Sequential Monte-Carlo Method -Introduction, implementation and application Fan, Xin

Probabilistic state estimation for a dynamic system Dynamic system, a system with changes over time What can SMC do -Economics, weather -Moving object, image -Generally speaking, anything in the world Extracting relevant information of the system through investigating the observations

State, hidden information to describe the system What can SMC do--State-Space Modeling -Kinematic characteristics in tracking Measurements, made on the system —observed noisy data -Image data available up to current time —Evolving over time (Dynamic model): —What we are interested -Intensities of pixels in image estimation -Intensities of the degraded image —Associated with states (Measurement Model):

State evolution is described in terms of transition probability What can SMC do--Probabilistic formulation How the given fits the available measurement is described in terms of likelihood probability Determining the belief in the state taking deferent values, given the measurements

Prediction: What can SMC do—Recursive Estimation Update with the innovative measurement Starting from, at time is estimated with available :

Why use SMC Only when all of the distributions are Gaussian, the posterior distribution is Gaussian and analytical solution exists -- Kalman filter Non-Gaussian process noise Nonlinear Dynamics --sudden and jerky motion Multiple targets tracking Partial occlusion

Implementation—Basic idea Use SAMPLES with associated weights to approximate posterior density Examples: --discrete probability: coin, galloping dominoes --continuous density sampling Gaussion density

Implementation—Basic assumptions No explicit assumptions on the forms of both transition and likelihood probabilities, SMC is applicable for nonlinear and non-Gaussian estimation Measurements are independent, both mutually and with respect to dynamical process: Markov Chain:

Implementation—A 1D nonlinear example We need to infer the state at the time with available measurements Measurements Model: Dynamic Model:

Implementation—Results

Implementation—Algorithm Initialization -Draw samples from -Set weights 1. Prediction: 2. Update: -Normalization - 3. Resample

Implementation—Results

Implementation—Discussion Relaxes : - Linearity of dynamic and measurement models - The forms of the distributions of process and measurement noise. Requires : - Initial prior density - The likelihood can be evaluated - State samples can be generated easily  Do not make use of any knowledge of the measurements  inefficient and sensitive to outliers

Implementation—Generic SIS Algorithm - Draw 1. Prediction: 2. Update: -Normalization - 3. Resample Introducing an Importance density to facilitate sampling and using observations

Application—Contour extraction Probabilistic state estimation formulation: Problem Definition: - Grouping edge points into continuous cures, represented by a series of control points. - The positions of the control points are the states, then a contour turns out to be a state sequence. - Edge points are those pixels with larger intensity gradients, which are used as measurements

Application—Contour extraction Definitions of the probabilities Likelihood: Dynamics: Importance density: Perform the standard procedure to estimate the states

Application—Some results

Summary of using SMC Define the probability densities Modeling problems as probabilistic estimation -States / what we want, but cannot observe directly -Measurements / observations - Likelihood / the relationship between states and measurements / functional form that can be evaluated - Transition / determine the evolution of the states over time / the prior knowledge of the system under investigation - Importance / employ the observations / easy for sampling

Future work Apply SMC to various problems - Vision tracking - Constrain the state space by using better dynamic model / incorporate more prior knowledge - Elaborate techniques for efficiently sampling / SA / move samples to density peaks - Data fusion - Image restoration/super-resolution - Digital communication High computational expense - Decompose a high dimensional problem to several lower dimensional ones…

Reference [4] P. Pérez, A. Blake, and M. Gangnet. JetStream: Probabilistic contour extraction with particles. Proc. Int. Conf. on Computer Vision (ICCV), II: , Contour extraction [3] Gordon, N., Salmond, D., and Smith, A.." Novel approach to nonlinear/non-Gaussian Bayesian state estimation". IEE Proc. F, 140, 2, the simple 1D example [1] Proceedings of the IEEE, vol. 92, no. 3, Mar Special issue [2] IEEE Trans On Signal Processing, Vol. 50, no. 2. Special issue [5] M. Isard and A. Blake, "Contour tracking by stochastic propagation of conditional density", ECCV96,pp ,1996. – Application to vision tracking, in which significant performance was achieved. [6] Jun S. Liu and Rong Chen, "Sequential Monte Carlo Methods for Dynamic Systems", Journal of the American Statistical Association, Vol. 93, No. 443, pp , – SMC from the point of statisticians