darryl morrellstochastic modeling seminar1 particle filtering

Darryl Morrell Stochastic Modeling Seminar 1

Particle Filtering


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 R&N

• Part II (PF from general filtering perspective– Uses the Arulampalam tutorial


Outcome: <Cloudy,~sprinkler,Rain,Wetgrass>

Darryl Morrell Stochastic Modeling Seminar 11Problem: may need many many samples if the required probability is very low..


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


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.


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)


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


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


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)


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.


Terrain Aided Navigationhttp://www.control.isy.liu.se/research/sensorfusion/

sensorfusion/sensorfusion.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.


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.


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 robot’s position is

estimated by correlating sensor data with known maps.


Tracking Articulated Objectshttp://www.dai.ed.ac.uk/CVonline/LOCAL_COPIES/

RINGER1/mocap_overview.html

• Observations are video sequences from two cameras.

• Unknowns are positions and velocities of model components


Tracking with Networks of Distributed Sensors

http://www.parc.xerox.com/spl/projects/cosense/

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


Other Applications

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


Fundamental Concepts

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


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(z|x)

models the relationship between Z and X.


Bayes Theorem

• The conditional distribution p(x|z) represents posterior information about X given Z.

)(

)()|()|(

zp

xpxzpzxp


Monte Carlo Samples (Particles)

• The posterior distribution p(x|z) may be difficult or impossible to compute in closed form.

• An alternative is to represent p(x|z) using Monte Carlo samples (particles):– Each particle has a value and a weight

x

x


Importance Sampling

• Ideally, the particles would represent samples drawn from the distribution p(x|z).– In practice, we usually cannot get p(x|z) in closed

form; in any case, it would usually be difficult to draw samples from p(x|z).

• We use importance sampling:– Particles are drawn from an importance

distribution.– Particles are weighted by importance weights.


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(x|z) is larger.

x

x


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


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:

xk+1 = fk (xk, wk)


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)


Objective-Find p(xk|zk,…,z1)

• The objective of the particle filter is to compute the conditional distribution

p(xk|zk,…,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.


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.


Initial State Distribution

x0

x0


State Update

x0

x1 = f0 (x0, w0)

x1

This is one way to sample from a proposal distribution.


Compute Weights

x1

x1

p(z1|x1)

x1

Before

After


Resample

x1

x1


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.


Particle Filter Demonstration

QuickTime™ and a Graphics decompressor are needed to see this picture.


Variations on this Simple Implementation

• Use a different importance distribution:– In this implementation, the importance

distribution is the predicted state distribution p(xk+1|zk,…,z1).

– Several papers have pointed out that this distribution may not be the best one can use.

– If the observation at time k+1 is available, significant improvement in performance can be obtained.


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.


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.


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.


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.


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.


A Foveal Sensor

Low Acuity Region

Foveal Region

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

Target Position


Mathematical ModelThe 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.

xk

zk

dk

Foveal Region


Before 1st Observation

Position

True Position Estimated Initial

Position


Observed

Collecting 1st Observation

Position

True

Predicted

1 Foveal sensor is configured using predicted values.

2 Observation is obtained

3 Position (and velocity) estimates are computed

Estimated


Observed

Collecting 2nd Observation

Position

True

Predicted




Estimated


Observed

Collecting 3rd Observation

Position

True

Predicted




Estimated


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.


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


Curves of Constant Error

1

10

100

1000

10000

0.001 0.01 0.1 1 10 100

Observation Noise Variance

Pro

cess

Noi

se V

aria

nce

Stat, Fixed Gain

PF, Fixed Gain

PF, Var. Gain


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.


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


Average Estimate Error

0.01

0.10

1.00

10.00

0 5 10 15 20

k

Ave

rage

Squ

ared

Pos

ition

Err

or

Adaptive GG = 0.25G = 1G = 4


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.

darryl morrellstochastic modeling seminar1 particle filtering

Documents

sequential monte carlo

arulampalam tutorial

engineering problems

general filtering perspective

hard problems

ieee transactions

class of problems

excellent tutorial paper