20111029 digital instrumentation 6 v1

8/2/2019 20111029 Digital Instrumentation 6 v1

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Janaka Wijayakulasooriya, Department of Electrical and Electronic Engineering, University of Peradeniya


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Janaka Wijayakulasooriya

PhD, MIEEE

Senior Lecturer

Department of Electrical and

Electronic Engineering

University of Peradeniya

[email protected]

Sensor Fusion

EE561

Are you ready for a 2nd Opinion ?


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Can you see anything ?


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With Thermal Sensor Fusion


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A new dimension to the vision


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You are no longer safe behind Bushes


7/41 Janaka Wijayakulasooriya, Department of Electrical and Electronic Engineering, University of Peradeniya

Sensor Fusion: Definition

Sensor Fusion is the combining of sensory

data or data derived from sensory data suchthat the resulting information is in some

sense (eg: accuracy, robustness) better than

would be possible when these sources were

used individually



Sensor Fusion Models: Complementary Type

sensors do not depend on each other

directly Can be combined to establish a more

complete picture of the phenomenon being

observed and hence the sensor datasets

would be complete

Example:

the use of multiple cameras each observing

different parts of a room four radars around a geographical region would

provide a complete picture of the area

surrounding the region

Fusion algorithm can be simply appending



Sensor Fusion Models: Competitive Type

Each sensor delivers independent measurements of

the same attribute or feature Fusion of the same type of data from different

sensors or the fusion of measurements from a single

sensor obtained at different instants is possible

Provide robustness and fault-tolerance becausecomparison with another competitive sensor can be

used to detect faults

Can provide a degraded level of service in the

presence of faults Competing sensors in this system do not necessarily

have to be identical



Sensor Fusion Models: Cooperative Type

Data provided by two independent sensors

are used to derive information that wouldnot be available from a single sensor

Eg: Stereoscopic vision system

By combining the two-dimensional (2D) images

from two cameras located at slightly different

angles of incidence (viewed from two image

planes), a three- dimensional (3D) image of the

observed scene can be determined

Cooperative sensor fusion is difficult to

design, and the resulting data will be

sensitive to the inaccuracies in all the

individual sensors.



Stereoscopic Camera



Levels of Fusion

Raw level Fusion

Directly the sensor outputs are combined

Eg: Averaging the temperature in a room

Feature level Fusion

Extract information from sensors before

combining

Eg: In fish classifier, features length and lightness

were extracted from the vision inputs

Decision/Action Level Fusion Fusion after taking the decisions based on

individual sensors



Fusion Methods

Probabilistic and statistical models:

Bayesian reasoning Evidence theory

Robust statistics

Recursive operators

Least-square (LS) and mean square methods:

Kalman Filter

Optimization

Regularization

Uncertainty ellipsoids

Heuristic methods

ANNs

Fuzzy logic

Approximate reasoning

Computer vision techniques



Raw data level fusion: Case Study

T ?T1 T2



Sensor Models

P(x|z)



Consider another sensor

P(x|z)



How can we combine ?

Let X = (1-W)z1+Wz2

E(Xhat) = ?

E() = ?



Combined Measurement

P(x|z1,z2)



Example



Which Feature(s) ?

Using length as a feature



Which feature(s) ?

Using Lightness as feature


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More features

Linearly separable decision boundary


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Overly complex decision boundaries

Over tuned decision boundary


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More complex decision boundaries

Optimized decision boundary


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Bayesian Decision Theory

Lets revisit the example of Sea Bass and

Salmon Let w represents the class:

w = w1 Sea Bass

w = w2 Salmon

Priori Probabilities P(w)

P(w1): The probability that next fish is Sea Bass

P(w2): The probability that next fish is Salmon

Reflect our prior knowledge of how likely weare to get a sea bass or salmon before the fish

actually appears

Can we make a decision only based on priori ?


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Improving the simple classifier

Can we use P(w1) > P(w2) or P(w1) < P(w2)

to decide next fish is Sea Bass or Salmon ? What if we have to predict many fish ?

Suppose, in addition to the priori we have:

Measurement vector (Feature vector)x={x1xN} on the subject

(Example: x1 = lightness, x2 = length)

Class-conditional probability density function

P(x|w)

Example: P(x2|w1) probability distribution of the

length of the fish, given that it is a Sea Bass

Bayes theorem can be used to calculate the

posterior probabilities P(w|x)


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Bayes Formula

How to find P(x) ?

Informally this can be expressed in English as


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Bayes Classifier

Bayes formula shows that by observing the

value of x we can convert the priorprobability P(j) to the a posteriori

probability (or posterior) probability P(j |x)

For the purpose of classification, what is

important is:

Likelihood

Priori

Evidence p(x) can be considered as just ascaling factor which make sure the sum of

each individual probabilities = 1


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Class-conditional PDF of the two classes

Normalized PDF = area under curve is 1


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Posteriors

Let priors P(w1)=1/3 and P(w2)=2/3


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Classification based on Posteriors

Now, it is sensible to set the decision

boundary as If P(w1|x) > P(w2|x) select class w1

If P(w1|x) < P(w2|x) Select class w2

In order to justify this decision, we have to

calculate the probability of error, whenever

we make a decision based on the

measurement x and priories


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Minimizing probability of error

The average probability of error is:

So, if P(error|x) can be minimized for eachindividual decision, P(error) can be

minimized

So, if we apply the Bayes decision rule then

the P(error|x) = min[ P(w1|x), P(w2|x) ]


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Bayes Decision Rule: elimination of p(x)

As, p(x) [ evidence ] in the Bayes just ensures

that P(w1|x) + P(w2|x) = 1, we can eliminateit from the decision rule

Hence, the decision can be made only based

on:

p(x|w1).P(w1) > p(x|w2). P(w2) Decide w1

p(x|w1).P(w1) < p(x|w2). P(w2) Decide w2

Special cases:

When p(x|w1) = p(x|w2) decision is onlybased on priories

When P(w1)=P(w2) decision is only based on

likelihood


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Generalization of Bayes Classifier

Consider what happens if:

More than one feature Feature vector

Feature space

More than two states of nature

Many classes

Allow other actions

Eg: Rejection

Introducing loss function more general than the

probability of error


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Fusion at Decision Level: Bayes Reasoning


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Example

Consider an air surveillance detector, which

can have 3 states (x):

Suppose a single sensor observes x and

return 3 values


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Sensor Model: Likelihood Matrix

Sensor can be modeled in the form of a

likelihood matrix This gives P(z|x)

Consider P(z|x) for two sensors


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Posterior Probability


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[email protected]

20111029 digital instrumentation 6 v1

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