USE OF KERNELS FOR HYPERSPECTRAL TRAGET DETECTION

Nasser M. Nasrabadi

Senior Research Scientist

U.S. Army Research Laboratory, Attn: AMSRL-SE-SE

2800 Powder Mill Road, Adelphi, MD 20783, USA

2

Outline

• Develop nonlinear detection algorithms.

• Exploiting higher order correlations.

• Why Kernels

• Kernel Trick

• Conventional matched filters

• Kernel matched filters

• Detection results

Nonlinear Mapping of DataExploitation of Nonlinear Correlations

• Nonlinear mapping

)x),x),((x)x 2211 ((:

FX

• Statistical learning (VC): Mapping into a higher dimensional space increases data separability

• However, because of the infinite dimensionality implementing conventional detectors in the feature space is not feasible using conventional methods

(y)(x),y)(x, k• Convert the detector expression into dot product forms Kernel-based nonlinear version of the conventional detector

Input space High dimensional feature space Input space

• Kernel trick :

4

Kernel Trick

221 ),(x Rxx

)2,(:),,(),(

x x)(,:22,21

2132121

232

xxxxzzzxx

RR

• Example of the kernel trick

),(:,)),)(,((

2),2,)(,2,()(),(22

2121

22

222121

21

21

2221

21

2221

21

yxyx

yx

kyyxx

yxyyxxyxyyyyxxxxT

T

dk yx,y)(x,

• Consider 2-D input patterns , where • If a 2nd order monomial is used as the nonlinear mapping

),(x 21 xx

• This property generalizes for and NRyx, Rd

(y)(x), y)(x, k

function kernel : , (y)(x), y)x, kk (

5

Examples of Kernels

)()()yx

exp()y,x( yx2 2

2

k1. Gaussian RBF kernel:

4. Polynomial kernel:dk ))((),( yxyx

yx

xy),( yxk3. Spectral angle-based kernel:

2. Inverse multiquadric kernel:22

y-x

1),(

ck

yx

Possible realization of

)x),x),(x) 2211 (((

Linear Spectral Matched Filter

• Spectral signal model

nsx:,nx:

aH

H1

0

:0a target present:s target spectral signature,:n background clutter noise

• Linear matched filter design (signal-to-clutter ratio),)( Tiiy xwx ],,,[ N21 xxxX

- Average output power of the filter for ix

Xofmatrix covariance the : C w, Cwwxxwx TT

)1

()(1 2

1

Tii

N

ii N

yN

- Constrained energy minimization:

,λE 1ws CwwW )()( TT sCs

sCw

1-T

-1

,sCs

xCsxwx)(

1-T

-1TT y aa

sCs

xCs1-

-1

of MLR:ˆT

T

:0a no target,nsx:

,nx:

aH

H1

0

:0a target present

• To stabilize the inverse of the covariance matrix, usually regularization is used, equivalent to minimizing:

• The regularized matched filter is given by:

Linear Spectral Matched Filter& Regularized Spectral Matched Filter

ww 1wsCwww TTT )()( βλE

,)(1-T

-1TT

sCs

xCsxwx y

• Linear matched filter is given as:

,)(

)((

sCIs

xCIsxwx)

1-T

-1TT

β

βy

8

Nonlinear Spectral Matched Filter

• In the feature space, the equivalent signal model

presentTarget )()(:

targetNo,)(:

1

0

nsx

nx

aH

H

• The equivalent matched filter in the feature space

T1-T

-1

where,)Φ()Φ(

)Φ(

XXC

sCs

sCw

• Output of the matched filter in the feature space

)()(

)()()())((

1T

1T

sCs

xCsxwx

-

-T

y

Kernelization of Spectral Matched Filter in Feature space

• Using the following properties of PCA and Kernel PCA

• Each eigenvector can be represented in terms of the input data

• Inverse Covariance matrix is now

• Kernel matrix spectral decomposition (kernel PCA)

],,,[ 21 M vvvV ,T11

VΛVC

],,,[ 21 MbbbB

,TT21 XBBΛXC

XxxxxKXXK jijiij k ,),(),(where,M

1 T22 BΛBK

,2/-1ΛBXVΦ

,, K

,)()(

)()())((

TT2-T

TT-2T

sXBBΛXs

xXBBΛXsx

y

• The kernelized version of matched filter

,)(2T

2T

s)k(X,Ks)k(X,

x)k(X,Ks)k(X,k

-

-

x y T

N21

T

N21

x))(x,x),(xx),,(x(x)k(X,s))(x,s),(xs),,(x(s)k(X,,,

,,kkkkkk

10

Conventional MF vs. Kernel MF

,(sCs

xCsxwx)

1-T

-1TT y

• Conventional spectral matched filter

• Nonlinear matched filter

• Kernel matched filter

)()(

)()()(

2T

2T

sX,kKsX,k

xX,kKsX,kk

-

-

x y

)()(

)()()())((

T

TT

sCs

xCsxwx

1-

-1

y

11

Matched Subspace Detection (MSD)

• Consider a linear mixed model:

• where and represent orthogonal matrices whose column vectors span the target and background subspaces and are unknown vectors of coefficients, is a Gaussian random noise distributed as • The log Generalized likelihood ratio test (GLRT) is given by

• where

)I,BζTθnBζTθy:

)I,Bζ,nBζy:2

1

20

(presentTarget

(absentTarget

H

H

T B

ζ θ n)I,( 20

1

0absent) signal |p(y

present) signal|p(y2

H

H

T

T

Ly)PI(y

y)PI(y)(

BT

By

,TBBPB ][]}[]]{[[ 1 BTBTBTBTPTB T

12

Kernel Matched Subspace Detection

• Define the matched subspace detector in the feature space• To kernelize we use the kernel PCA, and kernel function properties as shown below

βZZKβ

βZZKτ

τZZKβ

τZZKτΛ

),(

),(

),(

),(

BBT

BTT

TBT

TTT

1

),()(),,()(, TTTT yZkτyTandyZkβyBτ,ZTβZB TBTB

)()()(∴ TT y,Zkββy),k(ZyBBy BTT

B

y),k(Zβ

y),k(Zτ y),k(Zβy),k(Zτy)k(y,

y),k(Zββy),k(Zy)k(y,

B

B

BB

BBk

T

T1-

1

TTT

TT

2

-

-L

)Φ(][)Φ(

)Φ()()Φ(

)()()Φ(

)Φ()()Φ())((

Τ

Τ

Τ

Τ

Τ

ΤT

TT

T

T

2

yΒ

Τ

ΒΒ

ΒΤ

ΤΒ

ΤΤBTy

yBB-Iy=

yP-Iy

yP-Iyy

Φ

Φ

1-

ΦΦ

ΦΦ

ΦΦ

ΦΦΦΦ

ΦΦΦ

TBΦ

BΦ

ΦΦ

Φ

L

13

MSD vs. Kernel MSD

yPIy

yPIyy

BT

B

)(

)()(

T

T

2

L• GLRT for the MSD:

• Kernelized GLRT for the kernel MSD:

)()()(

)()()Φ())((

T

T

2 yPIy

yPIyy

TB

BΦ

L

• Nonlinear GLRT for the MSD in feature space:

y),k(Zβ

y),k(Zτ y),k(Zβy),k(Zτy)k(y,

y),k(Zββy),k(Zy)k(y,

B

B

BB

BBk

T

T1-

1

TTT

TT

2

-

-L

14

• The model in the nonlinear feature space is

• The MLE for in feature space is given as

• The kernel version of is given as

presentTargetsabsentTarget

1

0

nζB)y(:,nζB)y(:

μHH

)()PI()(

)y()PI()(

B

B

ss

sΦΦ

T

T

1

0

H

H

s),k(Zββs),k(Z-s)k(s,

y),k(Zββs),k(Z-y)k(s,=μ

BTT

B

BTT

Bk

Orthogonal Subspace Projector vs. Kernel OSP

15

• Consider a linear mixed model:

where U represent an orthogonal matrix whose column vectors

span the target subspace and C is the background covariance. is unknown vector of coefficients, is a Gaussian random noise distributed as • The log Generalized likelihood ratio test (GLRT) is given by

Adaptive Subspace Detection (ASD)

C)θUnθU:C),,n:

2

1

0

(presentTargetr0(absentTargetr

HH

θ),0( C

1

0

1T

1T1T1T

ˆ

ˆ)ˆ(ˆ

absent) signal(

present) signal()(

H

H

ASDD

rCr

rCUUCUUCr

|rp

|rpr

1

n

16

• The model in the nonlinear feature space is

• The GLRT ASD in feature space is given as

Where is a nonlinear function.• Substituting the following identities into the above Eq.

Nonlinear ASD

ASD

H

H

ASDD η)Φ(ˆ)Φ(

)Φ(ˆ)ˆ(ˆ)Φ()(

1

0

1T

1Φ

TΦΦ

1Φ

TΦΦ

1Φ

rCr

rCUUCUUCr

1T

presentTarget nθUr

absentTarget nr

ΦΦΦ1

Φ0

σ)(:

,)(:

Φ

Φ

H

H

τYXKX)K(XXxk

τYU XXX,KXBXBΛ X C

r

b

),(,),(

,,)(ˆ

2T

T221

k

17

ASD vs. Kernel ASD

• GLRT for the ASD:

• Kernelized GLRT for the kernel ASD:

• Nonlinear GLRT for the ASD in feature space:

ASD

H

H

ASDD η)Φ(ˆ)Φ(

)Φ(ˆ)ˆ(C)Φ()(

1

0

1T

1T1T1T

rCr

rCUUCUUr 1

1

0rr

rr1T

1T11T1T

r

H

H

ASDDC

CUU)CU(UC)(

),(),(),

]),(),(),[2T

T12TT

XrkXXKXk(r

K YXKXXKYK(X K (r)

b

rbr

KASDD

18

A 2-D Gaussian Toy Example

• Red dots belong to class H1, blue dots belong to H0

(a) MSD

(h) KSMF(b) KMSD

(c) ASD

(d) KASD

(e) OSP

(f) KOSP

(g) SMF

19

• Red dots belong to class H1, blue dots belong to H0

A 2-D Toy Example

(a) MSD

(h) KSMF(b) KMSD

(c) ASD

(d) KASD

(e) OSP

(f) KOSP

(g) SMF

20

Test Images

Forest Radiance I

Desert Radiance II

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Results for DR-II Image

(a) MSD

(h) KSMF(b) KMSD

(c) ASD

(d) KASD

(e) OSP

(f) KOSP

(g) SMF

22

ROC Curves for DR-II Image

23

(a) MSD

(h) KSMF(b) KMSD

(c) ASD

(d) KASD

(e) OSP

(f) KOSP

(g) SMF

Results for FR-II Image

24

ROC Curves for FR-II Image

25

SMF & KSMF Results for Mine Image

Mine Hyperspectral Image

KSMF for mine imageSMF for mine image

26

ROC Curves for Mine Image

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Conclusions

• Nonlinear target detection techniques are valuable.• Use of kernels and regularization in filter design.• Choice of kernels?• Nonlinear sensor fusion using kernels.