Center for Subsurface Sensing and Imaging Systems (CenSSIS)
A National Science Foundation Engineering Research Center
Research and Industrial Collaboration ConferenceNovember 13-15, 2000
This work was supported in part by the Engineering Research Center Program of the National ScienceFoundation under award number EEC-9986821.
Fundamentals of Underground Object Detection
Eric L. Miller, Northeastern University
Fundamentals of Signal Processing for Subsurface
Sensing
Eric MillerNortheastern University
Center for Subsurface Sensing and Imaging Systems (CenSSIS)
A National Science Foundation Engineering Research Center
Research and Industrial Collaboration ConferenceNovember 13-15, 2000
This work was supported in part by the Engineering Research Center Program of the National ScienceFoundation under award number EEC-9986821.
Fundamentals of Underground Object Detection
Eric L. Miller, Northeastern University
Outline• An overview of the problems
– Object detection– Object characterization
• A statistical framework – Models– Algorithms– Performance analyses
• Object detection methods• Object characterization
The Problem Defined
• Sensors (EMI, GPR, NQR, …) collect data as a function of:– Space– Time– Frequency
• Goal: use all sensor data to extract “information” about the subsurface object
Sensors
Object
What is “information?”• Here we look at two basic processing objectives• Detection: Given data, is there something there?• Characterization: Given data (and maybe a
detection) what is the structure of the thing?• Structure:
– An image– Geometric characteristics: size, shape, number of
anomalies– Classification
A Statistical Framework for Processing
• Naturally accommodates stochastic models for noise and clutter sources inherent in any sensing problem
• Easily incorporates physical models for the sensors
• Leads to well characterized processing methods (detection and estimation theory)
• Provides for quantitative performance analysis methods useful for e.g. offline analysis and sensor optimization
A Model for All Seasons
( ),y f x c n= +
Vector containingall collected data
Physical signalmodel
Additive sensor noise
Parameter vectordescribing the object(s)of interest
Parameter vectordescribing “clutter” sources
The GPR Problem
Earth
Target ofinterest
Air
Plane EM wave input:• Frequency diversity• Angle diversity
Receiver array
The GPR Model Defined• Object: A localized change in the Earth’s
electrical properties (conductivity and permittivity)
• Clutter: That part of the data arising from electrical properties of the Earth that are not the object– Roughness of the interface or soil inhomogeneities
• Model: The physics here is Maxwell’s equations which relate input fields to the Earth’s electrical parameters to the measured data
Model y=f(x,c)+n for GPR
• y = data from all sources at all receivers• x = pixels of subsurface, descriptors of the
mine, …• c = parameters describing volumetric
inhomogeneities, surface roughness• f = Maxwell’s equations that map electrical
properties into observed data• n = additive sensor noise (white and Gaussian)
The EMI Problem
• ynk = datum at sensor position n and frequency k• cnk = “clutter” arising from interaction of fields
with the air-Earth interface• For this problem, clutter is well modeled as an
additive disturbance• Modeling choices: use physics to describe c or
consider a simpler, ad hoc statistical model
, , ,T T
n k n k n n k n ky g R R f w c= Λ + +
Transmitter Receiver
Object
Clutter data from GEM3: 1kHzIn phase Quadrature
Data
Fitted Model
A Clutter Model
• Clutter basically smooth as a function of position• Model it using a polynomial regression• Deal with unknown coefficients later in the
processing.
( ) ,, 1
,j
Np q
k i j pq k ip q
c x y x yα=
= ∑
Model y=f(x,c) + n for EMI
• y = data at all positions and all frequencies• x = mine location (x0, y0, z0), mine
orientation (3 Euler angles) and mine dipole response (3 poles)
• c = vector of unknown clutter expansion coefficients
• n = additive sensor noise
The Detection Problem• Simplifying assumptions
– No clutter– The signal s = f(x) is known a priori– The noise, n, is Gaussian with zero mean and
covariance matrix R:
• The problem: Determine which is true– H0: y = n– H1: y = s+n
( )0,n N R:
The Detection Solution• So-called likelihood ratio test (LRT)• Statistical model for data
• Form log-likelihood ratio
• If Λ > threshold then say H1 else say H0
( ) ( )( ) ( )
0
0
| 0,
| ,
p y H N R
p y H N s R
:
:
( ) ( )( )
0
1
|log
|p y H
yp y H
Λ =
For Our Problem• After a bit of algebra, the test reduces to
• Known as a matched filter. – How much does the covariance adjusted data
resemble the signal
1Ts R y τ−declare H1
declare H0
Detection Performance Analysis• Probability of detection:
PD=Prob[declare H1 given H1 is true]
• Probability of false alarmPFA = Prob[declare H1 given H0 is true]
• Plotted on a graph called receiver operating characteristic (ROC) as a parametric function of τ
1
1
PD
PFA
τ small
τ→∞
What about those assumptions?• If there is clutter or if the signal is not totally
known– Develop statistical models for unknowns and use a
likelihood ratio that “averages” them out– Estimate unknowns and use in LRT– In special cases, can use alternate tests which are
invariant to the unknowns
• Non-Gaussian noise– Still use LRT, but does not take on matched filter form– Different analytical expression for Λ
Object Characterization• Characterization = statistical estimation• Determine x as best we can from y and statistics of
the noise• Assumptions
– No clutter– Gaussian noise as before– Say f(x) =Ax where A is matrix obtained by linearizing
the exact, nonlinear physical model about some reference x0
– Say we have a Gaussian prior model for how x behaves
( ),xx N Qµ:
Estimation Options• A couple of ways to determine x• Bayesian estimation (for squared error cost)
• Maximum a posteriori estimation
[ ] ( )ˆ | |Bx E x y Y x p x y dy= = = ∫
( ) ( )
[ ]
ˆ arg max |
arg max ln ( | ) ln ( )
MAPx
x
x y p x y
p y x p x
=
= +
For Our Problem• Under our assumptions, the two methods are equivalent
and the result is
• Increase R� Increase noise power � Decrease effects of data in estimate� Revert to prior guess: µx
• Increase Q�Increase uncertainty in x�Increase effects of data�For Q®¥, (or Q-1®0) obtain Maximum Likelihood
estimate i.e. ML = MAP with no prior information
( ) ( ) ( )11 1 1ˆ T TMAP x xx y A R A Q A R yµ µ
−− − −= + + −
Performance Analysis• Quantities used to judge and estimator• Bias: On average, how close is our estimate
to the true x• Error covariance: What is the “spread” of
the estimate– Smaller the spread implies more stable the
estimate• Often bias-variance tradeoff
Performance Analysis (cont)• For many problems, can be impossible to
explicitly find error covariance• Exists useful, computable lower bound indicating
the best MSE performance any unbiased estimator can ever hope to achieve
• Called the Cramér-Rao lower bound (CRLB)• For Gaussian models CRB=error covariance• As we lift the assumptions, do more work to find
CRB
Lifting the Assumptions• Clutter present:
– Estimate clutter parameters along with x– If possible, develop stochastic model for clutter and treat as an
“extra” noise source in Bayes/MAP procedures• Non-Gaussian statistics for noise or prior
– Generally Gaussian = least square optimization problem for determining estimate
– Non-Gaussian leads to other type of optimization problem. More of a practical headache (for MAP) than a conceptual one. Bayes gets more complicated.
• Non-linear forward model: same as above. Complicates the procedure for determining the estimate
EMI Object Characterization
, , ,T
n k n k n n k n k
Tk k
y g M f w c
M R R
∝ + +
= Λ
Datum atlocation nfrequency k
• 3-vectors of field distributions• Depend only on object position
Rotation matrix describing object orientation via 3angles
Diagonal magnetizationtensor holding dipolemoment spectra
AWGN
clutter
Overview of Algorithm
k k k ky B wµ= +
Rewrite model (ignore clutter for now)
Depends only on mine position, r0
• Unique elements of Mk
• Depends on λ and orientation
Strategy:1. Estimate µ’s and r0 first2. Use estimates of M’s to determine λ’s and orientation
Motivation:• Two small nonlinear problems solved faster than one
big one
Step One: Details
( )0 0ˆ ˆ, arg miny
k Rr y B rµ µ= −
( )( )( )( ) ( )
211 1
02
11 1
0
ˆ argmin
ˆ ˆfor
y y
y y
T T
T T
r I B B R B B R y
I B B R B B R y B B rµ
−− −
−− −
= −
= − =
Solution 3 parameternonlinear leastsquare problem
High dimensionallinear least squaressolution
Formulation
Step 2: Details
• In theory, one rotation matrix diagonalizes all Mk
• In practice: will not exactly diagonalize so use penalty (regularizer) to– Discourage off diagonal elements in Λk
– Encourage smoothness across frequencies in diagonal elements of Λk
• Same optimization structure as Step 1– Low dimensional non-linear least squares, θ– High dimensional linear least squares, Λk
( ) ( )ˆˆ ˆ, argmin penaltyTk k k Fk
R M Rθ θ θΛ = − Λ +∑
Calibration region
A Clutter Model
• Collect “calibration data” in region 0 and on boundary of region 1.
• Use both sets of data plus correction model linking the α’sto extrapolate clutter structure inside RUI.
• Extends current practice of subtracting RUI boundary data.
Region underinvestigation
Clutter 0 Clutter 1
Correlation model
( ) ,, 0,1j
p qk i j pq k ic x y x y kα= =∑
More Clutter Model• Vectorize clutter model
• Introduce correlation model
• To arrive at
0 0 0 0 1 1 1 1c X n c X nα α= + = +
1 0 2nα α= +
( )
00 0 0
1 11 1
2
1
00 0
or
0, c
nc X I X
nc X I
n
c D En c N R
α
α
= +
= + :
Complete statisticalcharacterization of clutter as a function ofunknown expansion coefficients over RUI
Clutter mitigation• Obtain linear least squares estimate of α1 based on
c0 and c1 on boundary and subtract from data
• Complete statistical model for cleaned data including effects of mitigation errors
{1
afteralgebra
1 1 1 1
ˆ
ˆ ˆc
y y X s c c w s Mc wα= − = + − + = + +
( )2, Tc wy N s MR M Iσ+:
Real Data Processing: NU Test Track
• Further look at clutter• Object estimation &
discrimination• Targets: VS50, Val 69,
Al sphere, striker• 5x7 grid, 3” on square
3”
3”
Localization resultsTruth Estimate
x (in) 0.00 0.87y (in) 0.00 1.01z (in) 8.00 8.65x (in) 0.00 0.72y (in) 0.00 0.33z (in) 9.00 9.05x (in) 6.50 7.91y (in) 0.00 0.29z (in) 9.00 9.88x (in) -6.50 -5.00y (in) 0.00 0.02z (in) 9.00 8.76
Striker
Sphere
VS 50
Val 69
Moment Estimates: Sphere
Real
Imag
Frequency (Hz)
Moment Estimates: Striker
Real
Imag
Frequency (Hz)
Orientation 1Orientation 2
Moment Estimates: VS50
Real
Imag
Frequency (Hz)
Moment Estimates: Val 69
Real
Imag
Frequency (Hz)
Classification results: 0 dB SCR Estimate
Al Sphere Striker VS50 VAL69
Al Sphere 72 % 7 21 7
Striker 2 90 6 2
VS50 15 4 81 0 Tru
th
Val 69 0 4 0 96
Summary• Presented an overview of statistical methods for
subsurface signal processing• Hypothesis testing for detection and estimation
methods for characterization• Advantages of a statistical aproach in terms of
physical models, description of clutter, and performance analyses
• Estimation example from EMI problem• Beyond this: more details, statistical extensions,
other processing methods (fuzzy, neural, …)