underwater target detection using multiple …...underwater target detection using multiple...

Underwater Target DetectionUsing Multiple Disparate Sonar

Platforms

Nick Klausner

Colorado State UniversityDepartment of Electrical and Computer Engineering

Advisor: Dr. Mahmood R. Azimi Sadjadi

Committee Members:Dr. Ali Pezeshki - ECE

Dr. Dan Cooley - Statistics

Master’s DefenseJuly 30, 2010

Klausner (CSU) Multi-Sonar Detection Master’s Defense 1 / 39

OutlineIntroduction

I MotivationsI Research Objectives

Detection ReviewI Gauss-Gauss Detection

Multi-Channel Coherence Analysis (MCA) DetectionI MCA ReviewI MCA-Based Gauss-Gauss Detection

Data Description and Experimental ResultsI Pre-ProcessingI Multi-Sonar Detection ResultsI Dual-Sonar Detection ResultsI Sensitivity Analysis Results

Gauss-Gauss Likelihood UpdatingI Full-Rank UpdatingI Reduced-Rank UpdatingI Simulation Studies

Concluding Remarks and Future Work


Problem Statement and MotivationsDetection of underwater targets from multiple disparate sources of information

Detection of underwater objects is complicated by various factors:I variations in operating and environmental conditionsI bottom features (sand formations, rough textures, vegetation) may obscure the target and confuse the detection

processI variations in target shape, composition, and orientationI lack of a priori knowledge of newly encountered target structures

In a surveillance area, there could be one or moreAutonomous Underwater Vehicles (AUVs) eachequipped with a wide variety of sensors

Final decision-making usually takes place at thecentral station using some type of decision-level orfeature-level fusion technique which will typicallylead to degraded, locally biased decisions resultingin poor performance at the fusion center

Analyzing the system as a whole by takingadvantage of the wealth of information supplied bythe disparate sensory systems will undoubtedlylead to better detection performance resulting indecisions with a higher degree of confidence


Research Objectives

To allow decision-making among multiple sonar platforms, it is essential todetect, isolate, and represent the coherent, or mutual information, among one ormultiple data sets

Research Objective: Develop and test an efficient and robust coherence-baseddetection system that can be applied to multiple sonar images

I Represent the set of observations in a coordinate system resulting from data processing algorithms that“discover” the linear relationships existing among the multiple sources of information

I Cast standard detection methods into the new coordinate system by reformulating the log-likelihood ratio andJ-divergence

I Test the proposed detection method on data captured from data acquisition systems deployed in the field andstudy its robustness to variables of disparity

I Theoretically analyze how adding an additional channel’s worth of data affects the standard detector


Gauss-Gauss Detection INeyman-Pearson lemma states that the most powerful test of size α (false alarm rate) is the likelihood ratio test

l(x) =pX|H1

(x|H1)

pX|H0(x|H0)

H1≷

H0η where P [l(x) > η|H0] = α

For this problem, we assume that realizations of our random vector, x ∈ Cn , are circular symmetric complex Gaussian

with zero mean and covariance R1 = EH1xxH under H1 versus R0 = EH0

xxH under H0

I We assume that our observations are proper, i.e. EH1xxT = EH0

xxT = O, if x = a + jb then

[ab

]∼ MVN

(0, 1/2

[Re R −Im RIm R Re R

])

I No specific structure is assumed for R0 and R1 with the only restriction that they be PD

Likelihood function for the ith hypothesis, i = 1, 2

pX|Hi(x|Hi) =

1πndetRi

e−xHR−1i x

Taking the logarithm of the likelihood ratio, we achieve the log-likelihood ratio

l(x) = ln

(pX|H1

(x|H1)

pX|H0(x|H0)

)= xH

(R−1

0 − R−11

)x

Observations are whitened through R−1/20 to produce y = R−1/2

0 x

l(y) = yH(

I − S−1)

y


Gauss-Gauss Detection IIS = R−1/2

0 R1R−H/20 is the signal-to-noise ratio matrix

Eigenvalue decomposition of this matrix, S = UΛUH , used to

produce z = UHR−1/20 x

l(z) = zH(

I −Λ−1)

z

J-divergence provides a tractable first-order characterization of theperformance of our detector (does not require a probability lawdescribing l(z) and is η-independent) by measuring the difference inthe means of l(z) under both hypotheses

J(S) = EH1l(z) − EH0

l(z)

= tr(−2I + S + S−1

)=

n∑i=1

(−2 + λi + λ−1

i

)

Rather than finding the best rank-p approximation of S in mean-squared sense (Eckart-Young Theorem, discard smallestλi’s), it is better to solve

maxrank(S)=p

J(S) ⇒ ensures we don’t inadvertently throw awayinformation that is important to the detector


Gauss-Gauss Detection IIIResort eigenvalues in a descending fashion such that

(λ1 + λ−1

1

)> · · · >

(λn + λ−1

n

)

U =[Up Up+1

]Λ =

[Λp OO Λp+1

]

Filter observations z = UHp R−1/2

0 x ∈ Cp , reduced-rank log-likelihood ratio and J-divergence

lp(z) = zH(

Ip −Λ−1p

)z

Jp(S) =

p∑i=1

(−2 + λi + λ−1

i

)

A lot of choices for choosing p, one possibility

min

q : Jq/J > 1 − ε

Special case of the “R0 vs. R1” test is a signal-plus-noise model

H1 : y = s + n

H0 : y = n

In this case, S = I + R−1/2nn RssR−H/2

nn leading to eigenvalues that are greater than one (pick largest λi’s)

Performing reduced-rank detection amounts to orthogonally projecting the data onto one-dimensional subspaces thathave high per-mode SNR


MCA Review IWe now assume that our observation is the composition of multiple measurements from N sources

z =[xH

1 · · · xHN

]H∈ Cd and z is still CN (0, Rzz)

Multi-Channel Coherence Analysis (MCA) searches for the ith multi-channel coordinate for the jth channel, vi,j = αHi,jxj ,

that satisfies the optimization problem

(αi,1 , . . . , αi,N

)= arg max

N∑j=1

N∑k=1

αHi,jRjkαi,k

s.t.N∑

j=1

αHi,jRjjαi,j = 1

with Rjk = RHkj = ExjxH

k

Letting ai =[αH

i,1 · · · αHi,N

]H, the solution is achieved by solving the generalized eigenvalue problem

Rzzai = λiDai

where λi =∑N

j=1∑N

k=1 Evi,jv∗i,k is the sum of the correlations of the mapped variates and D = blkdiag [R11 , . . . , RNN]

Solution can alternatively be solved using a standard evd

Epi = λipi

where E = D−1/2RzzD−H/2 is the multi-channel coherence matrix representing the covariance matrix of the whitenedvector w = D−1/2z and pi = DH/2ai


MCA Review IIMapping matrixΨj =

[p1,j p2,j · · · pd,j

](similar to F and G matrices in CCA) used to filter the observation from channel

j to its multi-channel coordinates µj =[v1,j v2,j · · · vd,j

]T= ΨH

j R−1/2jj xj

These MCA variates exhibit the following statistical properties

N∑j=1

EµjµHj = I

N∑j=1

N∑k=1

EµjµHk =Λ

Block diagonal matrix Ψ = blkdiag [Ψ1 ,Ψ2 , . . . ,ΨN] is used to resolve all N channels into their multi-channelcoordinates using

v =[µH

1 µH2 · · · µH

N

]H= ΨHD−1/2z


MCA Detection IFor this multi-channel problem, our detection hypothesis is that thepresence of target in multiple ROIs leads to a higher level ofcoherence than when those ROIs solely contain background

For the jth channel, we consider the signal-plus-noise model

H1 : xj = sj + nj

H0 : xj = nj

where EnjnHk = δj−kRnj , EsjsH

k = Rsjk , and EsjnHk = O for all

j, k = 1, . . . , N

Under H0 we have

EH0zzH = Rzz0 = D0 = blkdiag

[Rn1 , Rn2 , . . . , RnN

]Likewise, under H1 we have

EH1zzH = Rzz1 = Rs + D0

where [Rs]j,k = Rsjk and D1 = blkdiag[Rs11 + Rn1 , . . . , RsNN + RnN

]This leads to the eigenvalue decomposition under hypothesis H1

Rzz1 A1 = D1A1Λ1

resulting in the log-likelihood ratio

l(z) = zH(

R−10 − R−1

1

)z = zH

(D−1

0 − D−H/21 P1Λ1PH

1 D−1/21

)z


MCA Detection IIWe then whiten the data through the filter D1 producing the vector w = D−1/2

1 z

l(w) = wH(Σ− P1Λ

−11 PH

1

)w

where Σ = DH/21 D−1

0 D1/21 is a local SNR matrix with jth diagonal block Σj =

(Rsjj + Rnj

)H/2R−1

nj

(Rsjj + Rnj

)1/2

The whitened data is then mapped into theMCA coordinate system under H1

producing the vector v = PH1 D−1/2

1 z

l(v) = vH(

PH1 ΣP1 −Λ−1

1

)v

where v =[∑N

j=1 v1,j · · ·∑N

j=1 vd,j

]Tis a

vector of the sum of the MCA coordinatesunder H1

The J-divergence can then be written as

J = EH1l(v) − EH0

l(v)

=

d∑i=1

(−2 + pH

i

[λiΣ+ (λiΣ)−1

]pi

)


Data Description and Pre-Processing IGenerally, observations of targets are composed of two distinctstructures: highlight and shadow

These regions can depend on things like the type of target, range,aspect and grazing angles, etc. which can cause problems for mostmatched filter-based detection methods

Images are generated at the output of a coherent processor, thek-space or wavenumber beamformer, resulting in complex-valuedimages

Note that in situations where the local SNRs in each channel are verysmall but the coherence shared between pairs of channels issignificant for detection, we can approximate the log-likelihood ratioand J-divergence by the equations

l(v) ≈ v(

I −Λ−1)

v

J ≈d∑

i=1

(−2 + λi + λ−1

i

)Additionally, when performing low-rank detection we sort theeigenvalues of E in descending fashion such that λ1 > · · · > λd

P =[Pp Pp+1

]Λ =

[Λp OO Λp+1

]Filter the observation v =

[∑Nj=1 v1,j · · ·

∑Nj=1 vp,j

]= PH

p D−1/21 z

l(v) = v(

Ip −Λ−1p

)v

Jp =

p∑i=1

(−2 + λi + λ−1

i

)10 20 30 40 50 60 70

0

0.5

1

1.5

2Distribution of the Eigenvalues of S


Data Description and Pre-Processing II

To prepare the data for the MCA-baseddetector, each set of N images is partitionedinto coregistered ROIs with 50% overlap inboth the vertical and horizontal directions,72× 224 for HF and 24× 224 for BB

The set of N ROIs is then partitioned intonon-overlapping blocks of size 6× 4 for HFand 2× 8 for BB

Corresponding blocks form the compositeobservation vector z

An ensemble set is formed from all 336 blocks in each ROI and used to form a sample estimate of the compositecovariance matrix Rzz

Covariance estimate is then decomposed via MCA to form the log-likelihood ratio test statistic

Each observation from the ensemble set is applied to the LLRT to form a decision for that set of N blocks

If 50% or more of the blocks within an ROI set pass the LLRT, it is concluded that the ROI set contains a target

Forming detection decisions on individual blocks has several advantagesI Gives us multiple independent observations of the information contained in each ROI allowing us to make

detection decisions with much higher confidenceI Yields observations that sit in lower dimensional space which in turn facilitates the use of multiple sonar images

as the smaller the block sizes, the more sonar images we can add without processing an extremely highdimensional composite observation


Multi-Sonar Detection I

Data set consists of one HFside-scan sonar image as well asthree BB sonar images

Image database contains 59coregistered images with eachimage consisting of both port andstarboard-side images

Images are the direct result ofbeamforming (complex-valued)

2 4 6 8 10 12 14 161

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

i

λi

MCA Correlation Statistics for Target/Non-Target Set

Non-Target ROI

Target ROI

Database contains 53 targets with some images containing more than onetarget

HF sonar provides higher spatial resolution and better ability to capture targetdetails and characteristics while the BB sonar offers much better cluttersuppression ability with lower spatial resolution

Detectors were run with HF images along with 1-3 BB sonar imagesI Two Channel Detector: HF-BB1I Three Channel Detector: HF-BB1-BB2I Four Channel Detector: HF-BB1-BB2-BB3

2 4 6 8 10 12 14 161.2

1.4

1.6

1.8

2

2.2

i

λi


Non-Target ROI

Target ROI

2 4 6 8 10 12 14 161.4

1.6

1.8

2

2.2

2.4

2.6

2.8

i

λi


Non-Target ROI

Target ROI

Figures show the mean and standard deviation of the dominant 16 multi-channel correlations (λi , i = 1, . . . , 16)pertaining to all 53 target ROIs (212 in total) and 212 randomly selected background ROIs for all three detection cases

Confirms our hypothesis that there exists a higher amount of coherence among target ROIs compared to background ROIs


Multi-Sonar Detection IIDetector Targets Detected

(Out of 53 Targets)Average False

Detections per Image Knee-Point Pd Knee-Point Threshold

HF-BB1 51 7.48 96% 9.975HF-BB1-BB2 52 8.93 98% 11.725

HF-BB1-BB2-BB3 52 9.32 96% 10.825

When performing detection, only the dominant 16 multi-channelcorrelations are retained for all three cases

Using the same set of 212 target and background ROIs, a threshold of10.2 was experimentally determined for all three detection cases

Detectors perform well with Pd greater than 95% and less than 10false alarms per image

Targets missed by the detectors were faint in signature and hard tovisually discern in all images leading to low coherence

False alarms are generally attributed to dense clutter structureswhich visually present themselves in both the HF and BB images

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

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1

Probability of False Alarm

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Receiver Operating Characteristic Curves

HF-BB1

HF-BB1-BB

2

HF-BB1-BB

2-BB

3

Receiver-Operator Characteristic (ROC) curves generated from all 212 target ROIs and different randomly selected set of212 background ROIs (same background ROIs for each detector)

Lack of increase in performance when going from three to four sonar images seems to suggest that the point ofdiminishing return for this data set is a detector with three sonar images

This could be attributed to the fact that the three BB sonar images are bandpass filtered versions of the BB sonar data andtherefore essentially contain similar target information


Dual-Sonar Detection I

Data set consists of two images, one HF and one BB sonar image, has simplerclutter situations, and is real-valued, envelope data which is the result ofquantizing the magnitude of the beamformed images

Even if the data were truly CN to begin with, iid Normal random variablesresult in a Rayleigh distributed magnitude leading to a heavy-taileddistribution⇒More than likely there will exist a model mismatch

Performing detection with the magnitude also disregards the phaseinformation in the images

Image database contains over 1200 coregistered images with both port andstarboard-side images

Database contains 99 objects of interest which are further partitioned into 49target and 50 lobster trap objects

Figures display the mean and standard deviation of the dominant 16multi-channel correlations for all 99 objects (396 ROIs in total) and 396randomly selected background ROIs

Noticeable difference in the statistics of the features pertaining to targets andlobster traps

2 4 6 8 10 12 14 161

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

MCA Index i

λi

Statistics of MCA Correlations for Sample Target/Non-Target Set

Non-Target ROI

Target ROI

2 4 6 8 10 12 14 161

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

MCA Index i

λi

Statistics of MCA Correlations for Sample Lobster Trap/Non-Target Set

Non-Target ROI

Lobster Trap ROI


Dual-Sonar Detection IIDetector - Data Set Targets Detected

(Out of 49 Targets)Average False

Detections per Image Knee-Point Pd Knee-Point Threshold

MCA - Training N/A N/A 94% 0.5212CCA - Training N/A N/A 88% 231.8

MCA - Test 49 7.4 98% 0.6525CCA - Test 44 43.2 90% 232.1

A partial subset of images containing 50 objects of interest (25 targets, 25lobster traps) is extracted to determine the threshold

A ROC curve is generated from the 200 ROIs corresponding to the 50 objectsand 200 randomly selected background ROIs (same background ROIs for bothdetectors)⇒ Knee-point used to determine the threshold

Both the MCA and CCA-based methods are then applied to the remainingimages containing 49 objects of interest (24 targets, 25 lobster traps) and detectmore than 90% of the targets and lobster traps

Three of the five objects missed by the CCA-based detector were lobster traps

A ROC curve is generated from the 196 ROIs corresponding to the 49 objectsof interest and 196 randomly selected background ROIs (same backgroundROIs for both detectors)

At the knee-point of this ROC curve, the MCA-based detector misses a lobstertrap

Stark contrast in performance among the two methods may be attributed tothe simplistic assumptions made in developing the CCA-based detectionmethod

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.5

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MCA

CCA

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

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1

Probability of False AlarmP

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MCA

CCA


Sensitivity Analysis I

MCA-based detection method is applied to a data set consisting of (8-bit grayscale) snippets of simulated target andnon-target shapes embedded in synthetically generated background

Sonar snippets are generated with different resolutions, SNR values, ranges, and aspect angles

Only 1 and 3 in. resolutions are considered and SNR ranges from 0 to 15 dB in increments of 3 dB

Range values span from 10 to 120 m in increments of 1 m and aspect angle ranges from 0 to 360 in increments of 1

All non-target snippets are excluded and the subset of 1610 target snippets is further partitioned into three parts and used

to represent the H1 hypothesisI 138 cone-shape targetsI 736 cylinder-shape targetsI 736 trapezoid-shape targets

Snippets of background are used to represent the H0 hypothesis

Pre-processing the data to prepare it for the MCA-based detector is the same as before except there is no ROI partitioningof images and each snippet plays the role of an ROI

Each of the N snippets is partitioned into blocks of size dependent on the resolution


Sensitivity Analysis IITo mimic single platform, multiple sensory detection problems, a two-channel detectoris constructed where each channel consists of targets of the same type at the same range,aspect angle, and SNR

The two sonar snippets differ in resolution, one of high resolution (1 in.) and the other ofa lower resolution (3 in.)

A 4× 4 block size is used for high resolution snippets and a 2× 1 block size for those oflower resolution

The setup is run for all 1610 images at various ranges and aspect angles and the resultspartitioned on the basis of target type and SNR

Performance generally increases with increasing SNR for cylindrical and trapezoidalshaped targets

Lack of increase in performance for conical shaped targets may be attributed to clippingof the highlight

Target Type 0dB 3dB 6dB 9dB 12dB 15dBCone 91.30 % 94.93 % 96.38 % 89.86 % 81.16 % 87.68 %

Cylinder 83.70 % 85.19 % 82.20 % 85.33 % 89.67 % 94.02 %Trapezoid 84.24 % 84.51 % 84.65 % 85.73 % 90.08 % 93.75 %

0 0.1 0.2 0.3 0.4 0.5 0.60.4

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00 dB SNR

Cone Targets

Cylinder Targets

Trapezoid Targets

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06 dB SNR

Cone Targets

Cylinder Targets

Trapezoid Targets

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Cone Targets

Cylinder Targets

Trapezoid Targets

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Cone−Shape Targets

3 dB SNR

9 dB SNR

15 dB SNR

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Cylinder−Shape Targets

3 dB SNR

9 dB SNR

15 dB SNR

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Trapezoid−Shape Targets

3 dB SNR

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15 dB SNR


Sensitivity Analysis IIITo mimic multiple platform, single sensory detection problems, a two-channel detector is constructed where eachchannel consists of targets at the same resolution (1 in.), at ranges within±1 m of one another, and at an identical SNR of9 dB while the disparateness is in aspect angle separationThe two channels correspond to snippets of the same target at two aspect angles with separation θ such that ifφ1 andφ2 are their aspect angles then pairs of images are chosen such that |φ1 −φ2| ∈ [θ− δ,θ+ δ] where δwas chosen tobe 10

Block sizes of 4× 4 were used for both imagesImages of cone shaped targets were generated at the same aspect of 0 and excluded from the studyDetector remains fairly robust to disparateness in aspect angle separation as Pd at the knee-point never falls below 92%The J-divergence is also empirically estimated by measuring the difference in means of l(z) under H1 and H0 withexpectation taken over all pairs of images matching the aforementioned criteriaAgain, we can draw the same conclusion that the detector is robust to aspect separation as the difference betweenmaximum and minimum J-divergence never grows larger than 0.6

Target Type 0 30 60 90 120 150 180

Cylinder 94.40 % 96.14 % 97.83 % 98.33 % 97.47 % 96.82 % 94.92 %Trapezoid 92.86 % 96.29 % 97.21 % 97.66 % 97.57 % 96.09 % 93.14 %

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θ=0°

θ=30°

θ=90°

θ=150°

θ=180°

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θ=0°

θ=30°

θ=90°

θ=150°

θ=180°

0 30 60 90 120 150 1802.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

Aspect Angle SeparationE

mp

iric

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Div

erg

en

ce

Cylinder Targets

Trapezoid Targets


Full-Rank Updating IWish to take a closer look at what happens to the Gauss-Gauss detector when adding data

We assume we have measured the random vector zk ∈ Cm and form the likelihood ratio

l(zk) = zHk

(R−1

zkzk0− R−1

zkzk1

)zk

where Rzkzk0= EH0

zkzHk and Rzkzk1

= EH1zkzH

k

We then add the new observation xk+1 ∈ Cn to form the augmented measurement zk+1 =[zH

k xHk+1

]H

l(zk+1) = zHk+1

(R−1

zk+1zk+10− R−1

zk+1zk+11

)zk+1

where Rzk+1zk+10= EH0

zk+1zHk+1 and Rzk+1zk+11

= EH1zk+1zH

k+1

Structure inherent in the augmented observation implies the block matrix inversion identity

R−1zk+1zk+1

=

[R−1

zkzkO

O O

]+

[−WH

I

]Q−1 [ −W I

]

where W = Rxk+1zk R−1zkzk

and Q = Rxk+1xk+1 − Rxk+1zk R−1zkzk

RHxk+1zk

From here, it is easy to see that

∆l(zk+1 , zk) = l(zk+1) − l(zk) = eH0 Q−1

0 e0 − eH1 Q−1

1 e1


Full-Rank Updating II

where

e0 = xk+1 − W0zk

e1 = xk+1 − W1zk

W0 = Rxk+1zk0R−1

zkzk0

W1 = Rxk+1zk1R−1

zkzk1

Q0 = Rxk+1xk+10− Rxk+1zk0

R−1zkzk0

RHxk+1zk0

Q1 = Rxk+1xk+11− Rxk+1zk1

R−1zkzk1

RHxk+1zk1

The change in J-divergence then becomes

∆J(zk+1 , zk) = J(zk+1) − J(zk) = tr(−2I + Q−1

0 Q10 + Q−11 Q01

)where

Q10 = EH1

[e0eH

0

]= Rxk+1xk+11

− W0RHxk+1zk1

− Rxk+1zk1WH

0 + W0Rzkzk1WH

0

Q01 = EH0

[e1eH

1

]= Rxk+1xk+10

− W1RHxk+1zk0

− Rxk+1zk0WH

1 + W1Rzkzk0WH

1

As we are filtering with a sub-optimal smoother in such situations, we can make the following statements

xHQ10x > xHQ1x

xHQ01x > xHQ0x


Reduced-Rank Updating IIf the data we wish to add sits in a high dimensional space, we may wish toemploy low-rank methods to approximate the update

We begin by whitening the error vectors with the filter Q−1/20 , i.e.

w0 = Q−1/20 e0 and w1 = Q−1/2

0 e1

∆l(zk+1 , zk) = wH0 w0 − wH

1 Γ−1w1

where Γ = Q−1/20 Q1Q−H/2

0

We then take the evd of this matrix so that Γ = UΣUH , UHU = UUH = I andΣ = diag [σ1 , . . . ,σn], and form the new vectors y0 = UHw0 and y1 = UHw1

∆l(zk+1 , zk) = yH0 y0 − yH

1 Σ−1y1

Defining the two matrices

Γ10 = EH1w0wH

0 = Q−1/20 Q10Q−H/2

0

Γ01 = EH0w1wH

1 = Q−1/20 Q01Q−H/2

0

the change in J-divergence can be written as

∆J(zk+1 , zk) = tr(−2I + Γ10 + Γ−1Γ01

)=

n∑i=1

−2 + uHi

(Γ10 +σ−1

i Γ01

)ui


Reduced-Rank Updating IIUsing the two sub-optimal filtering statements, we can lower bound the change in J-divergence

∆J(zk+1 , zk) >n∑

i=1

−2 +σi +σ−1i > 0 ⇒ adding data can never have a negative impact on divergence

We want the best rank-p approximation of Γ that maximizes the change in J-divergence⇒ sort the coordinates such that

uH1

(Γ01 +σ−1

1 Γ10

)u1 > · · · > uH

n

(Γ01 +σ−1

n Γ10

)un

We would like to see σi small and uHi Γ10ui and uH

i Γ01ui largeI With respect to Q0 , σi measures the MSE in our estimate under H1 when using the right smoothing filter (W1),

natural that we would want this to be smallI Two quadratic terms measure the MSE when using the wrong smoothing filters in the one-dimensional subspace

spanned by uiI If we can generate data from one hypothesis and accurately estimate it with the smoother from the other then

that gives us some indication that there is no difference among models of the two hypotheses (or theirdistributions), thus the larger the MSE the better

Decompose the coordinate system

U =[Up Up+1

]Σ =

[Σp OO Σp+1

]Filter the error vectors yi = UH

p Q−1/20 ei

∆lp(zk+1 , zk) = yH0 y0 − yH

1 Σ−1p y1

∆Jp(zk+1 , zk) =

p∑i=1

−2 + uHi

(Γ10 +σ−1

i Γ01

)ui


Dynamical Structure Detection IWe wish to test whether or not an a priori model hasproduced the time series which we have measured,y[k]N−1

k=0

H1 : y[k] =

∞∑l=0

h[k − l]u[l] + n[k]

H0 : y[k] = n[k]

where Eu[k + l]u[k] = σ2uδ[l], En[k + l]n[k] = σ2

nδ[l],and Eu[k + l]n[k] = 0 ∀ l

We form the vector of measurements up to time k,zk = [y[0] · · · y[k]], and we wish to add the newmeasurement y[k + 1]

To update the likelihood ratio, we require the error inour measurement and the best estimate of themeasurement given observations up to time k (theinnovations sequence e[k + 1] = y[k + 1] − y[k + 1|k])and the variance of this error (σ2

ek+1)⇒ run a Kalman

filter, built from the state equation describing thedynamics of the system, along with the LLR updating toprovide this information for the H1 hypothesis

Log-likelihood ratio update

∆l(zk+1 , zk) = σ−2n y[k + 1]2 −σ−2

ek+1e[k + 1]2

Generated an arbitrary 4th-order, proper LTI system bychoosing random zero/pole locations with poles that lieinside the unit circle

A state error covariance matrix of P0|0 = 10I is alwaysused to initialize the system

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Ω

|H(ejΩ)|


Dynamical Structure Detection II

Log-likelihood ratio increases for bothhypotheses as time progresses but thedistance between values gets larger

We generate a finite sequence of data, y[k],k = 0, . . . , N − 1 with N = 300, from the H0hypothesis and run it through the timeupdating system

This is done 1500 times and a(time-dependent) threshold determinedcorresponding to a fixed false alarm rate of5%

The same is then done for the H1hypothesis and the Pd measured

Pd gets better the longer the system is runand the smaller the SNR, the longer thesystem must be run before we can say withany confidence that H1 is in force

Assumes that one hypothesis exclusivelygenerates the time series

Model is switched to the H0 hypothesisand subsequently switched back

Response looks the same up until themodel is switched where after ourconfidence in the H1 hypothesis begins todegrade

50 100 150 200 250 3000

50

100

150

200

250

300

k

l(z

k)

H0

H1:SNR = -5 dB

H1:SNR = 0 dB

H1:SNR = 5 dB

50 100 150 200 250 3000

50

100

150

200

250

300

k

l(z

k)

H1

H0

H1

H0

H1:SNR = -5 dB

H1:SNR = 0 dB

H1:SNR = 5 dB

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

k

PD

SNR = -5 dB

SNR = 0 dB

SNR = 5 dB

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

k

PD

H1

H0

H1

SNR = -5 dB

SNR = 0 dB

SNR = 5 dB


Multi-Array Detection IInterested in detecting the presence of a single,narrow-band source using multiple, 16-element uniformlinear arrays (ULAs) at half-wavelength spacing

We consider a situation where 20 platforms are orientedand traveling in the same along-track direction at thesame elevation

Platforms are uniformly spaced across a 40 m distancein cross-track and each platform is located 1 m behindthe platform to the left

We ignore the effects of direct path progagation and for

the kth observation from the lth array, consider thedetection problem

H1 : yl[k] = h(θl)αTl s[k] + nl[k]

H0 : yl[k] = nl[k]

where Enl[k]ni[k]H = δl−iσ2nI and

[h(θl)]i = e−j(i−1)π cosθl

αl =[(||rl|| + ||r1||)−1 · · · (||rl|| + ||rN ||)−1

]represents attenuation or fading

DOA θl estimated across a 750-observation windowusing MUSIC

||rl|| = ||rl|| + ε where ε ∼ N(

0, ||rl||(θl − θl

)2)With range, elevation, and DOA estimated, aspect angleφl determined by simple trigonometry

Signal vector is spatially correlated

Rs = σ2s

1 · · · ρ(φ1 −φN)

.

.

.. . .

.

.

.ρ(φN −φ1) · · · 1

Spatial correlation coefficient

ρ(φl −φi) = e−(φl−φi)

2

Ω


Multi-Array Detection IIWe want to study the effects ofadding channels to the detector

We measure the observationsfrom m arrays,

zm[k] =[y1[k]H · · · ym[k]H

]H

and we wish to update the teststatistic using the observationym+1[k]

We build a detector for Platform 1,update to account for Platform 2,and so on until the measurementsfrom all 20 platforms have beentaken into account

Figures display∆J and Pdcorresponding to a 5% false alarmrate for SNR values of −5, 0, and5 dB for 7500 Monte Carlosimulations

The point where the change indivergence reaches its maximumvalue signifies the point ofdiminishing return as it is afterthis point that divergenceincreases but at a decreasing rate⇒ increase in performance slowsdown

The change in J-divergence can bean effective tool for helping usdecide when “enough is enough”

5 10 15 200

5

10

15

20

25

30

35

40

45

Number of Platforms

∆ J

-5 dB SNR

-20 m

-10 m

0 m

10 m

5 10 15 200

5

10

15

20

25

30

35

40

45

Number of Platforms

∆ J

0 dB SNR

-20 m

-10 m

0 m

10 m

5 10 15 200

5

10

15

20

25

30

35

40

45

Number of Platforms

∆ J

5 dB SNR

-20 m

-10 m

0 m

10 m

5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Platforms

PD

-5 dB SNR

-20 m

-10 m

0 m

10 m

5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Platforms

PD

0 dB SNR

-20 m

-10 m

0 m

10 m

5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Platforms

PD

5 dB SNR

-20 m

-10 m

0 m

10 m


Multi-Array Detection III

We assume that we can only takeadvantage of 10 of the 20 platforms but weare given the opportunity to choose whichplatforms to use

Build a detector for Platform 1, add theobservation from the platform that yieldsthe highest increase in divergence, searchthrough all the remaining platforms andadd the observation that gives the largestincrease in divergence given theaugmented observation from the previousiteration, repeat until we’ve addedobservations from 9 other platforms

This selective platform allocation scheme iscompared to situation where platforms arechosen at random

2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PD

Number of Platforms

-20 m

2 4 6 8 104.5

4.65

4.8

4.95

5.1

5.25

5.4

5.55

5.7

5.85

6

∆ J

PD:Selective

PD:Random

∆ J:Selective

∆ J:Random

2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PD

Number of Platforms

0 m

2 4 6 8 1012

12.8

13.6

14.4

15.2

16

16.8

17.6

18.4

19.2

20

∆ JP

D:Selective

PD:Random

∆ J:Selective

∆ J:Random

2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PD

Number of Platforms

-10 m

2 4 6 8 107

7.25

7.5

7.75

8

8.25

8.5

8.75

9

9.25

9.5

∆ J

PD:Selective

PD:Random

∆ J:Selective

∆ J:Random

2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PD

Number of Platforms

10 m

2 4 6 8 1020

22.5

25

27.5

30

32.5

35

37.5

40

42.5

45

∆ J

PD:Selective

PD:Random

∆ J:Selective

∆ J:Random

Change in J-divergence averaged over 7500 Monte Carlo simulations

Can always expect better performance when selectively choosing platforms

Can achieve the same probability of detection with a smaller number of platforms, e.g. at -10 m it takes 9 platforms toachieve Pd = 80% when choosing at random whereas it only takes 6 when allocating platform selectively


Conclusions

A new multi-sensory, binary hypothesis detection method has been proposed by casting the standard Gauss-Gaussdetector into the Multi-Channel Coherence Analysis (MCA) framework

The log-likelihood ratio in this coordinate system becomes a quadratic detector in the MCA variates under the H1hypothesis

The detector is applied to two data sets consisting of one HF sonar image and one to three BB sonar images

Detectors perform well with a probability of detection above 95% and less than 10 false alarms per image

Detector is applied to a simulated data set and exhibits robustness to SNR, target type, and aspect angle separation

Updating the likelihood ratio when adding data inherently involves a linear estimation problem and can be useful inboth temporal and channel updating situations


Suggestions for Future Work

1 Study is limited to only a few runs and types of underwater targets. Future studies on the effects of different bottomtypes, target orientations, sonar aspect, resolution, SNR, different man-made objects, etc. should be conducted boththeoretically and experimentally

2 Developments of this work are focused on detection of underwater targets. Studies on the use of the multi-channelcorrelation features for classification of targets and non-targets should be conducted. If successful, we could buildsystems that carry out simultaneous detection and classification using only the extracted MCA coordinates andcorrelations without requiring a separate feature extraction system

3 Investigate the extension of the MCA-based detector to other sensing modalities such as magnetic, infared, optical, etc.

4 Extension to multi-hypothesis testing should be investigated to consider situations where targets exhibit weak sonarreturn in either the HF or BB images

5 Investigate distributed detection systems which include collaboration between local decision makers to yield globaldetection decisions with higher confidence

6 More investigation for the application of updating the log-likelihood ratio for both temporal and channel updatingprocedures. Extend the idea of analyzing the incremental increase in divergence to higher-order statistical measures,ideally the probability of detection and false alarm

7 Extension of a multi-channel Generalized Likelihood Ratio Test (GLRT) detector to multi-variate Gaussian time seriesand subsequent studies of the response of the detector to varying degrees of disparity


Neyman-Pearson Lemma

Classical Bayesian detection minimizes the expected risk involved inmaking a decision and leads to the likelihood ratio test

l(x) =pX|H1

(x|H1)

pX|H0(x|H0)

H1≷

H0

P0(C10 − C00)

P1(C01 − C11)

The a priori probabilities require knowledge of how densely targetsare spaced on the ocean floor even before sensors are deployed andassigning costs to the detection problem is heuristic anduser-dependent⇒ Characterizing these free parameters often boilsdown to forming educated guesses

In lieu of the impracticality of classical Bayesian detection, the Neyman-Pearson lemma looks to maximize the probabilityof detection while constraining the false alarm rate to a particular level (α) which again leads to a likelihood ratio test

l(x) =pX|H1

(x|H1)

pX|H0(x|H0)

H1≷

H0η

If pL(l) denotes the probability density function of the likelihood ratio, then the threshold is chosen to satisfy the equation

∫∞η

pL|H0(l|H0)dl = α


Rank Reduction (Eckart-Young Theorem)Consider the matrix H ∈ Cm×n (m > n wlog) which has singular value decomposition (SVD) H = UΣVH whereU ∈ Cm×n and V ∈ Cn×n are orthogonal and Σ = diag [σ1 , . . . ,σn]

We wish to find the best rank-p approximation of this matrix in mean-squared sense, i.e.

minrank(H)=p

||H − H||2F =

m∑i=1

n∑j=1

∣∣∣[H]i,j − [H]i,j

∣∣∣2

Norms are invariant to unitary transformations (“lengths” of matrices remain unchanged regardless of how you rotatethem) leading to an equivalent statement

minrank(H)=p

||Σ− UHHV||2F

Intuitively, to minimize this function it must be true that the matrix UHHV is diagonal implying that U and V are thesingular vectors of H such that H = UΣVH

It is now our objective to find Σ = diag[σ1 , . . . , σp

]. Mean-squared error is now written as

||Σ− Σ||2F =

p∑i=1

(σi − σi)2 +

n∑i=p+1

σ2i

which is clearly minimized when σi = σi , i = 1, . . . , p, and σ1 > · · · > σn

Three-step process

1 Decompose: H =∑n

i=1 σiuivHi

2 Sort: σ1 > · · · > σn3 Rebuild: H =

∑pi=1 σiuivH

i


CCA Review IConsider the composite data vector z consisting of two random vectors x ∈ Cn and y ∈ Cm . We will assume, w.l.g., thatn 6 m. The two-channel composite vector has composite covariance matrix

Rzz = E[zzH

]=

[Rxx RxyRH

xy Ryy

]

If x and y are replaced by their corresponding whitened vectors, then the composite vector ξ

ξ =

[µν

]=

[R−1/2

xx OO R−1/2

yy

][xy

]

has covariance matrix

Rξξ = E[ξξH

]=

[I C

CH I

]

Matrix C = R−1/2xx RxyR−H/2

yy is called the coherence matrix and has singular value decomposition C = FKGH withFFH = FHF = In and GGH = GHG = Im

The singular value matrix K =[Kn OT

]T∈ Rn×m with Kn = diag [k1 , . . . , kn] a diagonal matrix with canonical

correlations ki’s

Matrices F and G are used to map the whitened versions of each channel to their canonical coordinates

w =

[uv

]=

[FH OO GH

] [R−1/2

xx OO R−1/2

yy

][xy

]


CCA Review IIThe standard measure of linear dependence for the composite vector z is the Hadamard ratio

0 6detRzz∏m+n

i=1 [Rzz]ii6 1

where [Rzz]ii is the ith diagonal element of Rzz

By introducing a block Cholesky factorization for Rzz of the form

Rzz =

[I RxyR−1

yyO I

] [Qxx OO Ryy

] [I O

R−1yy RH

xy I

]

where Qxx = Rxx − RxyR−1yy RH

xy is a Schur complement representing the error covariance matrix when estimating x fromy

We can then decompose the determinant of the composite covariance matrix as

detRzz = detQxxdetRyy = detRxxdetQxxdetRxx

detRyy

yielding the decomposition of the Hadamard ratio

detRzz∏m+ni=1 [Rzz]ii

=detRxx∏n

i=1 [Rxx]iidet(I − KKH)

detRyy∏mi=1

[Ryy

]ii

Thus CCA decomposes the linear dependence of z into the linear dependence among x and y, detRxx∏ni=1[Rxx]ii

and

detRyy∏mi=1

[Ryy

]ii

, and the linear dependence shared between the two, det(I − KKH) =∏n

i=1(1 − k2i )


CCA DetectionWe assume that we are given the observation y ∈ Cn which is signal-plus-noise under H1 : y = x + n and noise under H0 :n with Rxx = ExxH and Rnn = EnnH

One can then recast the low-rank detector described previously into the Canonical Correlation Analysis (CCA)framework by forming the coherence matrix (with Ryy = Rxx + Rnn)

C = R−1/2xx RxyR−H/2

yy = RH/2xx (Rxx + Rnn)−H/2

SVD of coherence matrix C = FKGH , K = diag (k1 , . . . , kn), F and G orthonormalLog-likelihood ratio and J-divergence can then be rewritten as

l(y) = yH(

R−1nn − R−1

yy

)y

= yHR−H/2yy G

([I − K2

]−1− I)

GHR−1/2yy y

=

n∑i=1

∣∣∣gHi R−1/2

yy y∣∣∣2 ( k2

i1 − k2

i

)

J =

n∑i=1

k4i

1 − k2i

The function k4i /(1 − k2

i ) monotonically increases for ki ∈ (0, 1] so the rank-p detector that maximizes divergence is theone that uses those canonical coordinates associated with the dominant canonical correlationsThus the low-rank detector built previously can alternatively be implemented using CCA when the underlying model isa signal-plus-noise model


Two-Channel Detection Example IWe consider the two-channel detection problem

H1 :

x1 = s1 + n1x2 = s2 + n2

H0 :

x1 = n1x2 = n2

where Enjnk = δj−kσ2n , Es2

j = σ2sj

, and Es1s2 = ρσs1σs2

With ηj = σ2sj/σ2

n equal to the SNR of the jth channel, the local SNR and coherence matrices are given to be

Σ = DH/21 D−1

0 D1/21 =

[1 + η1 0

0 1 + η2

]E = D−1/2

1 Rzz1 D−H/21 =

[1 ξξ 1

]

where ξ =ρσs1σs2√(

σ2s1

+σ2n)(σ2

s2+σ2

n) is the cross-correlation among the whitened versions of the channels

Solving EP1 = P1Λ results in

Λ1 =

[1 + ξ 0

0 1 − ξ

]P1 = [p1 p2] =

1√2

[1 11 −1

]

⇒MCA decomposes the data using a 2-D Haar basis


Two-Channel Detection Example IIRetaining the coordinate associated with λ1 (low-frequency approximations), we find that

v1 = v1,1 + v1,2 = pH1 D−1/2

1 z =1√

2

[(σ2

s1+σ2

n

)−1/2x1 +

(σ2

s2+σ2

n

)−1/2x2

]pH

1 Σp1 = 1 + 1/2 (η1 + η2)

In contrast, we could form the SNR matrix

S = R−1/2zz0 Rzz1 R−H/2

zz0 =

1 + η1ρσs1σs2σ2

nρσs1σs2σ2

n1 + η2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ρ

J

J−Divergence versus Correlation Coefficient

MCA

SNR

Solving SU =ΛU results in the per-mode SNR and mapping matrix

λi = 1 +12

(η1 + η2)± 12

√(η1 − η2)2 + 4ρ2η1η2

U = [u1 u2] =1√

ρ2η1η2(λ1−1−η1)2 + 1

ρσs1σs2

σ2n(λ1−1−η1)

1

1 −ρσs1σs2

σ2n(λ1−1−η1)

Discarding the second coordinate we find

ζi = uH1 R−1/2

zz0 z =1

σn

√ρ2η1η2

(λ1−1−η1)2 + 1

[ρσs1σs2

σ2n (λ1 − 1 − η1)

x1 + x2

]

With η1 = η2 = 0 dB, figure suggests that the MCA-based detector may be better suited to situations where channelsexhibit low SNR yet there is large amount of coherence among the channels under H1


Divergence BoundWe are given the two sub-optimal filtering statements

xHQ10x > xHQ1x

xHQ01x > xHQ0x

Rewrite the change in divergence as

∆J(zk+1 , zk

)=

n∑i=1

−2 + uHi Γ10ui +σ−1

i uHi Γ01ui

=

n∑i=1

−2 + uHi Q−1/2

0 Q10Q−H/20 ui +σ−1

i uHi Q−1/2

0 Q01Q−H/20 ui

Letting x = Q−H/20 ui , the two inequalities above imply

∆J(zk+1 , zk

)>

n∑i=1

−2 + uHi Q−1/2

0 Q1Q−H/20 ui +σ−1

i uHi Q−1/2

0 Q0Q−H/20 ui

=

n∑i=1

−2 + uHi Γui +σ−1

i uHi Iui

Vector ui is orthonormal and an eigenvector of Γ , thus

∆J(zk+1 , zk

)>

n∑i=1

−2 +σi +σ−1i

The function −2 +σi +σ−1i is PSD for any σi > 0 leading us to conclude that

∆J(zk+1 , zk

)>

n∑i=1

−2 +σi +σ−1i > 0


underwater target detection using multiple …...underwater target detection using multiple...

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