Coherence Linearity and SKP-Structured Matrices in Multi-Baseline - - PowerPoint PPT Presentation

Stefano Tebaldini and Fabio Rocca

Politecnico di Milano Dipartimento di Elettronica e Informazione

Coherence Linearity and SKP-Structured Matrices in Multi-Baseline PolInSAR

IGARSS 2011, Vancouver

Introduction

The availability of Multi-baseline PolInSAR data makes it possible to decompose the signal into ground-only and volume-only contributions

ynwi

Track n Polarization wi



Re{yn(w1)} Re{yn(w2)} Re{yn(w3)} Im{yn(w1)} Im{yn(w2)} Im{yn(w3)}

Decomposition Track 1

HH HV VH VV HH HV VH VV

Track n Track N

HH HV VH VV

Volume-only contributions

200 600 1000 1400 1800 2200

• 10

10 20 30 40 50 60 200 600 1000 1400 1800 2200

• 10

10 20 30 40 50 60

Ground-only contributions

Properties of the vegetation layer

• Vertical structure
• Polarimetry
• Phase calibration
• Digital Terrain Model
• Ground properties

Polarimetric SAR Interferometry (PolInSAR)

• The coherence locus is assumed to be a straight line in the complex plane
• G/V decomposition is carried out by fitting a straight line in each interferometric pair

Introduction

Volume coherence Ground coherence Measured coherences

real part imaginary part real part imaginary part real part imaginary part

Algebraic Synthesis

• The data covariance matrix is assumed to be structured as a Sum of 2 Kronecker Products
• G/V dec is carried out by taking the first 2 terms of the SKP decomposition of the data covariance matrix

Scope of this work:

Compare the two approaches from the algebraic and statistical points of view

Model of the acquisitions

ynwi

Track n Polarization wi



Re{yn(w1)}

We consider a multi-polarimetric and multi-baseline (MPMB) data

• Monostatic acquisitions: up to 3 independent SLC images per track

Track 1

HH HV VH VV HH HV VH VV

Track n Track N

HH HV VH VV

Re{yn(w2)} Re{yn(w3)} Im{yn(w1)} Im{yn(w2)} Im{yn(w3)}

PolInSAR is based on the variation of the interferometric coherence w.r.t. polarization

Coherence linearity

Coherence linearity (*):

RVoG model => ESM coherences describe a straight line in the complex plane

 

j mm H j i nn H i j nm H i j i nm

w Σ w w Σ w w Σ w w w  , 

 

H m n nm

E y y Σ 

      

            HV y VV HH y VV HH y

n n n n

2 y

wi ≠ wj  Multiple Scattering Mechanisms (MSM) wi = wj  Equalized Scattering Mechanisms (ESM)

(*) Papathanassiou and Cloude, “Single Baseline Polarimetric SAR Interferometry

real part imaginary part Volume coherence Ground coherence Polarization depending factor Volume coherence Ground coherence ESM coherences

       

g v

          w w w w 1 ,

PolInSAR is based on the variation of the interferometric coherence w.r.t. polarization

Coherence linearity

Coherence linearity (*):

RVoG model => ESM coherences describe a straight line in the complex plane

 

j mm H j i nn H i j nm H i j i nm

w Σ w w Σ w w Σ w w w  , 

 

H m n nm

E y y Σ 

      

            HV y VV HH y VV HH y

n n n n

2 y

wi ≠ wj  Multiple Scattering Mechanisms (MSM) wi = wj  Equalized Scattering Mechanisms (ESM)

(*) Papathanassiou and Cloude, “Single Baseline Polarimetric SAR Interferometry

Volume coherence Ground coherence ESM coherences

Multiple baselines: one line per interferometric pair

real part imaginary part

n = 1 m = 2

real part imaginary part

n = 1 m = 4

real part imaginary part

n = 1 m = 3

The SKP structure

Without loss of generality, the received signal can be assumed to be contributed by K distinct Scattering Mechanisms (SMs), representing ground, volume, ground-trunk scattering, or other sk(n, wi) : contribution of the k-th SM in

Track n, Polarization wi

    





K k i k i n

n s y

1

;w w

Hp: the data covariance is structured as a Sum of Kronecker Products

Each SM is represented by a Kronecker Product

Covariance matrix among polarizations: EM properties

k-th Scattering Mechanism

WK  EyyH  

k1 K

Ck  Rk

Ck: polarimetric correlation of the k-th SM alone [3x3] Rk: interferometric coherences of the k-th SM alone [NxN]

Covariance matrix among tracks: Vertical structure

Note that Rk , Ck are positive definite

    





K k i k i n

n s y

1

;w w

The key to the exploitation of the SKP structure is the existence of a decomposition of any matrix into a SKP

W

SKP Dec





 

P p p p 1

V U W

Two sets of matrices Up, Vp such that: then, the matrices Uk, Vk are related to the matrices Ck, Rk via a linear, invertible transformation defined by exactly K(K−1) real numbers

   

2 1 2 1

1 1 V V R V V R b b a a

v g

                 

2 1 1 2 1 1

1 1 U U C U U C a a b a b b b a

v g

        

 

v v g g

R C R C W    

Corollary:

If only ground and volume scattering occurs, i.e: then, there exist two real

numbers (a,b) such that:





 

K k k k 1

R C W Theorem:

Let W be contributed by K SMs according to H1,H2,H3, i.e.:

The SKP decomposition

Forested areas: how many KPs ?

BIOSAR 2007 – Southern Sweden – P-Band BIOSAR 2008 – Northern Sweden – P –Band and L- Band TROPISAR – French Guyana – P-Band

Courtesy of ONERA

height

Height [m] 2000 2500 3000 3500 4000 4500 5000

• 10

10 20 30

P-Band - HV

Ground range [m] Height [m] 2000 2500 3000 3500 4000

• 10

10 20 30 Height [m] 2000 2500 3000 3500 4000

• 10

10 20 30

L-Band - HV

4500 5000 4500 5000 height [m] 200 600 1000 1400 1800 2200

• 10

10 20 30 40 50 60

HV

height [m] 200 600 1000 1400 1800 2200

• 10

10 20 30 40 50 60

HH

slant range [m] LIDAR Terrain Height

LIDAR Forest Height

HV

Height [m] 400 600 800 1000 1200 1400 20 40 60 Height [m] 400 600 800 1000 1200 1400 20 40 60

HH Slant range [m]

Forested areas: how many KPs ?

BIOSAR 2007 – Southern Sweden – P-Band BIOSAR 2008 – Northern Sweden – P –Band and L- Band TROPISAR – French Guyana – P-Band

Courtesy of ONERA

height

Height [m] 2000 2500 3000 3500 4000 4500 5000

• 10

10 20 30

P-Band - HV

Ground range [m] Height [m] 2000 2500 3000 3500 4000

• 10

10 20 30 Height [m] 2000 2500 3000 3500 4000

• 10

10 20 30

L-Band - HV

4500 5000 4500 5000 height [m] 200 600 1000 1400 1800 2200

• 10

10 20 30 40 50 60

HV

height [m] 200 600 1000 1400 1800 2200

• 10

10 20 30 40 50 60

HH

slant range [m] LIDAR Terrain Height

LIDAR Forest Height

HV

Height [m] 400 600 800 1000 1200 1400 20 40 60 Height [m] 400 600 800 1000 1200 1400 20 40 60

HH Slant range [m]

2 KPs account for about 90% of the information carried by the data in all investigated cases  2 Layered-models (Ground + Volume) are well suited for forestry investigations

Forested areas: how many KPs ?

BIOSAR 2007 – Southern Sweden – P-Band BIOSAR 2008 – Northern Sweden – P –Band and L- Band TROPISAR – French Guyana – P-Band

Courtesy of ONERA

height

Height [m] 2000 2500 3000 3500 4000 4500 5000

• 10

10 20 30

P-Band - HV

Ground range [m] Height [m] 2000 2500 3000 3500 4000

• 10

10 20 30 Height [m] 2000 2500 3000 3500 4000

• 10

10 20 30

L-Band - HV

4500 5000 4500 5000 height [m] 200 600 1000 1400 1800 2200

• 10

10 20 30 40 50 60

HV

height [m] 200 600 1000 1400 1800 2200

• 10

10 20 30 40 50 60

HH

slant range [m] LIDAR Terrain Height

LIDAR Forest Height

HV

Height [m] 400 600 800 1000 1200 1400 20 40 60 Height [m] 400 600 800 1000 1200 1400 20 40 60

HH Slant range [m]

Overview talk:

P-Band penetration in tropical and boreal forests: Tomographical results

Friday – 14:40 Room 1

Coherence linearity and 2KPs: Algebraic connections

• Polarimetric Stationarity (PS):
• Introduced by Ferro-Famil et al. to formalize the widely considered – RVoG consistent –

case where the scene polarimetric properties are invariant to the choice of the passage

• Always valid after whitening the polarimetric information of each image in such a way as:

 Always retained in the remainder

   

H m m mm H n n nn

E E y y Σ y y Σ   

n

nn

 

 3 3

I Σ

• Under the PS condition the ESM coherence can be decomposed into a weighted sum:





 

K k k k 1

R C W

 

 w

C w w C w w







K k k H k H k 1



   

 

k nm K k k nm

     

1

, w w w

 

 





  

K k k nm k H m n nm

E

1

 C y y Σ

(PS)

 

 

k nm nm k

  R

• 2 KPs  Coherence Linearity

v v g g

R C R C W    

Algebraic connections

• Coherence Linearity  N(N-1)/2 +1 KPs

     

g nm g v nm v nm

         w w w w,

 

 w

C C w w C w w

v g H v g H v g

 

, ,



       

w w w w w    , Re , Im

nm nm nm nm

b a  

 

 

 

 

nm k nm nm k nm

b a     Re Im    

 

k nm K k k nm

     

1

, w w w

ESM coherence decomposition SKP decomposition





 

K k k k 1

R C W

(PS)

 Each of the matrices Rk is fully specified by the real parts ( N(N-1)/2 ) plus one constant that multiplies the affine term bnm  There are at most N(N-1)/2 +1 linearly independent KPs Poor physical interpretation:

N(N-1)/2 +1 Scattering Mechanisms whose interferometric coherences are constrained to belong to the same line

Algebraic connections

• Coherence Linearity  N(N-1)/2 KPs

ESM coherence decomposition SKP decomposition (PS)

 Each of the matrices Rk is fully specified by the real parts ( N(N-1)/2 ) plus one constant that multiplies the affine term bnm  There are at most N(N-1)/2 +1 linearly independent KPs Poor physical interpretation:

N(N-1)/2 +1 Scattering Mechanisms whose interferometric coherences are constrained to belong to the same line

• Single baseline (N=2): perfect equivalence

2KPs  Coherence Linearity

• Multi-baseline (N>2): assuming 2KPs entails more algebraic

constraints than assuming coherence linearity: 2KPs Coherence Linearity

Algebraic connections

 In the multi-baseline case assuming 2KPs bring two advantages over coherence linearity:

v v g g

R C R C W    

1. Determination of physically valid solutions:

with Cg, Cv, Rg, Rv positive definite Assuming coherence linearity: Imposing pair-wise positive definitiveness results in physically valid ground and volume coherences to be ≤ 1 in magnitude Assuming 2KPs: The positive definitiveness constraint results in the regions of physical validity to shrink from the outer boundaries towards the true ground and volume coherences  The higher the number of tracks, the easier it is to pick the correct solution

Algebraic connections

 In the multi-baseline case assuming 2KPs bring two advantages over coherence linearity:

v v g g

R C R C W    

1. Determination of physically valid solutions:

with Cg, Cv, Rg, Rv positive definite

2. Coherence identification:

Assuming coherence linearity: Independent identification in each interferometric pair  2N(N-1)/2 possibilities Assuming coherence 2KPs: Joint identification on all interferometric pairs  2 possibilities

real part imaginary part

Ground – Volume OR Ground – Volume

???

Assuming coherence linearity: Imposing pair-wise positive definitiveness results in physically valid ground and volume coherences to be ≤ 1 in magnitude Assuming 2KPs: The positive definitiveness constraint results in the regions of physical validity to shrink from the outer boundaries towards the true ground and volume coherences  The higher the number of tracks, the easier it is to pick the correct solution

Simulated scenario:

Estimation from sample data

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4 imaginary part real part real part real part

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4 real part real part real part imaginary part

• 2 KPs:
• Number of tracks : N = 4
• Number of independent looks: L = {9 – 169}

v v g g

R C R C W    

• Case 1: High ground coherences
• Case 2: Low ground coherences

L2 norm minimization  fast but NOT optimal 1. Pair-Wise Estimator

Each pair is processed independently  Equivalent to assuming coherence linearity

2. Joint Estimator

All pairs are processed jointly

3. Preconditioned Joint Estimator

All pairs are processed jointly  As above, but the retrieved coherence matrices are

allowed to be slightly negative

2KP Estimators:

Note: coherence are assigned to ground or volume basing on knowledge of the true values  coherence identification is NOT considered

Criteria for coherence retrieval:

• Volume: existence of a ground-free polarization
• Ground: coherence maximization

Estimation from sample data

Case 1: High ground coherences

• L = 16
• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

Pair wise Joint

• Prec. Joint

True ground True volume Estimated ground Estimated volume

real part real part real part imaginary part real part real part real part real part real part real part imaginary part imaginary part

Estimated volume after ground phase compensation

Remarks:

Pair wise:

Ground coherence is algebraically bounded to belong to the unitary circle  Good accuracy when the true ground coherence is close to 1  Systematic bias for low ground coherences

Joint:

Ground coherence is NOT bounded to belong to the unitary circle  High coherence may be underestimated Improved accuracy over the Pair Wise approach for lower volume coherences

Preconditioned Joint:

Ground coherence underestimation is partly recovered

Estimation from sample data

Pair wise Joint

• Prec. Joint

True ground True volume Estimated ground Estimated volume

real part real part real part imaginary part real part real part real part real part real part real part imaginary part imaginary part

Estimated volume after ground phase compensation

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

Case 1: High ground coherences

• L = 49

Remarks:

Pair wise:

Ground coherence is algebraically bounded to belong to the unitary circle  Good accuracy when the true ground coherence is close to 1  Systematic bias for low ground coherences

Joint:

Ground coherence underestimation is mitigated by increasing the number of looks  Not a systematic bias Improved accuracy over the Pair Wise approach for lower volume coherences

Preconditioned Joint:

Underestimation of ground coherence is recovered

Estimation from sample data

Pair wise Joint

• Prec. Joint

True ground True volume Estimated ground Estimated volume

real part real part real part imaginary part real part real part real part real part real part real part imaginary part imaginary part

Estimated volume after ground phase compensation

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

• 1

1

• 1

1 n = 1 m = 2

• 1

1

• 1

1 n = 1 m = 3

• 1

1

• 1

1 n = 1 m = 4

Case 1: High ground coherences

• L = 100

Remarks:

Pair wise:

Ground coherence is algebraically bounded to belong to the unitary circle  Good accuracy when the true ground coherence is close to 1  Systematic bias for low ground coherences

Joint:

Ground coherence underestimation is mitigated by increasing the number of looks  Not a systematic bias Improved accuracy over the Pair Wise approach for lower volume coherences

Preconditioned Joint:

Underestimation of ground coherence is recovered

• 1

• 1

• 1

Estimation from sample data

Case 1: High ground coherences

Error on ground coherence Error on volume coherence Error on volume coherence after ground phase compensation Pair-wise Joint Preconditioned Joint

10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 2 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 3 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 4 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 2 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 3 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 4 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 2 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 3 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 4

10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 2 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 3 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 4

• 1

• 1

• 1

• 1

• 1

• 1

10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 2 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 3 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 4 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 2 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 3 10 100 10

• 1

100 Number of Looks RMSE n = 1 m = 4

Estimation from sample data

Case 2: Low ground coherences

Pair-wise Joint Preconditioned Joint Error on ground coherence Error on volume coherence Error on volume coherence after ground phase compensation

Conclusions

Single-baseline case: assuming Coherence Linearity is equivalent to assuming 2 KPs Multi-baseline: assuming 2KPs entails more algebraic constraints than assuming coherence linearity

• More accurate estimation of low-valued ground and volume coherence
• Simplifies the coherence identification problem to a single choice
• High ground coherence are underestimated if few looks (say < 50) are employed
• Underestimation is mitigated by pre-conditioning the problem

Estimators operating through L2 norm minimization  fast but not optimal The need for a pre-conditioning operator suggests that significant improvements could be achieved from the investigation of a statistically optimal multi-baseline estimator for the 2KP model