Probabilistic Tracking

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Probabilistic Tracking

• ECE 285 Class Presentation - 3
• Shankar Shivappa
• 3/2/2005

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Overview

• Introduction to Tracking Issues
• General Framework for Probabilistic Tracking
• Kalman Filter for Tracking
• Condensation Filter A Generalization
• Examples
• Performance Metrics for Tracking

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Estimation

• Inferring the value of a quantity of interest
  from the indirect, inaccurate and uncertain
  observations

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Tracking

• Estimating the State of a moving object
• What do we have?
• Measurements Noise Corrupted Observations

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Mathematical Framework

• State of the modeled object denoted by
• Its History is denoted by
• Measurements at time t
• And their history by

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Mathematical Framework

• State Dynamics form a Temporal Markov Chain
• This is same as claiming that
• For example -

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Kalman Filter in Stationary Case

• We need to estimate the unknown state from the
  measurements.
• We denote this by
• One way would be to collect all data and find the
  optimal estimate computationally intensive.
• Kalman Filter allows us to do this iteratively.

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Kalman Filter for Tracking

• Tracking - The state is changing with time
• Kalman Filter helps us follow the state using the
  measurements at each time-step.

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Kalman Filter
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Issues

• Extended Kalman Filter Deals with the system that
  is non-linear
• Works akin to Taylor Series Expansion, by using
  partial derivatives

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Issues

• Kalman Filter is optimal in the case where system
  noise and measurement noise are Gaussian. (EKF is
  Sub-optimal)
• In the non-Gaussian case, it is still the optimal
  linear filter.

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Issues

• What if the system dynamics are non-Gaussian?
• This happens in the presence of background
  clutter
• The observations are typically multimodal

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Particle Filters

• Clearly, what is needed is a framework to
  estimate the state in the general, non-Gaussian
  case
• General PDF Monte Carlo Method
• Use sample points to represent the PDF

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Particle Filters

• At each time step,
• If we can get an estimate of
• And update it according to the
• System Dynamics
• Measurements
• Then we can get

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CONDENSATION

• Conditional Density Propagation
• Note

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Factored Sampling

• Select based on
• Choose weights

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CONDENSATION

• Factored Sampling

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CONDENSATION
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CONDENSATION for Tracking

• Model
• Constrained B-Spline representation
• State Dynamics
• Learned by Maximum Likelihood Estimation
• Observation Model
• High Contrast points on the Normals

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CONDENSATION for Tracking
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Multimodal Distribution Case

• Multiple Hypothesis Tracking for free!

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Rapid Motion in Clutter
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Articulated Object Tracking

• High Dimensional Shape Space

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Performance Metrics
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Performance Metrics

• Displacement between 2 trajectories

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Performance Metrics

• Mean Displacement

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Performance Metrics

• What if Spatially separated?

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Performance Metrics

• What if temporally separated ?

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Performance Metrics

• Area Between Trajectories provides
  time-independent information
• Similar Metrics could be evolved for other cases

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References

• Optimal Filtering, Brian Anderson John Moore,
  Dover Publications INC.
• Multitarget-Multisensor Tracking Principles and
  Techniques, Yaakov Bar-Shalom Xiao-Rong Li.