Roll Invariant Target Detection based on PolSARClutter Models

Lionel Bombrun, Gabriel Vasile, Michel Gay,Jean-Philippe Ovarlez and Frederic Pascal

2010/08/28

Outline

1 Context

2 Roll-invariant target detection

3 Detection results

4 Conclusions et perspectives

2

Context Problem formulation

Problem formulation

Target detection in PolSAR imagery.Examples of steering vector for dipole and dihedral targets.

kdip =1√2

1cos(2ψ)sin(2ψ)

and kdih =

0cos(2ψ)sin(2ψ)

Influence of ψ for a dipole.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−1.5

−1

−0.5

0

0.5

1

1.5

ψ

SHH

+SVV

SHH

−SVV

2SHV

3

Context Problem formulation

Problem formulation

ψ is the orientation of the maximum polarization with respect tothe horizontal polarization

Estimation of ψ using SRR et SLL{SRR = (SHH − SVV + 2jSHV ) /2SLL = (SVV − SHH + 2jSHV ) /2

ψKrogager =[Arg(SRRS ∗LL) + π

]/4

Target Scattering Vector Model (TSVM, Touzi decomposition),Roll-invariant target decomposition.

4

Context Target Scattering Vector Model

The Kennaugh-Huynen characteristic decomposition

Con-diagonalization of the scattering matrix S.

S = R(ψ) T(τm) Sd T(τm) R(−ψ)

where R(ψ) are T(τm) are given by:

R(ψ) =[

cosψ − sinψsinψ cosψ

]and:

T(τm) =[

cos τm −j sin τm− j sin τm cos τm

]Sd is a diagonal matrix which contains the coneigenvalues µ1 and µ2 of S:

Sd =[

me2j (ν+ρ) 00 m tan2 γ e−2j (ν−ρ)

]=[µ1 00 µ2

]5

Context Target Scattering Vector Model

Target Scattering Vector Model

Projection in the Pauli basis.

−→eTSV = m|−→eT |me jΦs

1 0 00 cos(2ψ) − sin(2ψ)0 sin(2ψ) cos(2ψ)

× cosαs cos(2τm)

sinαse jΦαs

− j cosαs sin(2τm)

where αs and Φαs are derived from the coneigenvalues µ1 and µ2 by:

tan(αs) e jΦαs =µ1 − µ2

µ1 + µ2

Con-eigenvalue phase ambiguityRestriction of ψ to the interval [−π/4, π/4]

−→eTSV(Φs , ψ, τm ,m, αs ,Φαs ) = −→eT

SV(Φs , ψ ± π

2,−τm ,m,−αs ,Φαs )

6

Context Target Scattering Vector Model

Target Scattering Vector Model

Projection in the Pauli basis.

−→eTSV = m|−→eT |me jΦs

1 0 00 cos(2ψ) − sin(2ψ)0 sin(2ψ) cos(2ψ)

× cosαs cos(2τm)

sinαse jΦαs

− j cosαs sin(2τm)

where αs and Φαs are derived from the coneigenvalues µ1 and µ2 by:

tan(αs) e jΦαs =µ1 − µ2

µ1 + µ2

Computation of the tilt angle

ψTSVM =12

Arctan

2<e{

(S ∗HH + S ∗VV )SHV

}<e{

(S ∗HH + S ∗VV )(SHH − SVV )} .

6

Context Comparison between ψTSVM and ψKrogager

Link between ψTSVM and ψKrogager

According to the TSVM, the following relation between ψ andψKrogager is obtained:

ψTSVM = ψKrogager − 14

Arctan(

tan(αs) sin(Φαs )tan(αs) cos(Φαs ) + sin(2τm)

)+

14

Arctan(

tan(αs) sin(Φαs )tan(αs) cos(Φαs )− sin(2τm)

).

Comparison between ψTSVM and ψKrogager

αs = π/3 and Φαs= π/3 αs = π/3 and τm = π/8 Φαs

= π/3 and τm = π/8

7

Context Comparison between ψTSVM and ψKrogager

Comparison between ψTSVM ans ψKrogager

����

��

����

����

��������

��������

-45o Dihedral

Trihedral

Helix Right Screw Helix Left Screw

Sa

Sc [0 11 0

]αs45o Dihedral

[1 00 −1

]Dihedral

[0 −1−1 0

]

1

2

[1 −j−j −1

] Sb

[1 00 1

]

1

2

[1 jj −1

]

2τm

Poincare sphere for Φαs= 0

ψTSVM and ψKrogager are equal if:αs = 0αs = π/2Φαs = 0or τm = 0

It corresponds to a wide class of targets including trihedral, dihedral,helix, dipole, . . .

8

Roll-invariant target detection Binary hypothesis test

Binary hypothesis test {H0 : k = cH1 : k = αp + c

α : unknown scalar complex parameter.p : signal (”steering vector”)c : clutter, c =

√τz with z ∼ N (0, [M ])

Principle1 Computation of the tilt angle ψ and extraction of the

”roll-invariant” target vector k.2 Estimation of the covariance matrix [M ] of the clutter.3 Computation of the similarity measure between the steering vector

p and the ”roll-invariant” target vector k.4 Choice of the false alarm probability.5 Thresholding of the similarity image and conclude or not on the

detection. 9

Roll-invariant target detection Binary hypothesis test

Binary hypothesis test {H0 : k = cH1 : k = αp + c

Optimal detector under the SIRV hypothesis.

Λ ([M ]) =pk(k/H1)pk(k/H0)

=hp

((k− p)H [M ]−1(k− p)

)hp

(kH [M ]−1k

) H1

≷H0

λ

where the expression of density generator function is given by:

hp (x ) =

+∞∫0

1τp

exp(−xτ

)pτ (τ) dτ

10

Roll-invariant target detection GLRT-LQ detector

GLRT-LQ detector

Λ ([M ]) =|pH [M ]−1k|2

(pH [M ]−1p) (kH [M ]−1k)

H1

≷H0

λ

where [M ] is covariance matrix of the population under the nullhypothesis H0, i.e. the observed signal is only the clutter. λ is thedetection threshold.

False alarm probability

pfa =1

(1− λ)(1−p)

10 000 Monte-Carlo simulationswith η =

1(1− λ)p

11

Roll-invariant target detection GLRT-LQ detector

ML estimator of the normalized covariance matrix

The ML estimator of the normalized covariance matrix underthe deterministic texture case is solution of :

[M ]FP = f ([M ]FP ) =pN

N∑i=1

kikHi

kHi [M ]−1

FPki

.

The existence and the uniqueness, up to a scalar factor, of theFixed Point estimator of the normalized covariance matrix havebeen established.p-normalization.

GLRT-LQ detector

Replace [M ] by the fixed point covariance matrix estimator [M ]FP

Λ(

[M ]FP

)=

|pH [M ]−1FPk|2(

pH [M ]−1FPp

)(kH [M ]−1

FPk) H1

≷H0

λ

12

Roll-invariant target detection GLRT-LQ detector

GLRT-LQ detector

Λ(

[M ]FP

)=

|pH [M ]−1FPk|2(

pH [M ]−1FPp

)(kH [M ]−1

FPk) H1

≷H0

λ

False alarm probability

False alarm probability pfa

pfa = (1− λ)(a−1)2F1(a, a − 1; b − 1;λ)

with a =p

p + 1N − p + 2 and b =

pp + 1

N + 2.

N is the number of points used to estimate the covariance matrix[M ] with the fixed point covariance matrix estimator.2F1(·, ·; ·; ·) is the Gauss hypergeometric function.

13

Detection resultsRAMSES, P-band, Nezer

A: Dihedralψ = 0

B: Dihedralψ = π/5

C: Particular targetαs = Φαs = τm = π/3 and ψ = π/5

D: Trihedralψ = 0

E: Dipoleψ = π/11

F: Particular targetαs = π/4, Φαs = π/5, τm = π/8, ψ = π/6

Coloredcomposition[k ]2-[k ]3-[k ]1

Withoutdesying(ψ = 0)

ψKrogager

ψTSVM

14

Detection results

RAMSES, X-band, Toulouse

Colored composition in the Pauli basis [k ]2-[k ]3-[k ]1

dihedralGLRT-LQ ψ αs Φαs τm

Krogager 0.912 0.761TSVM 0.956 0.770 -1.453 0.450 -0.178

Pure target 1.571 ∞ 0

pfa = 5× 10−3 → λ = 0.931

narrow diplaneGLRT-LQ ψ αs Φαs τm

Krogager 0.828 -0.023TSVM 0.849 -0.026 1.210 -0.172 0.052

Pure target 1.249 0 0

15

Conclusions et perspectives

Conclusions

Roll-invariant target detection.Comparison between ψTSVM and ψKrogager .Results on both synthetic and RAMSES PolSAR images.

Perspectives

Optimal detectors for heterogeneous clutter.Extension of the TSVM to the bistatic case.

Two orientation angles: ψR and ψE .See my poster on Friday morning (FRP1.PH.8).

16