NONLINEAR RCM COMPENSATION METHOD FOR SPACEBORNEAIRBORNE FORWARD-LOOKING BISTATIC SAR.pdf

Polarimetric methods for tomographic imaging

of natural volumetric media

L. Ferro-Famil1,2, Y. Huang1, A. Reigber3 and M. Neumann4

1 University of Rennes 1, IETR lab., France

2 University of Tromsø, Physics and Technology Dpt., Norway

3 DLR, Airborne SAR Dept., Munich, Germany

4Caltech, JPL, Pasadena, CA, USA

[email protected]

Polarization Coherence Tomography [Cloude 2006, 2007]

Principle

◮ Born’s approximation (Common to most SARTOMtechniques)

γ(kz) =z0+hv∫

z0

f (z) ejkz zdz/

z0+hv∫

z0

f (z)dz

◮ Decomposition of f (z) over Legendre Polynomials

f (z ′) =No∑

i=0

aiPi(z′) ⇒ γ(kz) =

No∑

i=0

aigi (z0, hv , kzi )

◮ Reconstruction γ(kz) → ai → f (z ′) =No∑

i=0

aiPi(z′)

L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

Polarization Coherence Tomography [Cloude 2006, 2007]

Principle

◮ Born’s approximation (Common to most SARTOMtechniques)

γ(kz) =z0+hv∫

z0

f (z) ejkz zdz/

z0+hv∫

z0

f (z)dz

◮ Decomposition of f (z) over Legendre Polynomials

f (z ′) =No∑

i=0

aiPi(z′) ⇒ γ(kz) =

No∑

i=0

aigi (z0, hv , kzi )

◮ Reconstruction γ(kz) → ai → f (z ′) =No∑

i=0

aiPi(z′)

Basic steps

1. Set volume limits : z0, z0 + hv

2. Compute polynomial integrals at order NoPCT

3. Select a polarization channel

4. Invert linear relation setL. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

Outline

Tomography using Orthogonal Polynomials (TOP)

Fully Polarimetric TOP

Perspectives for Adaptive TOP


Proposed TOP LS approach (example with Legendre polynomials)

MB-inSAR coherence decomposition using OP

◮ Normalization : γ(kzj ) = ejkzjzm

1∫

−1

f ′(z) ejkzj

hv

2z

dz, ,1∫

−1

f ′(z)dz = 1





1∫

−1

f ′(z) ejkzj

hv

2z

dz, ,1∫

−1

f ′(z)dz = 1

◮ Decomposition over N0 + 1 OP and integration

f ′(z) =No∑

i=0

aiPi(z) + r(z) and gij = ejkzjzm

1∫

−1

Pi(z) ejkzj

hv

2z

dz

◮ Coherence : γ(kzj ) =No∑

i=0

ai gij + r(kzj )





1∫

−1

f ′(z) ejkzj

hv

2z

dz, ,1∫

−1

f ′(z)dz = 1

◮ Decomposition over N0 + 1 OP and integration

f ′(z) =No∑

i=0

aiPi(z) + r(z) and gij = ejkzjzm

1∫

−1

Pi(z) ejkzj

hv

2z

dz

◮ Coherence : γ(kzj ) =No∑

i=0

ai gij + r(kzj )

◮ Vertical profile reconstruction using M inSAR images, Nth0 order OP

◮ From γ ∈ CM−1 estimate a ∈ RNo+1

◮ Reconstruct estimated profile f ′(z) =No∑

i=0

ai Pi (z)



Least-Square TOP compact solution (generalized order PCT)

◮ LS criterion at arbitrary order No : c = ||γ − Ga||2, a ∈ RNo+1, [G]ij = gij

◮ Compact LS solution : a = arg mina c = [a0, a′T ]T

a0 = 0.5 ⇒1∫

−1

f ′(z)dz = 1, a′ = (X + XT )−1(v + v∗)

G = [g0Gr ], X = G†r Gr , v = G

†r (γ − 0.5go)






a0 = 0.5 ⇒1∫

−1

f ′(z)dz = 1, a′ = (X + XT )−1(v + v∗)

G = [g0Gr ], X = G†r Gr , v = G

†r (γ − 0.5go)

Selecting polynomial order

◮ M MB-inSAR images : NoPCT= 2(M − 1) DoF

◮ No = NoPCT: PCT ”inversion”. No < NoPCT

: LS fitting






a0 = 0.5 ⇒1∫

−1

f ′(z)dz = 1, a′ = (X + XT )−1(v + v∗)

G = [g0Gr ], X = G†r Gr , v = G

†r (γ − 0.5go)

Selecting polynomial order

◮ M MB-inSAR images : NoPCT= 2(M − 1) DoF

◮ No = NoPCT: PCT ”inversion”. No < NoPCT

: LS fitting

Reasons for choosing No < NoPCT

◮ Finite No values may not explain the whole tomographic content

γ(kzj ) =No∑

i=0

aigij + r(kzj ) + γn

◮ High No value (PCT inversion) sensitive to mismodeling

◮ Volume limits, z0, z0 + hv need to be estimatedL. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

TOP LS estimator : influence of the selected order No

−1 −0.5 0 0.5 1 1.5 2 2.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

estimated f’(z)

z

N

o= 1

No= 2

No= 3

No= 4

No= 5

No= 6

No= 7

No= 8

No= 9

No=10

f’(z)

1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

No

rel rmse

direct fitstabilized fit

Configuration

◮ M = 6 images

◮ NoPCT= 10

◮ NoTRUE= 3

Behavior w.r.t. No

◮ No < Ntrue : underfitting

◮ NoTRUE≤ No ≤ NoPCT

− 2 : correct fit

◮ NoPCT− 1 ≤ No ≤ NoPCT

: severe instability


TOP LS estimator : influence of orthogonal terms

◮ M = 3 images, NoPCT= 4

◮ NoTRUE= 4 → estimate a

◮ Add terms at orders 5 and 6 → anew

◮ RMSE for lower order terms

1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

No

rmse


TOP LS estimator : influence of orthogonal terms

◮ M = 3 images, NoPCT= 4

◮ NoTRUE= 4 → estimate a

◮ Add terms at orders 5 and 6 → anew

◮ RMSE for lower order terms

1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

No

rmse

Conclusions on TOP order selection

◮ NoPCT: maximal order for a unique solution

◮ No = NoPCT⇒ severe instabilities

◮ No < NoPCT− 2 : stable estimates, low sensitivity to higher order terms

◮ Number of images :◮ M = 2 → No = 1 : phase center under sinc approx.◮ M > 2 : profile parameters may be estimated, with No ≪ 2(M − 1)

◮ Volume limits, z0, z0 + hv need to be estimated


TOP LS estimator : influence of volume limit selection

◮ M = 6, dkz = 0.2

◮ TOP solutions for various No

◮ Adaptive solution

a, z0, hv = arg min ||γ − Ga||2−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20

5

10

15

20

25

30

reflectivity

z

f’(x)case a, N

o=5

case b, No=5

case a, No=N

oPCT

case b, No=N

oPCT

adaptive, No=5

Conclusions

◮ Volume extent underestimation → severe instabilities

◮ Volume extent overestimation → consistent estimates◮ Adaptive nonlinear Least Squares :

◮ Slightly improve estimation performance◮ Correct solution may be ”missed”+ high complexity


TOP LS estimator : application to real data

◮ TROPISAR Campaign over FrenchGuyana

◮ ONERA/SETHI MB-POLinSAR dataat P band

◮ Resolutions : δr = 1m, δaz = 1.245m

◮ M = 6 images



◮ TROPISAR Campaign over FrenchGuyana

◮ ONERA/SETHI MB-POLinSAR dataat P band

◮ Resolutions : δr = 1m, δaz = 1.245m

◮ M = 6 images

Estimation of volume limits

◮ Capon HH tomogram◮ SB-PolinSAR volume bounds

◮ hv may be underestimated (tunable)◮ z0 overestimated

◮ POLTOMSAR boundssee Huang et al. IGARSS 2011

50 100 150 200 250 300 350−60

−40

−20

0

20

40

60

range bins

z

POLinSAR boundsLIDAR boundsPOLTOMSAR bounds



TOP LS, HH

z

range bins50 100 150 200 250 300 350

−60

−40

−20

0

20

40

60

◮ M = 6

◮ No = 6 ≪ NoP CT

◮ Fixed bounds, −40m ≤ z ≤ 40m

Capon tomogram

50 100 150 200 250 300 350−60

−40

−20

0

20

40

60

z

range bins

◮ Similar global features

◮ Similar z-resolution (color coding)

◮ Legendre polynomials⇒ high intensity at estimation bounds



z

range bins50 100 150 200 250 300 350

−60

−40

−20

0

20

40

60

◮ No = 6

◮ −40m ≤ z ≤ 40m

z

range bins50 100 150 200 250 300 350

−60

−40

−20

0

20

40

60

◮ No = 6

◮ −60m ≤ z ≤ 60m

z

range bins50 100 150 200 250 300 350

−60

−40

−20

0

20

40

60

◮ No = 4

◮ POLTOMSAR bounds

High sensitivity to the volume bounds, possibly unstable


Outline





Full rank POLTOM : FR-P-CAPON

Coherence tomography and spectral estimators

γ(kzj ) =z0+hv∫

z0

f (z) ejkzj

zdz/

z0+hv∫

z0

f (z)dz = ejkzj

zm

1∫

−1

f ′(z) ejkzj

hv

2z

dz

◮ Equivalent to a normalized, or ”whitened” covariance matrix

◮ f (z) = aδ(z − z1), optimal estimation by BF

◮ f (z) =∑

iaiδ(z − zi) , optimal estimation by CAPON

◮ . . .




γ(kzj ) =z0+hv∫

z0

f (z) ejkzj

zdz/

z0+hv∫

z0

f (z)dz = ejkzj

zm

1∫

−1

f ′(z) ejkzj

hv

2z

dz



◮ f (z) =∑


◮ . . .

Classical CAPON estimators

◮ SP-CAPON : f (z) = (a(z)†R−1a(z))−1

◮ P-CAPON : f (z), ωopt(z) : limited rank-1 POLSAR information (H = 0)




γ(kzj ) =z0+hv∫

z0

f (z) ejkzj

zdz/

z0+hv∫

z0

f (z)dz = ejkzj

zm

1∫

−1

f ′(z) ejkzj

hv

2z

dz



◮ f (z) =∑


◮ . . .

Classical CAPON estimators

◮ SP-CAPON : f (z) = (a(z)†R−1a(z))−1

◮ P-CAPON : f (z), ωopt(z) : limited rank-1 POLSAR information (H = 0)

Full-rank CAPON estimator

◮ Full rank POLSAR coherency matrix : T(z) (H 6= 0)

◮ 3-D POLSAR MLC information : T(az, rg , z)◮ Decompositions , classifications, . . .usual POLSAR tools◮ Undercover parameter estimation◮ . . .L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

FR-P-CAPON over the Paracou tropical forest at P band


FR-P-CAPON over the Paracou tropical forest at P band


FR-P-CAPON over the Dornstetten temperate forest at L band








Full rank polarimetric tomography : FP-TOP

Classical TOP

◮ TOP (and PCT) : single channel technique, e.g. fHH(z)

◮ Pol. basis are not equivalent : fHH(z), fHV (z), fVV (z) ; fP1 (z), fP2 (z), fP3 (z)

FP-TOP using MB-ESM optimization

◮ Joint diag. (Ferro-Famil et al. 2009, 2010)

◮ Basis Maximizing the MB-POLinSAR information,

ω1, ω2, ω3 = arg maxω1⊥ω2⊥ω3

∑i,j

|γ(ωi , kzj ))|2

◮ Estimate fω1(z), fω2 (z), fω3(z)

◮ Compute fpq(z) from fωi , i = 1 . . . 3, using U3 = [ω1ω2ω3]


Outline





Adaptive TOP

γ(kzj ) =z0+hv∫

z0

f (z) ejkzjz

dz/z0+hv∫

z0

f (z)dz = ejkzjzm

1∫

−1

f ′(z) ejkzj

hv

2z

dz

Principle of adaptivity

◮ Classical spectral estimator tracks zm, OP set characterizes the structureof the volume

◮ Joint OP-spectral cost function : fast semi-analytical solutions

◮ z0, hv and No are adaptively estimated


Conclusion


◮ LS fit using any polynomial order No : analytical compact solution◮ Stability and robustness assessment :

◮ Order selection, Volume bounds◮ No ≪ NoPCT


◮ Full Rank P-CAPON : T(az, rg , z)

◮ 3-D polarimetry, under-cover soil moisture . . .

◮ FP-TOP


◮ OP based spectral estimators

◮ Adaptive order and volume bound selection
