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, …)