Bayesian Filtering for Location Estimation D. Fox, J. Hightower, L. Liao, D. Schulz, and G....
-
date post
22-Dec-2015 -
Category
Documents
-
view
219 -
download
3
Transcript of Bayesian Filtering for Location Estimation D. Fox, J. Hightower, L. Liao, D. Schulz, and G....
Bayesian Filtering for Location Estimation
D. Fox, J. Hightower, L. Liao, D. Schulz, and G. Borriello
Presented by: Honggang Zhang
Outline
• Basic idea of Bayes filters• Several types of Bayes filters• Some applications
Bayes Filters
1( , )
( , )t t t t
t t t t
x f x w
z g x v
System state dynamics
Observation dynamics
1( ) ( | , , )t t tBel x p x z z
We are interested in: Belief or posterior density
Estimating system state from noisy observations
1:( 1) 1 1where , ,t tz z z
1:( 1) 1, 1:( 1) 1 1:( 1) 1( | ) ( | ) ( | )t t t t t t t tp x z p x x z p x z dx
From above, constructing two steps of Bayes Filters
1:( 1)1:( 1) 1:( 1)
1:( 1)
( | , )( | , ) ( | )
( | )t t t
t t t t tt t
p z x zp x z z p x z
p z z
Predict:
Update:
1 1 1( ) ( | ) ( )t t t t tp x p x x p x dx ( | ) ( )
( | )( )
t t tt t
t
p z x p xp x z
p z
Recall “law of total probability” and “Bayes’ rule”
1:( 1) 1, 1:( 1) 1 1:( 1) 1( | ) ( | ) ( | )t t t t t t t tp x z p x x z p x z dx
1:( 1)replace ( | , ) with ( | )t t t t tp z x z p z x
Predict:
Update:
Assumptions: Markov Process
1 1: 1 1replace ( | , ) with ( | )t t t t tp x x z p x x
1:( 1)1:( 1) 1:( 1)
1:( 1)
( | , )( | , ) ( | )
( | )t t t
t t t t tt t
p z x zp x z z p x z
p z z
1:( 1) 1:( 1)( | , ) ( | ) ( | )t t t t t t t tp x z z p z x p x z
Bayes Filter
1:( 1) 1 1 1:( 1) 1( | ) ( | ) ( | )t t t t t t tp x z p x x p x z dx
1( | )
( | )t t
t t
p x x
p z x
How to use it? What else to know?
Motion Model
Perceptual Model
Start from: 0 00 0 0
0
( | )( | ) ( )
( )
p z xp x z p x
p z
Example 1
10 0( ) or ( )Bel x p x
Step 0: initialization
0 0 0
0 0 0 0
( ) or ( | )
( | ) ( )
Bel x p x z
p z x p x
Step 1: updating
Example 1 (continue)
1 1 1
1 1 1 0 0
( ) or ( | )
( | ) ( | )
Bel x p x z
p z x p x z
Step 3: updating
12 2 1
2 1 1 1 1
( ) or ( | )
( | ) ( | )
Bel x p x z
p x x p x z dx
Step 4: predicting
11 1 0
1 0 0 0 0
( ) or ( | )
( | ) ( | )
Bel x p x z
p x x p x z dx
Step 2: predicting
Several types of Bayes filters
• They differs in how to represent probability densities– Kalman filter– Multihypothesis filter– Grid-based approach– Topological approach– Particle filter
Kalman FilterRecall general problem
1( , )
( , )t t t t
t t t t
x f x w
z g x v
Assumptions of Kalman Filter:
1 , where (0, )
, where (0, )t t t t t t
t t t t t t
x A x w w N Q
z C x v v N R
( ) ( : , )t t t tBel x N x Belief of Kalman Filter is actually a unimodal Gaussian
Advantage: computational efficiencyDisadvantage: assumptions too restrictive
Multi-hypothesis Tracking
• Belief is a mixture of Gaussian
• Tracking each Gaussian hypothesis using a Kalman filter
• Deciding weights on the basis of how well the hypothesis predict the sensor measurements
• Advantage: – can represent multimodal Gaussian
• Disadvantage:– Computationally expensive– Difficult to decide on hypotheses
( ) ~ ( : , )i i it t t t t
i
Bel x w N x
Grid-based Approaches
• Using discrete, piecewise constant representations of the belief
• Tessellate the environment into small patches, with each patch containing the belief of object in it
• Advantage:– Able to represent arbitrary distributions over the
discrete state space
• Disadvantage– Computational and space complexity required to
keep the position grid in memory and update it
Topological approaches
• A graph representing the state space– node representing object’s location (e.g.
a room)– edge representing the connectivity (e.g.
hallway)• Advantage
– Efficiency, because state space is small • Disadvantage
– Coarseness of representation
Particle filters
• Also known as Sequential Monte Carlo Methods
• Representing belief by sets of samples or particles
• are nonnegative weights called importance factors
• Updating procedure is sequential importance sampling with re-sampling
( ) ~ { , | 1,..., }i it t t tBel x S x w i n
itw
Example 2: Particle Filter
Step 0: initialization
Each particle has the same weight
Step 1: updating weights. Weights are proportional to p(z|x)
Example 2: Particle Filter
Particles are more concentrated in the region where the person is more likely to be
Step 3: updating weights. Weights are proportional to p(z|x)
Step 4: predicting.
Predict the new locations of particles.
Step 2: predicting.
Predict the new locations of particles.
Compare Particle Filter with Bayes Filter with Known Distribution
Example 1
Example 2
Example 1
Example 2
Predicting
Updating
Comments on Particle Filters
• Advantage:– Able to represent arbitrary density– Converging to true posterior even for non-
Gaussian and nonlinear system– Efficient in the sense that particles tend to
focus on regions with high probability
• Disadvantage– Worst-case complexity grows exponentially
in the dimensions
ComparisonKalman Multihypothesi
s TrackingGrid Topolog
yParticle
Belief Unimodal
Multimodal Discrete
Discrete Discrete
Accuracy + + 0 - +Robustness
0 + + + +
Sensor Variety
- - + 0 +
Efficiency + 0 - 0 0Implementation
0 - 0 0 +
+ : good; 0 : neutral; - : weak
• Particle Filters (unconstrained)• Particle Filters (constrained)• Combination of Particle Filters and
Kalman Filters
Example Applications
Sensors
• Ultra sound and infrared Sensors:– Less accurate but certain with identification
– Laser range finder– Accurate but anonymous
Example Indoor Environment
Red circles: ultra-sound ID sensors
Blue squares: infrared ID sensors
Using Particle Filters (unconstrained)
• Due to high noise level of ultrasound and infrared sensors, we use particle filters
• Whenever detect the person, updating particles
Using Particle Filters (unconstrained)Another Example
Using Particle Filters (unconstrained)Another Example
Using Particle Filters (constrained)A more efficient way to use particle filters
• constraining the state space to locations on a Voronoi graph (a structure similar to a skeleton of an environment’s free space)
Combine Particle and Kalman FiltersTo Solve Data Association Problem
Area covered by ID sensors
Data Association Problem
In area 3 and 4, identities of A and B are known
In area 5 and 6, resolving ambiguity, but need additional hypotheses
Laser range finder
• Track individual people using Kalman filters (using laser range data)
• A particle filter maintains multiple hypothesis wrt identities of people
Combine Particle and Kalman FiltersTo Solve Data Association Problem
Conclusion
• “The Location Stack”: a general framework with publicly available implementation
• Probabilistic techniques have tremendous potential for inference problems
Questions?