POLinSAR2009 1/22

Polarimetric Target Detector by the use of the Polarisation Fork

Armando Marino¹Shane R Cloude²

Iain H Woodhouse¹

The University of Edinburgh

The University of Edinburgh

¹The University of Edinburgh, Edinburgh Earth Observatory (EEO), UK

²AEL Consultants, Edinburgh, UK

POLinSAR2009 2/22

Mathematical formulation

POLinSAR2009 3/22

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛=

VVVH

HVHH

SSSS

S

Scattering matrix:

[ ]( ) [ ]TkkkkSTracek 4321 ,,,21

=Ψ=

Scattering vector:

Single (coherent) target

kk=ωScattering mechanism:

6 Huynen parameters:

[ ]( ) [ ]TkkkSTracek 321 ,,21

=Ψ=

[ ] ( )[ ] ( )[ ] ( )ξντφγ

ντφ jUmUS mmT

mm exp,,tan0

01,, *

2*

⎥⎦

⎤⎢⎣

⎡=

Backscattering & reciprocity

Nulls pol-X , 21 =XXNulls pol-Co , 21 =CC

Max pol-X , 21 =SS

PolarisationFork:

POLinSAR2009 4/22

Partial target: Target Coherency Matrix

+⋅= kkC ][ 3

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=2

3*23

*13

*32

22

*12

*31

*21

21

3

kkkkk

kkkkk

kkkkk

C

The second orderstatistics are necessary.

Paulibasis

Lexicographic basis

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=2**

*2*

**2

2

222

2

VVHVVVHHVV

VVHVHVHHHV

VVHHHVHHHH

L

SSSSS

SSSSS

SSSSS

C

[ ]( )( ) ( )

( )( ) ( )( ) ( ) ⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+

−−+−

+−++

=2**

*2*

**2

222

2

2

HVVVHHHVVVHHHV

HVVVHHVVHHVVHHVVHH

HVVVHHVVHHVVHHVVHH

P

SSSSSSS

SSSSSSSSS

SSSSSSSSS

C

Classical formulations:

POLinSAR2009 5/22

[ ]321 ,, kkkk =

Polarimetric Detector

)()()()(

)()(

2*

21*

1

2*

1

ωωωω

ωωγ

iiii

ii

⋅⋅

⋅=

( ) 1kki TT =⋅= +ωω[ ]0,0,1=Tω

[ ]cbaP ,,=ω1≈a 0≈b 0≈c

1) A change of basis where the target to detect is one axes

( ) ki jj ⋅= +ωω

Ccba ∈,,

( ) 321 ckbkakki PP ++=⋅= +ωω

2) The Polarisation Fork (or Huynen parameters) is slightly changed to obtain:

Polarimetric coherence:

2,1=jWhere:

Demonstration:

In the new basis

Pseudo target:

Target:

POLinSAR2009 6/22

Polarimetric Detector

[ ]0,0,1=Tω

[ ]cbap ,,=ω( ) [ ]

[ ]( ) [ ]( )ppTT

pTpT

CC

C

ωωωω

ωωωωγ

33

3,

++

+

=

[ ] *31

*21

213

*21 kkckkbkaCii pT ++==⋅ + ωω

[ ]

( ) ( ) ( ) ( ) ( ) ( )*23

**31

**21

*

23

222

221

23

*22

222 kkcbkkcakkba

kckbkaCii pp

ℜℜ+ℜℜ+ℜℜ+

+++==⋅ + ωω

[ ] 213

*11 kCii TT ==⋅ + ωω

Ccba ∈,,

[ ]321 ,, kkkk =

3) Evaluation of the polarimetric coherence

( ) TpT >ωωγ ,Detector (first attempt):

POLinSAR2009 7/22

( )2

12

*22

21

*31

21

*211

,

kaii

k

kkac

k

kkab

pT⋅

++

=ωωγ

( ) ( ) ( ) ( ) ( ) ( )2

1

*23

2

*

21

*31

2

*

21

*21

2

*

21

23

2

2

21

22

2

2

21

22 22221

k

kk

acb

k

kk

aca

k

kk

aba

k

k

a

c

k

k

a

b

kai ℜℜ

+ℜℜ

+ℜℜ

+++=

Where:

Polarimetric detector

•If the components of the scattering vector are uncorrelated, the cross products correspond to a “noise” residual terms (biasing low coherence).

•If the components are correlated the cross product is not 0 and the coherence is biased up/down depending on how they sum with phase.

After normalisation

for:2

1ka

POLinSAR2009 8/22

Bias removal

[ ][ ] [ ] PPTT

PTd

PP

P

ωωωω

ωωγ

++

+

⋅= [ ]

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=2

3

22

21

00

00

00

k

k

k

P

( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛++

=

21

23

2

2

21

22

2

2

1

1,

k

k

ac

k

k

ab

pTd ωωγ

( ) ( )23

222

221

221

21

,kckbkak

kapTd

++=ωωγ

Detector:

4) Definition of a new operator that works on target powers

Where:

( ) TpTd >ωωγ ,

POLinSAR2009 9/22

Threshold selection

POLinSAR2009 10/22

Sample detector

[ ]xEx ≈Approximation: ( ) [ ]

[ ][ ][ ]⎟⎟⎠

⎞

⎜⎜

⎝

⎛++

=

21

23

2

2

21

22

2

2

1

1,

kEkE

ac

kEkE

ab

pTd ωωγ

[ ][ ]2

2

21

2kE

kESCR =

[ ][ ]2

3

21

3kE

kESCR =

3

2

2

2 111

1

SCRac

SCRab

d

++

≈γSa

mpl

e de

tect

or a

mpl

itude

SCRSCRSCR == 32

1=ab

12.0=ab

2.0=ab

5.0=ab

POLinSAR2009 11/22

Detector: random variable

Det

ecto

r am

plitu

de250

realisations

Average window 5x5

5.0==ac

ab

SCR

SCR

Det

ecto

r am

plitu

de

Standarddeviation

Randomcoherence

ic2 ˜ ( )2ˆ,0 σNrc2 ( )2ˆ,0 σN˜

ir jcck 222 +=

SCRSCRSCR == 32

POLinSAR2009 12/22

Validation

POLinSAR2009 13/22

Full-polarimetric Dataset

L-bandDLR: E-SAR

SARTOM projectLandsberg

POLinSAR2009 14/22

Detection

[ ]THVVVHHVVHHP SSSSSk 2,,21 −+= [ ]TVVHVHHL SSSk ,2,=

Odd-bounce Even-bounce Vertical dipoleHorizontal dipole

Multiple reflection Oriented dipole

POLinSAR2009 15/22

5x5

Open field: multiple reflection

TrihedralCR

TrihedralCR

Wolf1

Metallicnet

Tree

VVHHHVVVHH SS:Blue ;2S:Green ;SS :Red +−

L-band

Red: Even-bounceGreen: 0Red: Odd-bounce

POLinSAR2009 16/22

5x5

Open field: oriented dipoles

TrihedralCR

TrihedralCR

Wolf1

Metallicnet

Tree

VVHVHH S:Blue ;S2:Green ;S :Red

L-band

Red: Horizontal dipoleGreen: 0Red: Vertical dipole

POLinSAR2009 17/22

Forested area: multiple reflection5x5

VVHHHVVVHH SS:Blue ;2S:Green ;SS :Red +−

L-band

Red: Even-bounceGreen: 0Red: Odd-bounce

POLinSAR2009 18/22

Forested area: oriented dipole5x5

VVHVHH S:Blue ;S2:Green ;S :Red

L-band

Red: Horizontal dipoleGreen: 0Red: Vertical dipole

POLinSAR2009 19/22

Comparison with Polarimetric Whitening Filter

(PWF)

L. M. Novak, M. C. Burl, and M. W. Irving, "OptimalPolarimetric Processing for Enhanced Target Detection," IEEE

Trans. Aerospace and Electronic Systems, vol. 20, pp. 234-244,1993.

POLinSAR2009 20/22

5x5

Open field

PWF

L-band

Red: Even-bounceGreen: 0Red: Odd-bounce

POLinSAR2009 21/22

Forest: POLSAR5x5

PWF

L-band

Red: Even-bounceGreen: 0Red: Odd-bounce

POLinSAR2009 22/22

Conclusions

• A target detector was developed based on the unique polarimetricfork (PF) of the single target (similarly the Huynen parameters can be used).

• The mathematical formulation carried out is general, and so can be

applied for any single target of interest (as long as the PF is

known).

• The validation was achieved over two categories of targets: multiple

reflection and oriented dipoles, with results in line with the expected

physical behaviour of the targets.

• A supplementary theoretical validation is carried out, where the

algorithm is compared with the Polarimetric Whitening Filter (PWF).

POLinSAR2009 23/22

Thank you very much for your attention!

POLinSAR2009 24/22

Uniqueness of detection

[ ]321 ,, kkkk = 3Ck∈

We pass with a projection to the space of Power: +→ 33 RC

332211 ˆˆˆ ekekekk ++=

Defined a basis the scattering vector in is represented by:

The projective space is obtained with the operator:2*

iT

ii ekP ⋅=

ie

This is a sujective operation, hence the vector in P is uniquelly defined once we select the vector in the 3-D complex space (and we set a basis).

+→ 33 RC

The detection is unique since the detection rule is defined on the power (SCR or peak) and the Power space is uniquelly related with the target space (we need only 3 real numbers).

3C

POLinSAR2009 25/22

TTT kkkkkkP *33

*22

*11 ⋅+⋅+⋅=

[ ]Tkk 0,,0 22 =

[ ]Tkk 0,0,11 =

[ ]Tkk 33 ,0,0=

Uniqueness of detection

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=2

3

22

21

00

00

00

k

k

k

P

POLinSAR2009 26/22

Coherence: random variable

SCR

SCR

Coh

eren

ce a

mpl

itude

Coh

eren

ce a

mpl

itude

Standarddeviation

250realisations

Randomcoherence

5.0==ac

ab

5x5

SCRSCRSCR == 32

POLinSAR2009 27/22

Entropy estimation