use of kernels for hyperspectral traget detection nasser m. nasrabadi senior research scientist u.s....
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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
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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 :
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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
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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
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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
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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
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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
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• 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
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• 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
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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
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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
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• 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
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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
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(a) MSD
(h) KSMF(b) KMSD
(c) ASD
(d) KASD
(e) OSP
(f) KOSP
(g) SMF
Results for FR-II Image
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ROC Curves for FR-II Image
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SMF & KSMF Results for Mine Image
Mine Hyperspectral Image
KSMF for mine imageSMF for mine image
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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.