Information Content of Ultrawideband Scattered Fields in Random Media: a Bayesian Approach via Kullb

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Information Content of Ultrawideband Scattered
Fields in Random Media a Bayesian Approach via
Kullback-Leibler Distances
by

• Burkay Donderici, Ph.D. Student
• Fernando Teixeira, Advisor

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Background

• M.S.
• FDTD method
• FDTD method extensions (such as subgridding)
• ADI-FDTD method
• Ph.D.
• Some more extensions to FDTD
• Inverse Problems
• Detection and Estimation

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Inverse Problems
Inverse Problem
Forward Problem
t1
t1
transmitted signal
?
transmitted signal
tranciever
tranciever
scatterer
scatterer
t2gtt1
t2gtt1
?
?
tranciever
tranciever
scatterer
scatterer
received signal
received signal
received signal
Problem Parametrization
Forward Solution (FDTD, FEM, MoM, etc)
Problem Parametrization
received signal
Inverse Solution (ill-posed, regularization, optim
ization)
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Deterministic vs Statistic Approaches
Deterministic Approach
Deterministic (point in received signal space)
Deterministic (function)
Deterministic (point in parameter space)
Deterministic (point in received signal space)
Deterministic (set of points in parameter space)
Deterministic (ill-posed function)
Statistical Approach
Statistic (distribution in received signal space)
Statistic (distribution in received signal space)
Deterministic (function)
Statistic (distribution in parameter space)
Statistic (distribution in parameter space)
Deterministic (ill-posed function)
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Bayes Theorem
FOCUS Statistical approach to the inverse
problem
is not available and cannot be computed directly
from .
BAYES THEOREM Write in terms
of , which can be calculated by
using the relation .
Probability of having a certain specific
parameter set PRIOR to receiving the signal data.
Probability of having a certain parameter
set given a specific signal data.
Probability of having a certain signal data given
a specific parameter set.
a-priori pdf
a-posteriori pdf
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Bayes Theorem (Example 1)
QUESTION
The probability that a PhD student graduates in
more than five years given he has good/bad
relations with the advisor is 0.3/0.9
respectively. If an old student came by and said
that he graduated in more than five years, what
is the probability that he had bad relations with
the advisor? (Assume probability of having
good/bad relations with advisors in general as
0.5/0.5)
Use Bayes Theorem
ANSWER
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Bayes Theorem (Example 2) page 1
PROBLEM
Given an image, find the probability of having
symbol A or B in the image. (Assume
probability of having A/B as 0.5/0.5)
Use Bayes Theorem
ANSWER
not given
not given, assume
Multivariate Gaussian model
and find model parameters using image samples
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Bayes Theorem (Example 2) page 2
To be used in the calculation of
To be used in the calculation of
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Bayes Theorem (Example 2) page 3
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Bayes Theorem (Example 2) page 4
NOW ready to apply Bayes Theorem for the images
given below.
B detected with very high certainty
A detected with very high certainty
Unsure about the weird letter
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Bayes Theorem (Example 2) page 5
SUMMARY
- Before we move on to our application, need a
measure to quantize uncertainty.
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Entropy
COMMENT Amount of certainty/uncertainty in the
results can be identified by inspecting the
probability distribution. But, how to quantify
mathematically?
A detected with very high certainty
B detected with very high certainty
Unsure about the weird letter
ANSWER Entropy (is the amount of uncertainty in
a probability density function)
A-symbol 0.43
B-symbol 0.70
weird-symbol 0.99
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Differential Entropy
NOTE We may already have some certainty even
before the measurement (Ex. radar
application).
BEFORE THE MEASUREMENT
AFTER THE MEASUREMENT
Ex.
Target present 0.0001
Target present 0.4352
Target not present 0.9999
Target not present 0.5648
NOTE Interested in the change in the
uncertainty, rather than the absolute value after
the measurement.
SOLUTION Differential Entropy. Although many
different definitions are present, most
popular is the Kullback-Leibler
(KL) distance given for a continuous pdf
in a single experiment
averaged over experiments
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Problem Setup
PROBLEM SETUP 2-D Half space with air above,
soil below
Scatterer Model (Plastic Mine)
(size 0.11 x 0.05 m 11 by 5 grids)
Soil Model (Sandy Soil)
Antenna Model Soft Magnetic Point Source
with UWB excitation (Blackman-Harris 1st
derivative with central frequency equal to
481 Mhz. NOTE Since the problem is UWB, it
will be treated in time-domain.
a s log-normal distribution exponent
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Transfer Matrix
DATA ACQUISITION
FULL-MATRIX MODE
tranciever 1
tranciever 2
tranciever 3
transmitted signal
reflected signal
ILLUSTRATION for DATA in TIME
M t,c
DIAGONAL-MATRIX MODE
response for channel 2
response for channel 1
t
Variable to use in the Bayes theorem (Analogous
to the symbol images in Ex.2)
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Solution to the Inverse Problem
- As a reminder
Forward Problem
Inverse Problem
Transfer Matrix
Transfer Matrix
FDTD
X
Random Media
Target Location
Target Location
Random Media
quantity that we are after
- Applying the Bayes Theorem to this problem gives
a-priori distribution of target location
a-posteriori distribution of target location
- can be calculated
from Ns samples of by
assuming a
Multivariate Gaussian model on . Same
approach that we have
followed for the A/B symbols problem.
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Dimensionality Problem
PROBLEM
- We need Ns samples of
for each Nk candidate target locations. Since
evaluation of the above equation once requires a
FDTD run, total number of FDTD runs for
modelling the probability density at all
locations is Ns x Nk.
- If we choose FDTD grid centers as candidate
target locations, a practical value for Nk is
100 x 100 10000. If Ns500 samples are used at
each location, total number of FDTD runs
becomes 5.000.000.
SOLUTION
- If the target is a electrically small PEC or
dielectric, then transfer matrix for different
target locations can be calculated without any
FDTD runs, if we have the greens function at
the tranciever locations (can be obtained by
FDTD).
- Electrically small PECs or dielectrics can be
replaced by a current source.
cylindrical PEC object with radius a
Dielectric object with permittivity et
- Greens function carries out the effect of the
effective current source to the tranciever
locations.
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Estimator Accuracy Tests 1
Single Tranciever
Multiple Trancievers
- Plots show the average a-posteriori distribution
- Each point is a possible target location
NOTE Target location is chosen WLG as the mass
center of the mine.
Low Soil Randomness
- Color at each point shows the probability of
having the target at that location.
variance 25
variance 25
- Cross shows the target location of the test
data.
- Star shows the location of the trancievers.
High Soil Randomness
- Information gain of each case is shown below.
variance 7.5
variance 15
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Estimator Accuracy Tests 2
- Two more tests, before moving on.
GAUSSIANITY The probability distribution was
calculated by assuming a multi-variate Gaussian
distribution. Its hard to test for MVG. Test for
G on marginals.
CONVERGENCE The probability distribution was
calculated by using a number of samples of
. How many do we need for accurate
results?
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Tranciever Configuration Case
Information Gain with tranciever count and array
width
Information Gain with target depth and array width
yt0.10m
Tr3
Full-Matrix
Diag.-Matrix
Water Content 1
Water Content 2
- Increases with the array width first, then
decreases. - Increases with Tr. - Higher for FM
mode.
- Decrease with increased target depth - Decrease
is more pronounced for higher water content. -
Smaller array width performs better.
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Soil Moisture Effect
- Increases first, then decreases. - Increase is
due to increased contrast of the plastic mine to
the soil. - Decrease is due to increase
randomness in the soil parameters. - Idea
previously used by J. Jenwatanavet and J. T.
Johnson An Analytical Model for Studies of Soil
Modification Effects on Ground Penetrating
Radar, IEEE TAP, June 2001. - Full matrix gives
higher information gain.
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Optimization of Tranciever Locations
- So far, we have been conducting case
studies. - Can also look for optimum sensor
configurations. - Since the dimensions of the
optimization problem is low, do regular
search. - At each step move one of the
trancievers and see if it improves the
information gain. If improves, keep it.
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Conclusion
- Forward vs Inverse problems - Deterministic
vs Statistical inverse problems - Bayes theorem
to solve statistical inverse problems. -
Entropy and Differential Entropy. -
Application to detection of a mine in random
medium. - Dimensionality problem and solution in
our application. - Estimator Accuracy tests. -
Case Studies - Optimization