1

ECE 194C Target Classification in Sensor

Networks

www.ece.ucsb.edu/Faculty/Iltis/ece194c

• Problem

– Given signature of a target, e.g. seismic, acoustic,

video.

– Determine which category the signature belongs to

• Fundamental problem in pattern recognition.

– Design classifiers to use temporal, FFT,

autoregressive model parameters to minimize error

rate.

2

Seismic Target SignaturesRef: Li, Wong, Hu and Sayeed “Detection, Classification and Tracking of Targets,” IEEE

Signal Proc. Mag. 2002.

Seismometer

)(nx

FFT

∑−

=

−1

0

/2)(N

n

Nnkienx

π

2||Power spectral

density (PSD)

3

Video Sensing – Pedestrian Avoidance

Ref. D. Gavrila, “Sensor-based pedestrian protection, IEEE Intelligent Syst. 2001

Daimler-Chrysler

Chamfer System

Problem is to detect

pedestrians in

cluttered imagery.

Classify pedestrian vs.

tree vs. building vs.

car.

4

Birdcall ClassificationRef: Wang, Estrin, Girod, “Preprocessing … for habitat monitoring,” EURASIP Journal on

Applied Sig. Proc. 2003

5

Bayes Classifier

Optimal classifier to minimize probability of classification error.

)|(max

argˆ xi

i

i P ωω

ω =

∑−

=

−=

−

=1

0

/2

2

2

2

)()(,

|)1(|

|)1(|

|)0(|

N

n

NknienxkX

NX

X

X

πx

Feature vector example -- PSD

x Classifier iω̂

6

Bayes Classifier – Gaussian

Distribution

Special case – Assume x is Gaussian with known mean vector for

each class and constant covariance.

Use Bayes’ rule

)(ln||)(||min

argˆ

)(||)(||2

1exp

)2(

11

)()|()(

1)|(

2

2

22/

ii

i

i

ii

n

N

n

N

iii

P

Pc

PPP

P

ωωω

ω

ωωσσπ

ωωω

+−=

⇒

−−=

=

mx

mx

xx

x

7

Bayes Classifier – Known Gaussian

Distribution – Equal Priors

Matlab voronoi(x,y) plots Voronoi regions. Region i is the set of points closer to

(xi,yi) then any other point. Points (xi,yi) are means of x in each category for the

Gaussian case.

1m

2m

NmClassify as

ω1 for all x in

this region.

1ω

2ω

Nω

8

Nearest-Neighbor Classifier

Assume Ni prototype vectors x are available with known classification (e.g.

seismic FFTs of known vehicles.)

1ω 2ω 3ω

1

2

1

1

1

1Nx

x

x

1

2

2

1

1

2Nx

x

x

3

3

2

3

1

3Nx

x

x

For each test vector y, find prototype x that is closest and classify as

associated iω

||||,

minargˆ

i

n

i

in

xy −=ω

ω

9

Nearest-Neighbor Classifier

1ω

1ω

2ω

2ω

3ω

3ω

3ω

y

y

y

1

1x

3

1x

10

Gaussian Classifier/K-Means

Algorithm

Assume prototypes and test vectors come from the same mixture Gaussian

density.

)(||||2

1exp

)2(

11

)()|()|(

1

2

,22/ i

M

k

ik

nn

N

iii

pc

ppp

i

ωσσπ

ωωω

∑=

−−

=

mx

xx

)...(#

)(#)(ˆ

21 N

i

ipxxx

x

∪∪=ω

Simplest estimate of prior probability of class i is relative frequency of prototypes

in class i.

11

K-Means AlgorithmProblem: Prototypes form Mi clusters in class i. For Gaussian mixture modeling,

want to find best “separation” between clusters to choose mean vectors.

11m

21m

12m

22m

12

K-Means Algorithm within a Class

Step 1: Place k points randomly into space represented by class i. These are

the initial centroids

11m

21m

Step 2: Assign each xi to group whose centroid is

closest.

11m

21m

Step 3: Recompute centroids

11m

21m

13

Matlab kmeans(x,2) Function

Centroids

Points are x = randn(5,1), y = randn(5,1) + 2. Note centroids are close to true means [0,0]

and [2,2].

11m

21m

14

Gaussian Mixture ClassifierStep 1: Run k-means algorithm on prototypes x.

Step 2: Set Gaussian mean vectors to centroids.

Step 3: (Compute covariance matrix of prototypes corresponding to each centroid.)

ijallforp

p

if

j

M

k

jk

nn

N

i

M

k

ik

nn

N

i

i

i

≠

−−

>

−−

→

∑

∑

=

=

)(||||2

1exp

)2(

1

)(||||2

1exp

)2(

1

1

2

,22/

1

2

,22/

ωσσπ

ωσσπ

ω

mx

mx

y

15

Neural Network Interpretation of Gaussian

Mixture Classifier

x

2

1||||

imx −

exp()

Σ

2

2||||

imx −

exp()

2

1||||

jmx −

exp()

2

2||||

jmx −

exp()

Σ<

>iω̂

16

Examples of Gaussian Mixture Classification – 2

Dimensional Feature Vector

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

i1m i2m

17

Seismic Classification Results

Ref: Li, Wong, Hu and Sayeed “Detection, Classification and Tracking of Targets,” IEEE

Signal Proc. Mag. 2002.

18

Distance Metric – Euclidean Distance

Vs. Normalized Correlation

||||||||||''||

||||||||22''''2''||''||

||||

1,

1

||||

1'

2

2

yx

yxyx

yx

yxyyyxxxyx

yy

yxxx

xx

x

T

TTTT

T

−∝−⇒

−=+−=−

===

∑∑

∑−

=

−

=

−

===1

0

1

0

1

0

)()(

)()(

|||||||| N

n

N

n

N

n

T

xy

nynx

nynx

yx

yxρ

19

Cross-Correlation Classification

Bird call examples Loon 2 Loon 4

Loon 2 using specgram(loon2a,256,fs2) Loon 4

20

Autocorrelation Loon 2

),(),(

),(

1),(

1

0

1

01

0

1

0

2

jminxmnx

mnx

jiN

n

N

mN

n

N

m

xx −−= ∑∑∑∑

−

=

−

=−

=

−

=

ρ

1max =ρ

21

Cross-Correlation – Loons 2 and 4Use Matlab xcorr2 function on spectrogram image. Normalized cross-correlation

coefficient.

),(),(

),(),(

1),(

1

0

1

01

0

1

0

21

0

1

0

2

jminymnx

mnymnx

jiN

n

N

mN

n

N

m

N

n

N

m

xy −−= ∑∑∑∑∑∑

−

=

−

=−

=

−

=

−

=

−

=

ρ

5671.max =ρB1 = specgram(loon,256,fs)

B1 = abs(B1)

xcorr2(B1,B2)/(norm(B1,’fro’)*…

norm(B2,’fro’))

22

Cross-Correlation Loon/Heron

Heron Loon 2

23

Cross-Correlation Loon/Heron

4191.max =ρ

24

Support Vector Machine for SAR

ClassificationRef: Q. Zhao and J. Principe “Support vector machines for SAR automatic

target recognition,” IEEE Trans. AES 2001

SAR imaging

(www.sandia.gov/radar)

25

SAR Imaging/Pose Ref: Q. Zhao and J. Principe “Support vector machines for SAR automatic target

recognition,” IEEE Trans. AES 2001

Problem: Classify type of

object (T72 Tank) with

orientation uncertainty.

T72

BTR70

BMP2

26

Two-Class Perceptron LearningPrevious clustering techniques did not minimize an error metric. Use

supervised perceptron learning to classify.

Assume training set is available of vectors x (e.g. SAR images, bird calls,

fingerprints, etc.)

Two-class problem (i.e. Tank, not-Tank)

1,1

},...,,{

},....,,{

21

21

−=

−=

i

i

m

m

y

imagethi

yyy

x

xxx

Perceptron – can classify linearly separable prototypes (Rosenblatt, 1957)

)sgn()( bgT += xwx

27

Perceptron Architecture

=

Nx

x

x

2

1

xΣ

b−

1w

1w

Nw

1x

2x

3x

Σ

xwT

)sgn()( bgT += xwx

28

Perceptron Criteria

w

bxw −=T

1)sgn( =+ bxwT

0)sgn( =+ bxwT

∑∈

+=xi E

i

TJ

Minimize

x

bxww ||)(

29

Perceptron Learning Rule

Define error due to new input. (Note b is like a weight with unity input.)

∈>+−

∈<+

+

=

+−=

0

1

,1,1

,1,1

,0

)sgn()(

Rb

Rb

classifiedcorrectlyb

btne

T

T

T

T

x

xxw

xxw

xw

xw

)))()1(sgn(()()1(

)1()))()1(sgn(()()1(

nbntnbnb

nnbntnn

T

x

T

x

++−+=+

+++−+=+

xw

xxwww

30

Perceptron Learning

)(nw

)()( nbnT −=xw

x is in R0, but tx = 1. e(n) = +1

1)()1( +=+ nbnb

)1()1( +−=+ nbnT xw

1R

1R

0R

31

Matlab Perceptron Demo (NN Toolbox)

Neural Network DESIGN Decision Boundaries

Move the perceptron

decision boundary by

dragging its handles.

Try to divide the

circles so that none

of their edges are red.

The weights and bias

will take on values

associated with the

chosen boundary.

Drag the white and

black dots to define

different problems.

Chapter 4W= 1.55 1.27 b= -2.22

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

W

32

Matlab Peceptron – Feature Space

Classification

-1

1

-1

1

-1

1

shape

Input Space

texture

weig

ht

Neural Network DESIGN Perceptron Classification

Click [Go] to send a

fruit down the belt

to be classified by a

perceptron network.

The calculations for

the perceptron will

appear to the left.

Chapter 3

Fruit

SHAPE: +0.78 TEXTURE: +0.85 WEIGHT: -0.26

Neural

Network

Oranges

Apples

W = [0 1 0]

b = 0

p = [0.78;0.85;-0.26]

a = hardlims(W*p+b)

a = hardlims(0.85)

a = 1

Fruit = Apple

33

Neural Network – Mean-Square Error

Minimization

))(exp(1

1),(

)),(()(1

2

bg

gyJ

T

m

i

ii

−−+=

−=∑=

xwxw

xww

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

y=1/(1+exp(-x))

Class 1

Class -1

Problem: The NN should learn based on prototypes xi. Learning

means selecting the w and b vectors to minimize J(w) based on

adaptation to prototypes and desired responses.

34

Delta Rule (Backpropagation) for MSE

Classifier

∑

∑

∑

=

+

−−

−−

=

+

=

−=

+−−=

∂

∂−=

−−+=

−=

m

i

ii

nnn

b

bm

i

ii

n

n

n

nn

T

m

i

ii

iT

iT

e

egy

J

bg

gyJ

1

1

2)(

)(

1

1

1

2

),(2

)1()),((2

)(

))(exp(1

1),(

)),(()(

xxwww

xxww

ww

ww

xwxw

xww

xw

xw

δµ

µ

µ

35

Neural Network MSE Classifier

(Single Layer)

x

w

Σ

b−

)exp(1

1

u−+Σ

1±=iy

0<

> Dec.

Adapt

Delta-rule

36

Support Vector Machine

+= ∑

∈

bKyxgSVi

iii ),(sgn)( xxα

Example kernel functions

−−= 2

2||||

2

1exp),( xxxx iiK

σ

01 ,1,,1 RyRy iiii ∈−=∈+= xx

37

Support Vector Machine

2

1|||| xx −−

exp()

Σ

2

2|||| xx −−

2

1||||

−−−

Nxx

exp()

2||||

Nxx −−

exp()

Σ<

>iω̂

x

exp()

11 yα22 yα

NN yα11 −− NN yα

38

Optimization for SVMProblem is to select alpha coefficients to maximize separation

between classes, while keeping fixed error rates.

Problem in convex optimization, Lagrangian primal and duals.

( )

2||||

1)(

Constraint

)()(sgn),(sgn)(

w

xw

xxxx

Minimize

by

bybKyxg

i

T

i

SVi

T

iii

SVi

iii

≥+

+=

+= ∑∑

∈∈

φ

φφαα

∑

∑

∈

∈

=

=

SVi

ii

SVi

iii

y

xy

Solution

0

)(

α

φαw

39

ADATRON Learning Rule for SVM

WhileEnd

ybbIf

xf

patternxChoose

tMWhile

tthresholdbInitialize

xfM

bxxkyyxf

iiiiiii

iADi

i

AD

j

iADAD

jij

m

j

jiiAD

)6

,,0)()5

))(1()4

)3

)2

,)1

)(min

),()(1

δδααδα

ηδ

ηα

α

+=+=>+

−=

<

=

+= ∑

=

Ideally should

be 1, or > 0.

Zero if fAD = 1 (like perceptron

rule)

Tries to drive fAD to 1.

40

SVM for SAR Imagery

SVM

T72

80 x 80 pixel

SAR image

(pose corrected)

SVM

BTR70

SVM

BMP2

Ref: Q. Zhao and J. Principe “Support vector machines for SAR automatic target

recognition,” IEEE Trans. AES 2001

Threshold

Test (Clutter

Rejection)

Threshold

Test (Clutter

Rejection)

Threshold

Test (Clutter

Rejection)

x1,1 −

1,1 −

1,1 −

41

SAR Imagery Confusion MatricesRef: Q. Zhao and J. Principe “Support vector machines for SAR automatic target

recognition,” IEEE Trans. AES 2001

42

Problems in Image Classification

• Color

• Correct for orientation (pose), translation, scale.

– Use invariant transforms, e.g. FFT magnitude is invariant to translation (phase shift.)

• Feature extraction (identify object in background)

• Choice of nearest-neighbor, k-means clustering/Bayes classifier, or neural-net methods.

– Long training times for images.