components of Sequential Bayesian Filter Derivation of Sequential Bayesian Filtering : : posterior likelihood prior : process transition probability Process model (dynamic model) Observation model,, noises Markov assumption for process model,,, Conditional independence in,,, Joint distribution of all variables,,,,,, We should estimate the marginal posterior : sequentially. 7 8

3 Derivation of Sequential Bayesian Filtering Examples of Sequential Bayesian Filtering : : :, :, : :, : : : :, : : : : Kalman filter Extended Kalman filter Unscented Kalman filter Particle filter Sequential Importance Sampling (SIS) Sampling Importance Resampling (SIR) Regularized particle filter Auxiliary particle filter : : normalization constant 9 10 Examples of Sequential Bayesian Filtering They are different in Flexibility of density representation Flexibility of dynamic model : : Posterior density Process Model Observation Model KF Gaussian Linear Linear Kalman Filter and Its Extension Kalman filter is optimal if The initial posterior is Gaussian The process model is linear The density is Gaussian Real world problems hardly meet the conditions. Limitations of (extended) Kalman filter They are limited to handle non linear process models They cannot model multi modal posterior density functions EKF Gaussian Approximation Linear Approximation Linear Approximation Critical limitations for real world problems PF Arbitrary Arbitrary Arbitrary 11 12

4 Basic idea Monte Carlo Approximation Sample based The more we draw samples, the more accurate the estimation is. Useful when analytical solution is unknown or hard to compute MC in sequential Bayesian filtering Estimating unknown probability distribution using a set of samples MC can easily be used to compute marginal posterior distribution, : It does not require any assumption on the underlying distribution. Particle Filter Most flexible implementation of sequential Bayesian filtering Representation of the state with (location, weight) pair of each sample No restriction of posterior density representation No restriction of process and measurement models Also known as Sequential Monte Carlo (SMC) The posterior is estimated in a sequential manner Advantages The posterior is estimated by the population of samples. Actually, there are some other SMC techniques other than PF. Able to handle multi modal (arbitrary) posterior density functions Able to handle non linear process model Very simple implementation Posterior Representation Procedure Sequential Importance Sampling (SIS) probability posterior density weighted sample : state transition... state :, : Unknown measurement function 15 16

5 Degeneracy problem Limitations Most particles have very small and negligible weights. Most of the weights are concentrated on a few particles. Most of particles are useless. Density estimation becomes inaccurate. Condensation Algorithm Giving more diversity in samples by resampling (SIR) resampling : state transition : : Resampling Particle Filter for Visual Tracking Properties Identical sample weights More sample diversity Less degeneracy Condensation algorithm animation Sample a set of particles (samples) from the prior. Perform an for each particle. Obtain the target state However, resampling does not solve the degeneracy p roblem completely. initial sample Isard and Blake, COPIES/ISARD1/condensation.html frame t 1 sample after prediction frame t 19 20

6 Observation: Color Histogram Particle Filter for Visual Tracking Likelihood computation for each particle By histogram comparison exp, where, 1 ; P. Perez, C. Hue, J. Vermaak, and M. Gangnet. Color Based Probabilistic Tracking, ECCV, Characteristics Tracking Results It is practically (almost) impossible to find the proper dynamic model. A suggestion: Auto Regressive (AR) model, ~0, Random walk is frequently used. Tracking control and are independent. Any reasonable technique can be integrated into the model (e.g., histogram comparison) exp, where, 1 ; 23 24

7 Computer vision Applications of Particle filter Visual tracking Dynamic parameter estimation Robotics SLAM: Simultaneous Localization And Mapping Signal processing Financial engineering Prediction of stock price Tutorial on particle filter Reference M.S. Arulampalam, S. Maskell, N. Gordon, T Clapp, "A tutorial on particle filters for online nonlinear/non Gaussian Bayesian tracking," IEEE Trans. on Signal Processing 50(2), , 2002 Particle filtering for visual tracking M. Isard and A. Blake, Condensation Conditional Density Propagation for Visual Tracking, Int l Journal of Computer Vision 29(1), 5 28, 1998 P. Pérez, C. Hue, J. Vermaak and M. Gangnet, Color Based Probabilistic Tracking, ECCV Tracking by Detection Combination of tracking and detection techniques Exploits the recent advance of object detection techniques Typically needs to design online classifiers Requires to handling outliers and noises effectively Advantages More convenient to handle initialization problem More robust to abrupt changes in lighting condition and motion More rigorous to deal with occlusion and entrance/leaving event Examples Online boosting STRUCK TLD (Tracking Learning Detection) etc

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