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Remote Estimation of the Temperature and Size. This is part 3.
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Lecture 13: Remote Estimation of the Temperature and Size of an Unresolved Object in
Space
Part III: Bayesian Problem Formulation
Up-Front Simplification 1
1 1 1 1 2
22 2 2 2 2
;
;
Jr s n s
rJ
r s n sr
= + =
= + =
Recall the measurement model
Since the range r is known to the sensor, one can alterna6vely consider formula6ng the problem in terms of the radiances of the source. Simply mul6ply both of the above equa6ons by r2 to obtain: ' 21 1 1 1 1
' 22 2 2 2 2
( , )
( , )
r J r n J T A m
r J r n J T A m
ε
ε
= + = +
= + = +
Since n1 and n2 are both distributed N(0,σ2), mul6plying them by the constant r2 means that m1~N(0,r2σ2) and m2~N(0,r2σ2)
Up-Front Simplification
Prac6cally, this simply means that one mul6plies sensor measurements by r2 before proceeding which is what is now assumed from here forward.
' 21 1 1 1 1
' 22 2 2 2 2
( , )
( , )
r J r n J T A m
r J r n J T A m
ε
ε
= + = +
= + = +
This transforma6on simplifies the problem by working directly in target radiance space vs. the sensor measurements (irradiances):
Bayes’ Theorem Now re-‐assign the variables r1 and r2 to now correspond to the transformed noisy sensor measurements. The Bayesian determina6on of object temperature and emissive-‐area is given by:
1 21 2
1 2
1 2
1 2
( , | , )( , | , ) ( , )
( , )( , ) ( , | , )
( , ) ( , | , )
p r r T Ap T A r r p T A
p r rp T A p r r T A
p T A p r r T A d AdT
εε ε
ε ε
ε ε ε
=
=∫
Bayes’ Theorem
1 21 2
1 2
1 2
1 2
( , | , )( , | , ) ( , )
( , )( , ) ( , | , )
( , ) ( , | , )
p r r T Ap T A r r p T A
p r rp T A p r r T A
p T A p r r T A d AdT
εε ε
ε ε
ε ε ε
=
=∫
Note that the integral normaliza6on factor in the denominator can be evaluated given knowledge of the prior probability and likelihood func6on in the numerator. We will evaluate the normaliza6on factor using Monte Carlo Integra6on.
A Priori Probabilities
( ) ( )1 1
( , )u l u l
p TT T A A
εε ε
=− −
g
For sake of both simplicity and our state of ignorance (uncertainty) of the object temperature T and emissive-‐area εA are both uniform and given by
where Tu and Tl are the upper and lower limits on possible temperatures T and emissive-‐areas εAl and εAu .
Likelihood Probability
1 2 1 2( , | , ) ( | , ) ( | , )p r r T p r T p r Tε ε ε=
The likelihood or measurement probability describes the probability distribu6on of measurements r1 and r2 condi6oned on a given state-‐of-‐nature which in this case is the object emissive-‐area εA and temperature T. This probability is dominated by the nature of the noise effect of the non-‐ideal sensor. In this case this probability is given by What assump6on was made that jus6fies the factoriza6on above?
Likelihood Probabilities
( )222
( , )1 1( | , ) exp
22i i
i
r J T Ap r T A
εε
σπσ
⎡ ⎤−= −⎢ ⎥
⎢ ⎥⎣ ⎦
J
The noisey radiance measurement are Gaussian with mean J about which fluctua6ons occur. Note the dependency of Ji on temperature T and emissive-‐area εA which in turn are determined by the a priori probabili6es on these parameters.
Complete Bayesian Formulation
[ ] [ ]( ) ( )2 2
1 1 2 21 2 2 2 2
1 2
( , ) ( , )1 1 1 1 1 1( , | , ) exp exp
( , ) 2 2 2u l u l
r J T A r J T Ap T A r r
T T A A p r rε ε
εε ε πσ σ σ
⎡ ⎤ ⎡ ⎤− −= − −⎢ ⎥ ⎢ ⎥
− − ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
The complete Bayesian formula6on for es6ma6ng temperature and emissive-‐areas from noisey measurements is given below.
Remember that the MAP es6mator means finding the T and εA that maximize the a posteriori probability p(T,εA) or equivalent, log p(T,εA). Taking the logarithm of the above and elimina6ng terms that are no func6onally dependent on T and εA means solving
( ) ( )2 21 1 2 2
1 2 2 2,
( , ) ( , )1 1argmax log ( , )
2 2T A
r J T A r J T Ap r r
ε
ε ε
σ σ
⎡ ⎤− −− − −⎢ ⎥⎢ ⎥⎣ ⎦
Part IV: Homework and MATLAB Code Overview
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