Particle Filtering

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Particle Filtering
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Organization of Slides

• Part I (PF from Dynamic Bayes Net Perspective)
  Understand particle filtering as a likelihood
  monte carlo sampling method on DBNs
• Review of likelihood sampling
• Uses RN
• Part II (PF from general filtering perspective
• Uses the Arulampalam tutorial

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Outcome ltCloudy,sprinkler,Rain,Wetgrassgt
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Problem may need many many samples if the
required probability is very low..
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Particle Filters a Solution to Hard Problems in
Navigation, Target Tracking, and Perception

• Darryl Morrell Ya Xue
• Department of Electrical Engineering
• Arizona State University
• Portions of this work supported by AFOSR under
  award number F49620-00-1-0124

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References for More Information

• M. Arulampalam, S. Maskell, N. Gordon, and T.
  Clapp. A tutorial on particle filters for online
  nonlinear/non-Gaussian Bayesian tracking. IEEE
  Transactions on Signal Processing, 50(2)174-188,
  February 2002.
• This is an excellent tutorial paper-read this
  first.
• A. Doucet, N. de Freitas, and N. Gordon, editors.
  Sequential Monte Carlo Methods in Practice.
  Springer-Verlag, 2001.
• This is a broad ranging collection of articles
  that will introduce your to most of the important
  particle filter developments.

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Introduction

• The importance of Monte Carlo methods for
  inference in science and engineering problems has
  grown steadily over the past decade. This growth
  has largely been propelled by an explosive
  increase in accessible computing power. it has
  become clear that Monte Carlo methods can
  significantly expand the class of problems that
  can be addressed practically.
• (Introduction to Feb 2002 IEEE Transactions on
  Signal Processing special issue on Monte Carlo
  Methods)

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Sequential Monte Carlo Techniques

• Sequential Monte Carlo techniques have been
  developed in a wide range of disciplines, and go
  under many names
• Bootstrap filtering
• The condensation algorithm
• Particle filtering
• Interacting particle approximations
• Survival of the fittest

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Presentation Outline

• Applications of particle filters
• Fundamental concepts
• Anatomy of a simple particle filter
• Variations on the simple particle filter
• Pros and Cons of particle filters
• Application to configuration of a foveal sensor
• Conclusions

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Applications of Particle Filters

• Particle filters have provided solutions to
  problems from many disciplines
• image processing and understanding
• tracking complex objects (e.g. people) in video
  sequences
• robot navigation
• tracking and identifying complex military targets
  (e.g. vehicle convoys)

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Some Specific Applications

• Terrain aided navigation
• Car positioning using map information
• Robot navigation
• Tracking of articulated targets using video
• Tracking of complex targets using distributed
  sensors.

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Terrain Aided Navigationhttp//www.control.isy.li
u.se/research/sensorfusion/sensorfusion/sensorfus
ion.html

• Observations are measured ground clearance.
• Unknowns are aircraft position and velocity.
• The particle filter is needed because measured
  ground clearance does not uniquely determine
  position.

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Car Positioning Using Map InfoGustafsson et al.,
Particle Filters for Positioning, Navigation,
and Tracking, IEEE Transactions on SP, Feb 2002

• Observations are yaw rate and speed information
  computed from wheel speed sensors.
• Vehicle position is unknown.
• The map provides constraints on the vehicle
  position.

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Mobile Robot Localizationhttp//www.cs.washington
.edu/ai/Mobile_Robotics/mcl/2

• Observations are sensor data (image, video,
  sonar, laser rangefinder, etc.)
• Robot position is unknown.
• The robots position is estimated by correlating
  sensor data with known maps.

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Tracking Articulated Objectshttp//www.dai.ed.ac.
uk/CVonline/LOCAL_COPIES/RINGER1/mocap_overview.h
tml

• Observations are video sequences from two
  cameras.
• Unknowns are positions and velocities of model
  components

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Tracking with Networks of Distributed
Sensorshttp//www.parc.xerox.com/spl/projects/cos
ense/

• Targets are tracked using an ad hoc network of
  distributed micro-sensors.

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Other Applications

• Channel equalization
• Estimation of parameters of multiple chirp
  signals
• Multiple target tracking
• Bearings-only target tracking
• Track before detect target tracking
• Image segmentation

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Presentation Outline

• Applications of particle filters
• Fundamental concepts
• Anatomy of a simple particle filter
• Variations on the simple particle filter
• Pros and Cons of particle filters
• Application to configuration of a foveal sensor
• Conclusions

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Fundamental Concepts

• Bayesian inference
• Monte Carlo samples
• Importance Sampling
• Resampling

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Bayesian Inference

• X is unknown-a random variable or set (vector) of
  random variables
• Z is observed-also a set of random variables
• We wish to infer X by observing Z.
• The probability distribution p(x) models our
  prior knowledge of X.
• The conditional probability distribution p(zx)
  models the relationship between Z and X.

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Bayes Theorem

• The conditional distribution p(xz) represents
  posterior information about X given Z.

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Monte Carlo Samples (Particles)

• The posterior distribution p(xz) may be
  difficult or impossible to compute in closed
  form.
• An alternative is to represent p(xz) using Monte
  Carlo samples (particles)
• Each particle has a value and a weight

x
x
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Importance Sampling

• Ideally, the particles would represent samples
  drawn from the distribution p(xz).
• In practice, we usually cannot get p(xz) in
  closed form in any case, it would usually be
  difficult to draw samples from p(xz).
• We use importance sampling
• Particles are drawn from an importance
  distribution.
• Particles are weighted by importance weights.

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Resampling

• In inference problems, most weights tend to zero
  except a few (from particles that closely match
  observations), which become large.
• We resample to concentrate particles in regions
  where p(xz) is larger.

x
x
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Presentation Outline

• Applications of particle filters
• Fundamental concepts
• Anatomy of a simple particle filter
• Variations on the simple particle filter
• Pros and Cons of particle filters
• Application to configuration of a foveal sensor
• Conclusions

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Anatomy of a Simple Particle Filter

• A simple particle filter requires the following
• A system state evolution model
• An observation model
• Particle computation processes
• Propagate forward in time
• Compute weights given observations
• Resampling

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System State

• The state represents the unknown whose value we
  want to infer. For example,
• Position (and velocity) of a robot, car, plane,
  ...
• Position of articulated model components.
• The system state at (discrete) time k is denoted
  xk.
• The state evolves according to the following
  dynamics equation
• xk1 fk (xk, wk)

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Observation Model

• The observation zk may be an image, a frame of
  video, a radar or sonar measurement, etc.
• The relationship between the observation and the
  state is given by the conditional probability
  distribution p(zk xk).
• This distribution may be derived from a
  functional relationship between zk and xk
• zk hk (xk, vk)

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Objective-Find p(xkzk,,z1)

• The objective of the particle filter is to
  compute the conditional distribution
• p(xkzk,,z1)
• To do this analytically, we would use the
  Chapman-Kolmogorov equation and Bayes Theorem
  along with Markov model assumptions.
• The particle filter gives us an approximate
  computational technique.

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Particle Filter Algorithm

• Create particles as samples from the initial
  state distribution p(x0).
• For k going from 1 to K
• Sample each particle from a proposal
  distribution.
• Compute weights for each particle using the
  observation value.
• (Optionally) resample particles.

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Initial State Distribution
x0
x0
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State Update
x1 f0 (x0, w0)
x1
This is one way to sample from a proposal
distribution.
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Compute Weights
p(z1x1)
x1
Before
x1
After
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Resample
x1
x1
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Particle Filter Demonstration

• A target moves from left to right.
• Two sensors
• Each measures the distance from itself to the
  target.
• Sensors at (30,0) and (0,50)
• 4000 Particles were used to track the target.
• The animation on the following slide shows the
  particles, the true target position, and the
  estimated target position.

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Particle Filter Demonstration
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Presentation Outline

• Applications of particle filters
• Fundamental concepts
• Anatomy of a simple particle filter
• Variations on the simple particle filter
• Pros and Cons of particle filters
• Application to configuration of a foveal sensor
• Conclusions

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Variations on this Simple Implementation

• Use a different importance distribution
• In this implementation, the importance
  distribution is the predicted state distribution
  p(xk1zk,,z1).
• Several papers have pointed out that this
  distribution may not be the best one can use.
• If the observation at time k1 is available,
  significant improvement in performance can be
  obtained.

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Variations

• Use a different resampling technique
• Resampling adds variance to the estimate several
  resampling techniques are available that minimize
  this added variance.
• Our simple resampling leaves several particles
  with the same value methods for spreading them
  are available.

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Variations

• Reduce the resampling frequency
• Our implementation resamples after every
  observation, which may add unneeded variance to
  the estimate.
• Alternatively, one can resample only when the
  particle weights warrant it. This can be
  determined by the effective sample size.

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Variations

• Rao-Blackwellization
• Some components of the model may have linear
  dynamics and can be well estimated using a
  conventional Kalman filter.
• The Kalman filter is combined with a particle
  filter to reduce the number of particles needed
  to obtain a given level of performance.

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Presentation Outline

• Applications of particle filters
• Fundamental concepts
• Anatomy of a simple particle filter
• Variations on the simple particle filter
• Pros and Cons of particle filters
• Application to configuration of a foveal sensor
• Conclusions

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Advantages of Particle Filters

• Under general conditions, the particle filter
  estimate becomes asymptotically optimal as the
  number of particles goes to infinity.
• Non-linear, non-Gaussian state update and
  observation equations can be used.
• Multi-modal distributions are not a problem.
• Particle filter solutions to inference problems
  are often easy to formulate.

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Disadvantages of Particle Filters

• Naïve formulations of problems usually result in
  significant computation times.
• It is hard to tell if you have enough particles.
• The best importance distribution and/or
  resampling methods may be very problem specific.

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Presentation Outline

• Applications of particle filters
• Fundamental concepts
• Anatomy of a simple particle filter
• Variations on the simple particle filter
• Pros and Cons of particle filters
• Application to configuration of a foveal sensor
• Conclusions

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A Foveal Sensor

• A foveal sensor has a high acuity area (similar
  to the fovea of the eye) that can be steered
  towards a desired location.

Foveal Region
Target Position
Low Acuity Region
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Mathematical Model

• The foveal sensor is modeled mathematically as
• zk tan-1(Ck(xk-dk))
• xk is the target position.
• dk controls the location of the center of the
  foveal region.
• Ck controls the width of the foveal region.

Foveal Region
zk
xk
dk
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Before 1st Observation
Position
Estimated Initial Position
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Collecting 1st Observation
1 Foveal sensor is configured using predicted
values.
Position
2 Observation is obtained
3 Position (and velocity) estimates are computed
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Collecting 2nd Observation
1 Foveal sensor is configured using predicted
values.
Position
2 Observation is obtained
3 Position (and velocity) estimates are computed
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Collecting 3rd Observation
1 Foveal sensor is configured using predicted
values.
Position
2 Observation is obtained
3 Position (and velocity) estimates are computed
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Implementation

• We implemented a particle filter to estimate the
  target position from observations.
• The foveal region is centered on the predicted
  target position.
• The gain is either set to a constant value or
  adjusted to include a certain percentage of the
  particles in the foveal region.
• The implementation took a few hours.
• Tuning the filter has taken a few weeks.

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Comparison with Previous Foveal Sensor

• A two dimensional linear system dynamics model is
  used.
• The system state transition matrix is stable.
• The following plot shows curves of constant
  estimation error as a function of process and
  observation noise variance
• Stat fixed gain-Kalman filter implementation of
  fixed gain sensor
• PF fixed gain-Particle filter implementation of a
  fixed gain sensor
• PF Var. gain-Particle filter implementation of an
  adaptive gain sensor

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Curves of Constant Error
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Discussion of Results

• Adaptive acuity gives better performance than
  fixed acuity.
• The particle filter implementations do not
  perform well with very small observation noise
  variances.
• The number of particles is too small for very
  sharply peaked observation densities-few
  particles fall within the peaks.
• Several approaches to improve the performance for
  small observation variances are currently under
  investigation.

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Fixed vs. Adaptive Acuity

• The foveal sensor collects observations of
  position.
• The acuity of the foveal region is adjusted so
  that 80 of the predicted particle positions fall
  into the foveal region.
• The gain of the foveal region is smoothed using a
  low-pass filter with an exponentially decaying
  impulse response.
• We show plots of the average squared estimate
  error as a function of time for
• Fixed acuity foveal sensor for gains of 0.25, 1,
  and 4
• Adaptive acuity foveal sensor

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Average Estimate Error
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Presentation Outline

• Applications of particle filters
• Fundamental concepts
• Anatomy of a simple particle filter
• Variations on the simple particle filter
• Pros and Cons of particle filters
• Application to configuration of a foveal sensor
• Conclusions

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Conclusions

• Particle filters (and other Monte Carlo methods)
  are a powerful tool to solve difficult inference
  problems.
• Formulating a filter is now a tractable exercise
  for many previously difficult or impossible
  problems.
• Implementing a filter effectively may require
  significant creativity and expertise to keep the
  computational requirements tractable.