EQUIVALENCE OF DIFFERENT FORMAT RADAR POLARIMETRIC DATA FOR COHERENCY MATRIX ESTIMATION

Riccardo Paladini, Fabrizio Berizzi, Marco Martorella and Amerigo Capria

(1) University of Pisa, VIA CARUSO 16, 56122 PISA, Italy , Email: riccardo.paladini@iet.unipi.it, f.berizzi@iet.unipi.it, marco.martorella@iet.unipi.it, amerigo.capria@iet.unipi.it

OUTLINE

• Introduction• Coherency Matrix T Definition• Data Formats• Coherency Matrix Estimation Algorithms• Equivalence Demonstration• Real Data Analysis • Conclusions

INTRODUCTION•• HermitianHermitian Coherency matrix is largely employed for many Coherency matrix is largely employed for many PolPol--SAR application :SAR application :a)a) Speckle FilteringSpeckle Filteringb)b) Detection StudiesDetection Studiesc)c) Classification Classification d)d) Inversion StudiesInversion Studiese)e) InterferometryInterferometry•• Multi look effect have been recently investigated.Multi look effect have been recently investigated.•• Signal formats considered for coherency matrix estimation are heSignal formats considered for coherency matrix estimation are heterogeneous:terogeneous:a)a) Time format (Time format (multilookmultilook averaging processing)averaging processing)b)b) Spatial format (Boxcar filtering)Spatial format (Boxcar filtering)c)c) Hybrid format (Hybrid format (multilookmultilook + Spatial filter)+ Spatial filter)

•• Goal:Goal: Coherency matrix estimation obtained from different four data formats are identical

•• ValidationValidation is performed by real data analysis, in ISAR scenariois performed by real data analysis, in ISAR scenario

•• The target 4x4 coherency matrix The target 4x4 coherency matrix TT has been defined by S. R. has been defined by S. R. CloudeCloudeextending the Wiener definition of wave 2x2 coherency matrix :extending the Wiener definition of wave 2x2 coherency matrix :

[ ]2 *

2

*

*

*

* *

*

2

*

*

**

* 2

12 HV

THH HVHV VHHHH H

VHVH

H

VVV

VVV

b S S

b

g S S

g

g

g g g g

r S S

r

r r r r

r

r

S Sn jS jS

S S

n

n

n

n n n n

b b b

b

b

b g

⎡ ⎤= → = = −⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥

= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= ==+ + +S k

T kk

( )1 b

adt

b a− ∫T = T = T ( )1

1 N

nn

tN =

= ∑T T

•• PolarimetricPolarimetric signals, can be considered random processes, and signals, can be considered random processes, and T elements can be estimated by introducing a time averaging operelements can be estimated by introducing a time averaging operator ator

...

DEFINITION OF THE TARGET COHERENCY MATRIX

A) Transmitted Signal: Stepped Frequency WaveformsA) Transmitted Signal: Stepped Frequency Waveforms

Target Axis Direction  $x  

$yz$

( )( ) ( )( ) ( )

( )

, ,

cos coscos sin

sin

EL AZ

EL AZ

EL

r x y z

x ry r

z r

ε αε α

ε

⎧ =⎪ =⎨⎪ =⎩

( )( )1sin / cosAZ ELy rα ε−=

( )1sin /EL z rε −=

0

1nf f n f

n N= + Δ

≤ ≤

( )( )

0

0

az k

El k

kα τ α α

ε τ ε

= + Δ

=

tau1 H tau1 V tau2 H tau2 V tau3 H tau3 V

Fmin

Fmax

Time t

Freq

uenc

y f

Time-Frequency behavior of the trasmitted signal

ASSUMPTIONS

( )1

1 N

nn

fN =

= =∑T T

B) Uniform Azimuth sampling and radial motion compensated signaB) Uniform Azimuth sampling and radial motion compensated signalsls

Cross Range imaging is synthesized by Cross Range imaging is synthesized by Fast Fourier Transform Fast Fourier Transform

RADAR PARAMETERSCentral Frequency f0 =10 GHz

Frequency Step Δf = 50 MHzTx frequencies for Sweep N = 80

Tx Sweeps for Pol. K = 20

Azimuth Angle +- 5°

Azimut Step δα = 0.5°Tx/Rx Pol. H,V

Range Resolution Δr = 3.75cmCross Range Resolution Δcr = 9 cm

Elevation Angle 16°

Pol-ISAR UWB System in Controlled Environment:4 Combination of elementary targets over Polystyrene Turntable in Anechoic Chamber (CEPAMiR Adelaide)

EXPERIMENTAL REFERENCE DATA

Target Axis Direc�on �x

( )( )1sin / cosAZ ANTENNA ELy rα ε− −5°+0.5°⋅k, k∈[1, 20]= =

�y

z�

( )1 16 (FIXEsin / D)EL Antennaz rε − °= =

( ), ,ANTENNA ANTENNA ANTENNAA x y z��

( ) ( )( ) ( )

( )2 2 2

cos coscos sin

sin

EL AZ

EL AZ

EL

x Ay A

z A

A r x y z

ε αε α

ε

⎧ =⎪ =⎨⎪ =⎩

= = + +

Turntable Rotation 0.5° for each Tp

( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

1 1 1 2 1

2 1 2 2 2

1 2

,

, , . . ,, , . . ,

. . . . .

. . . . ., , . . ,

1 ;1 ; , ; ,

TR n k

TR TR TR K

TR TR TR K

TR N TR N TR N K

S f

S f S f S fS f S f S f

S f S f S f

n N k K T H V R H V

τ

τ τ ττ τ τ

τ τ τ

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦≤ ≤ ≤ ≤ = =

( ) ( )( )

( ) ( )( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

2 1 1 /1

1

1 1 1 2 1

2 1 2 2 2

0 1

, ,

1 ,

, , . . ,, , . . ,

. . . . .

. . . . ., , . . ,

1 ;1 ; , ; ,

TR i k f TR n k

Nj n i N

TR n k i Nn

TR TR TR K

TR TR TR K

TR N TR N TR N K

S r IDFT S f

S f eNS r S r S rS r S r S r

S r S r S r

i N k K T H V R H V

π

τ τ

τ

τ τ ττ τ τ

τ τ τ

− −≤ ≤

=

= =

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦≤ ≤ ≤ ≤ = =

∑( ) ( ) ( )( )

( ) ( )( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

2 1 1 /1

1

1 1 1 2 1

2 1 2 2 2

1 2

, , ,

1 ,

, , . . ,, , . . ,

. . . . .

. . . . ., , . . ,

1 ;1 ; , ; ,

TR i j TR i j i j

Kj k j K

TR i k k Kj

TR TR TR K

TR TR TR K

TR N TR N TR N K

S r Xr DFT S r U r

S r eK

S r Xr S r Xr S r XrS r Xr S r Xr S r Xr

S r Xr S r Xr S r Xr

i N j K T H V R H V

τ

π

τ τ

τ − −≤ ≤

=

= =

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦≤ ≤ ≤ ≤ = =

∑

POLARIMETRIC DATA FORMATS

( ) ( ) ( )( )( ) ( )( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

2 1 1 /1

1

1 1 1 2 1

2 1 2 2 2

1 2

, , ,

1 ,

, , . . ,, , . . ,

. . . . .

. . . . ., , . . ,

1 ;1 ; , ;

TR n j TR n j n j

Kj k j K

TR n j k Kj

TR TR TR K

TR TR TR K

TR N TR N TR N K

S f Xr DFT S f Xr U f Xr

S f Xr eK

S f Xr S f Xr S f XrS f Xr S f Xr S f Xr

S f Xr S f Xr S f Xr

i N j K T H V R

τ

π − −≤ ≤

=

= =

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦≤ ≤ ≤ ≤ = =

∑

,H V

Color Coded Raw Samples (f,τ)

Slow time τ [s/PRF]

Fre

quen

cy f

[G

Hz]

5 10 15 20

9

10

11

12

Color Coded Range Profiles (r,τ)

Slow time τ [s/PRF]

Ran

ge r

[cm

]

5 10 15 20

75

150

225

300

Color Coded Crossrange Profiles ( f,Xr)

Crossrange Xr [cm]

Fre

quen

cy f

[G

Hz]

-80 -40 0 40 80

9

10

11

12

Color Coded ISAR Images (r ,Xr)

Crossrange Xr [cm]

Ran

ge r

[cm

]

-80 -40 0 40 80

75

150

225

300

Format-I : / frequency time

Format-II : / range time

Format-IV : /frequency crossrange

Format-III : /range crossrange

ReBl

dGreen

ue

H

HV

HH VV

HV

H VVS SS S

S S=⎧ +−

==

+

⎪⎨⎪⎩

COHERENCY MATRIX ESTIMATION ALGORITHMS

1.1. Projection of Projection of polarimetricpolarimetric channels on Pauli scattering vector channels on Pauli scattering vector k2.2. Outer Product of scattering vectors (every pixel is represented Outer Product of scattering vectors (every pixel is represented by a Rankby a Rank--1 1 TT matrix)matrix)

3.3. 2D average operation 2D average operation

( )

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

1 1

,

1 ,

, , ,

12

, ,

1 ;

,

,

,

,,

1

I HV n k VH n k

N K

I n kn k

Hn k

I HH

I HH n k VV

n k

I n k I n k

I

I HV

VV n

n k VH n k

k

n k

b S

fNK

f f f

n jS f jS f

n N

r S f

k

g S f S ff S f

S f

K

τ

τ

τ τ

τ

τ

τ

τ

τ τ

τ

τ

= =

= =

=

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥

= −⎢ ⎥⎣ ⎦≤ ≤ ≤ ≤

== −

+

= +

∑∑T T Τ

T k k

k

( )

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

1 1

, ,

1 ,

,

, , ,

12

, ,

,,

1 ;1

,

N K

II i ki k

Hi k II i k I

I

II HH

II HH i k VV i k

I HV i

i k VV

I i k

i

II

II HV i k HV i

k H

k

V i

k

k

r S r

rNK

r r r

n jS r jS r

i

gS r

b S r S r

k

r

K

S S

N

r τ ττ τ

τ

τ

τ

τ τ

τ τ

τ

= =

= =

=

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥

= −⎢ ⎥⎣ ⎦≤ ≤ ≤

= −

≤

= +

= +

∑∑T T T

T k k

k

( )

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

1 1

1 ,

, , ,

12

, ,

1 ;

,

, ,

,

, ,

1

III HH i j VV i

III HV i j VH i

N K

III i ji j

H

i j III

III HH i j VV

i j III i j

III

III HV i j

i j

V

j

j

H i j

r S r Xr S r Xr

g S r Xr

r XrNK

r Xr r Xr r Xr

n jS r Xr jS r Xr

i N j

b S r

S r X

Xr S r

K

r

Xr

= =

= =

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥=

= +

=

−⎣ ⎦≤ ≤ ≤

= −

≤

+

∑∑T T T

T k k

k

( )

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

1 1

,

1 ,

, , ,

12

, ,

,

1 ;1

, ,

,

,IV HV n

N K

III n j

IV HH n j VV n j

IV

n j

H

i j IV n

j V

HH n

j III n j

IV

IV HV n

j VV n

j VH n

n j

j

j

H

r S f

f XrNK

r Xr f Xr f Xr

n jS f Xr

g S f Xr S f X

X

jS f

b S f Xr S f Xr

r S f Xr

Xr

K

r

i N j

= =

= =

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥= −

= +

⎣

=

⎦≤ ≤ ≤ ≤

= −

+

∑∑T T T

T k k

k

Color Coded Raw Samples (f,τ)

Slow time τ [s/PRF]

Fre

quen

cy f

[G

Hz]

5 10 15 20

9

10

11

12

Color Coded Range Profiles (r ,τ)

Slow time τ [s/PRF]

Ran

ge r

[cm

]

5 10 15 20

75

150

225

300

Color Coded Crossrange Profiles ( f,Xr)

Crossrange Xr [cm]

Fre

quen

cy f

[G

Hz]

-80 -40 0 40 80

9

10

11

12

Color Coded ISAR Images (r,Xr)

Crossrange Xr [cm]

Ran

ge r

[cm

]

-80 -40 0 40 80

75

150

225

300

ALGO-I : /frequency time

ALGO-IV : / frequency crossrange

ALGO-II : /range time ALGO-III : / range crossrange

ReBl

dGreen

ue

H

HV

HH VV

HV

H VVS SS S

S S=⎧ +−

==

+

⎪⎨⎪⎩

EQUIVALENCE BETWEEN RANGE AND FREQUENCY AVERAGES

( ) ( )1 1

1 1, k k

N Nk kn nI

n nf f

N Nτ =

=

= == =∑ ∑T Τ Τ

( ) ( )1 1

1 1,N Nk k

II i ik ki i

r rN N

τ=

== =

= = =∑ ∑T T T T

( ) ( )

( )

1 11 1 1

1 1

1 1 1 1 1, ,

1 ,

K K N K Nk kII II i bk k k k

i nk k kK N

Ib kk n

r fK K N K N

fKN

τ τ

τ

=

= == == = =

= =

= = = =

=

∑ ∑ ∑ ∑ ∑

∑∑

T T T T

T T

Color Coded Raw Sam ples [f,τ]

Slow tim e τ [s /PRF]

Fre

que

ncy

f [G

Hz]

5 10 15 20

9

10

11

12

Color Coded Range Profi les [r,τ ]

Slow tim e τ [s /PRF]

Ran

ge H

RR

[cm

]

5 10 15 20

75

150

225

300

- for a single column:ALGO I

- for a single columnALGO II

The two above sentences are equivalent by applying the Bessel –Parseval Theorem to the matrices elements.By averaging K columns of estimates of ALGO-I and ALGO-II, identical coherency matrix are obtained.identical coherency matrix are obtained.

EQUIVALENCE BETWEEN CROSS-RANGE AND SLOW TIME AVERAGES

( ) ( )1 1

1 1,K Ki i

II k ki ik k

rK K

τ τ=

== =

= = =∑ ∑T T T T

( ) ( )1 1

1 1,i i K K

III j ji ij j

r Xr XrK K

=

== =

= =∑ ∑T T T

( ) ( ) ( )1 1 1 11 1 1

1 1 1 1 1 1 , , ,N N K N K K Ni i

II II IIIk j i ji i i ik j i ji i i

r r X r XN N K N K KN

τ τ τ=

= == = = == = =

= = = = =∑ ∑ ∑ ∑ ∑ ∑∑T T T T T T

Color Coded Range Profi les [r,τ ]

Slow tim e τ [s /PRF]5 10 15 20

75

150

225

300Color Coded ISAR Im ages [r,Xr]

Cros s Range Xr [cm ]

Range

HR

R [

cm]

-80 -40 0 40 80

75

150

225

300 - for a fixed range binALGO II

- for a fixed range binALGO III

The two above sentences are equivalent by applying the Bessel –Parseval Theorem to the matrices elements.By averaging N rows of estimates of ALGO-II and ALGO-III, identical coherency matrix are obtainedidentical coherency matrix are obtained.

EQUIVALENCE BETWEEN THE FOUR DATA FORMATS

( )

( )

1 1

1 1

1 ,

1 ,

N K

K Average n kn k

PARSEVAL

N K

K Average i ki k

fNK

rNK

τ

τ

−= =

−= =

⇔ =

⇔ =

∑∑

∑∑

I

II

T T

T T

c

Color Coded Raw Samples ( f,τ)

Slow time τ [s/PRF]

Fre

quen

cy f

[G

Hz]

5 10 15 20

9

10

11

12

Color Coded Range Profiles (r ,τ)

Slow time τ [s/PRF]

Ran

ge r

[cm

]

5 10 15 20

75

150

225

300

Color Coded Crossrange Profiles ( f,Xr)

Crossrange Xr [cm]

Fre

quen

cy f

[G

Hz]

-80 -40 0 40 80

9

10

11

12

Color Coded ISAR Images (r ,Xr)

Crossrange Xr [cm]

Ran

ge r

[cm

]

-80 -40 0 40 80

75

150

225

300

( )1 b

adt

b a− ∫T = T = T

( )

Time-Sampling

1

1 N

nn

tN =

= ∑T T

c

( )

1

1

Stepped Frequency Wavefor

n

m

N

n

fN =

= ∑T T

c

( )1

1PARSEVAL

N

ii

rN =

= ∑T T

c

( )

( )

1 1

1 1

1 ,

1 ,

AZ

AZ

K K

PARSEVAL n jn j

PARSEVAL

N K

PARSE

MOTION

MOTION VAL i ji j

f XrNK

r XrNK

α

α

= =

= =

∧ ⇔ =

∧⇔ ⇔ =

⇔ ∑∑

∑∑

IV

III

T T

T T

c

EXPERIMENTAL VALIDATION OF THE THEORY

2

67 3 38 1 7 03 38 74 12 6 0

101 7 12 6 17 0

0 0 0 0

ALGO I ALGO II ALGO III ALGO IV

i ii i

i i

− − − −

−

= = =

− − +⎡ ⎤⎢ ⎥− + − −⎢ ⎥=⎢ ⎥− − +⎢ ⎥⎣ ⎦

T T T T

T

ReBl

dGreen

ue

H

HV

HH VV

HV

H VVS SS S

S S=⎧ +−

==

+

⎪⎨⎪⎩

Color Coded Raw Samples ( f,τ)

Slow time τ [s/PRF]

Fre

quen

cy f

[G

Hz]

5 10 15 20

9

10

11

12

Color Coded Range Profiles (r ,τ)

Slow time τ [s/PRF]

Ran

ge r

[cm

]

5 10 15 20

75

150

225

300

Color Coded Crossrange Profiles ( f,Xr)

Crossrange Xr [cm]

Fre

quen

cy f

[G

Hz]

-80 -40 0 40 80

9

10

11

12

Color Coded ISAR Images (r ,Xr)

Crossrange Xr [cm]

Ran

ge r

[cm

]

-80 -40 0 40 80

75

150

225

300

CONCLUSIONS:Equivalence Between different domains of

integration of the signals has been proved in the special case.

A relationship between temporal, spectral and spatial estimate has been traced

, ,

, ,

SFW

M

sampling

n

n K average n k PARSEVAL n j

PARSEVAL PARSEVAL PARSEVAL

i K average i k PARSE

OTION MODEL

MOTION MOD VAL iE jL

t

t

f f f Xr

r r r Xr

τ

τ

−−

− −

⇔ ∧⇔

⇔ ∧⇔

⇔

⇔

c

c

c

c c

Color Coded Raw Samples ( f,τ)

Slow time τ [s/PRF]

Fre

quen

cy f

[G

Hz]

5 10 15 20

9

10

11

12

Color Coded Range Profiles (r ,τ)

Slow time τ [s/PRF]

Ran

ge r

[cm

]

5 10 15 20

75

150

225

300

Color Coded Crossrange Profiles (f,Xr)

Crossrange Xr [cm]

Fre

quen

cy f

[G

Hz]

-80 -40 0 40 80

9

10

11

12

Color Coded ISAR Images (r,Xr)

Crossrange Xr [cm]

Ran

ge r

[cm

]

-80 -40 0 40 80

75

150

225

300

APPLICATION OF THE THEORY:We have been extending the results of our classification schemWe have been extending the results of our classification scheme based on ICTD [1]:e based on ICTD [1]:[1] [1] PaladiniPaladini, , MartorellaMartorella, , BerizziBerizzi, , ““Incoherent ISAR decomposition for target classificationIncoherent ISAR decomposition for target classification””, , Proc. Proc. EuradEurad 2008 Amsterdam.2008 Amsterdam.1)Training is made by windowed ISAR noisy images2) Testing is made with the Pol-ISAR, Pol -HRR, and frequency/time data on AWGN

-10-9.5 -9 -8.5 -8 -7.5 -7-6.5 -6-5.5 -5-4.5 -4 -3.5 -3 -2.5 -2-1.5 -1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of Correct Identification SNR train -2:3:22 dB ISAR MODE

SNR(dB)

ISAR

HRR

F τ

5.41 dB3 dB

T1 T2 T3 T4

Applying A=23 by B=10 pixel window ISAR provides greater performances.

Other data formats work correctly but with lower performances in noisy scenario.

In AWGN ISAR Gain = (NK)/(AB)N/A= 80/23 = 5.41 dBK/B= 20/10 = 3 dB

Thank you For The Attention!

• Appendices: A review of Bessel-Parseval Theorem.( ) ( )

( ) *

1

, , , be two complex function of real discrete variablethe complex cross-correlation of the two signals can be evaluated on the range domain:

( ) ( )

The second term can be numericc

i i

N

i i ii

Let x r y r

z r x r y r=

= ∑

( )( ) ( ) ( )( )

( )( ) ( ) ( )( )

( )

* 2 1 1 /1

1

* 2 1 1 /1

1

1

ally transformed by means of DFT:1( )

The transformation is reversible by the IDFT:1( )

by substitution we obtain:

( )

Nj i n N

n i i r Ni

Nj i n N

i n n r Nn

i ii

Y f DFT y r y r eN

y r IDFT Y f Y f eN

z r x r

π

π

− −≤ ≤

=

− − −≤ ≤

=

=

= =

= =

=

∑

∑

( )( )

( ) ( )( )

( ) ( )

( )

2 1 1 /*

1 1

2 1 1 /*

1 1 1

* *

1 1

* *

1

1 ( )

by inverting the two series operators:

1( ) ( )

( ) ( )

therefore

( ) ( ) ( ) ( )

N Nj i n N

nn i N

N Nj i n N

i n in i f N

N N

n n n nn n

N

i i i n ni n

Y f eN

z r Y f x r eN

Y f X f X f Y f

z r x r y r X f Y f

π

π

− − −

= ≤ ≤

− − −

= = ≤ ≤

= =

=

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤= =⎢ ⎥

⎣ ⎦

=

= =

∑ ∑

∑ ∑

∑ ∑

∑1

N

=∑