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Adaptive Rao-Blackwellized Particle Filter and It’s Evaluation for Tracking in Surveillance

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### Adaptive Rao-Blackwellized Particle Filter and It’s Evaluation for Tracking in Surveillance

Xinyu Xu and Baoxin Li, Senior Member, IEEE

Abstract

- In this paper, by proposing an adaptive Rao-Blackwellized Particle Filter (RBPF) for tracking in surveillance, we show how to exploit the analytical relationship among state variables to improve the efficiency and accuracy of a regular particle filter (PF).

Introduction

- Visual tracking is an important step in many practical applications.
- Generally, suppose we have an estimator depending upon 2 variables R and L, the RB theorem reveals its variance satisfies:

Non-negative

For the visual tracking problem, let denote the state to be estimated and the observation, with subscript t the time index.

- The key idea of RBPF is to partition the original state-space into two parts and .
- The justification for this decomposition follows from the factorization of the posterior probability

RBPF for tracking in surveillance

- a) Partition the state space

The scale change is related to its position alone y axis, so the scale change can be estimated conditional on the location components. The 8-D state space can separate into 2 groups

Root variables containing the motion information.

Leaf variables containing the scale parameters.

b) Overview of the method

- In this work, root variables are propagated by a first order system motion model defined by
- Conditional on the root variables, the leaf variables forms a linear-Gaussian substructure specified by

transition matrix

random noise

Gaussian random noise

A function encoding the conditional relation of L

Since both color histogram and gradient cues do not follow a linear-Gaussian relationship with state variable, the observation model is given in a general form:

- The observations form a linear relationship with state L

Image observation

Random noise

Nonlinear function

Gaussian random noise

Just like regular PF, RBPF represents the posterior density by a set of weighted particles:

- Each particle is represented by a triplet .
- The proposed RBPF algorithm will sample the motion using PF, while apply Kalman filter to estimate the scale parameters and conditional on the motion state.

(1)Propagate samples

- a) Sample the object motion according to

After this step, we have

minus sign is denotes the corresponding variable is a priori estimate

- b) Kalman prediction for leaf states according to

According to the Kalman filter model(4)and(6), we project forward the state and error covariance using:

- After this step, we have

Prediction for the mean of the leaf variables

Covariance for leaves

Observationprediction

(2)Evaluate weight for each particle

- a) Compute the color histogram for each sample ellipseΓ characterized by ellipse center and scale
- Pixels that are closer to the region center are given higher weights specified by

Kronecker delta function

b) Compute the gradient for each sample ellipseΓ characterized by ellipse center and scalethe gradient of a sample ellipse is computed as an average over gradients of all the pixels on the boundarywhere the gradient at pixel is set to the maximum gradient by a local search along the normal line of the ellipse at location

A simple operator is used to compute the gradient in x and y axis for pixelfinally, the gradient at point is computed as

c) Compute the weight

- one isbased on color histogram similarity between the hypothetical region and the target modelp stands for the color histogram of a sample hypothesis in the newly observed image, and q represents the color histogram of target model.

Another is based on gradient

- Notice that all the sample is divided by the maximum gradient to normalize into range[0,1], the final weight for each sample is given by

(3)Select samples

- Resampling with replacementthe latest measurements will be used to modify the prediction PDF of not only the root variables but also the leaf variables.
- After this step,

(4)Kalman update for leaf variables

- Kalman update is accomplished by
- After this step, we have

(5)Compute the mean state at time t

- Since resampling has been done, the mean state can be simply computed as the average of the state particles

(6)Compute the new noise variance

- We found that when velocity is small and constant, we only need a small noise variance to reach the smallest MSE, if velocity changes dramatically, we need a much larger noise variance to reach the lowest MSE.
- The noise variance is computed by

Evaluation of the RBPF algorithm

- Evaluate the performance between RBPF and PF.

Discussion

- Failure cases:when camera is not mounted higher than the target object…
- Computation cost:the same level of estimation accuracy, RBPF needs far fewer particles than PF dose; hence, it is more efficient than PF.

Conclusion

- Comparative studies using both simulated and real data have demonstrated the improved performance of the proposed RBPF over regular PF.
- Future working: to find a proper dependency model from a large number of state variables.

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