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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?