This document describes polarization coherence tomography (PCT) techniques for tomographic imaging of natural volumetric media using polarimetric synthetic aperture radar (PolSAR) data. It presents the principle of PCT using Born's approximation and decomposing the scattering function over Legendre polynomials. It then describes the basic steps of PCT including setting volume limits, computing polynomial integrals, and inverting the relation to reconstruct the scattering function. It also discusses tomography using orthogonal polynomials (TOP) and a least squares approach for TOP, including selecting the polynomial order and estimating volume limits. Finally, it applies TOP to real PolSAR data from a tropical forest campaign.

1. 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
4
Caltech, JPL, Pasadena, CA, USA
Laurent.Ferro-Famil@univ-rennes.fr

2. Polarization Coherence Tomography [Cloude 2006, 2007]
Principle
Born’s approximation (Common to most SARTOM
techniques)
z0 +hv z0 +hv
γ(kz ) = f (z) ejkz z dz/ f (z)dz
z0 z0
Decomposition of f (z) over Legendre Polynomials
No No
f (z ) = ai Pi (z ) ⇒ γ(kz ) = ai gi (z0 , hv , kzi )
i =0 i =0
No
ˆ ˆ
Reconstruction γ (kz ) → ai → f (z ) =
ˆ ˆi Pi (z )
a
i =0
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

3. Polarization Coherence Tomography [Cloude 2006, 2007]
Principle
Born’s approximation (Common to most SARTOM
techniques)
z0 +hv z0 +hv
γ(kz ) = f (z) ejkz z dz/ f (z)dz
z0 z0
Decomposition of f (z) over Legendre Polynomials
No No
f (z ) = ai Pi (z ) ⇒ γ(kz ) = ai gi (z0 , hv , kzi )
i =0 i =0
No
ˆ ˆ
Reconstruction γ (kz ) → ai → f (z ) =
ˆ ˆi Pi (z )
a
i =0
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 set
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

4. Outline
Tomography using Orthogonal Polynomials (TOP)
Fully Polarimetric TOP
Perspectives for Adaptive TOP
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

5. Proposed TOP LS approach (example with Legendre polynomials)
MB-inSAR coherence decomposition using OP
1 k z hv
j
1
Normalization : γ(kzj ) = ejkzj zm f (z) ej 2
z
dz, , f (z)dz = 1
−1 −1
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

6. Proposed TOP LS approach (example with Legendre polynomials)
MB-inSAR coherence decomposition using OP
1 k z hv
j
1
Normalization : γ(kzj ) = ejkzj zm f (z) ej 2
z
dz, , f (z)dz = 1
−1 −1
Decomposition over N0 + 1 OP and integration
No 1 k z hv
j
f (z) = ai Pi (z) + r (z) and gij = ejkzj zm Pi (z) ej 2
z
dz
i =0 −1
No
Coherence : γ(kzj ) = ai gij + ˜(kzj )
r
i =0
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

7. Proposed TOP LS approach (example with Legendre polynomials)
MB-inSAR coherence decomposition using OP
1 k z hv
j
1
Normalization : γ(kzj ) = ejkzj zm f (z) ej 2
z
dz, , f (z)dz = 1
−1 −1
Decomposition over N0 + 1 OP and integration
No 1 k z hv
j
f (z) = ai Pi (z) + r (z) and gij = ejkzj zm Pi (z) ej 2
z
dz
i =0 −1
No
Coherence : γ(kzj ) = ai gij + ˜(kzj )
r
i =0
th
Vertical profile reconstruction using M inSAR images, N0 order OP
From γ ∈ CM−1 estimate ˆ ∈ RNo +1
ˆ a
No
Reconstruct estimated profile ˆ (z) =
f ˆi Pi (z)
a
i =0
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

8. Proposed TOP LS approach (example with Legendre polynomials)
Least-Square TOP compact solution (generalized order PCT)
LS criterion at arbitrary order No : c = ||ˆ − Ga||2 , a ∈ RNo +1 , [G]ij = gij
γ
Compact LS solution : ˆ = arg mina c = [ˆ0 , a T ]T
a a ˆ
1
ˆ0 = 0.5 ⇒
a f (z)dz = 1, ˆ = (X + XT )−1 (v + v∗ )
a
−1
G = [g0 Gr ], X = G† Gr , v = G† (γ − 0.5go )
r r
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

9. Proposed TOP LS approach (example with Legendre polynomials)
Least-Square TOP compact solution (generalized order PCT)
LS criterion at arbitrary order No : c = ||ˆ − Ga||2 , a ∈ RNo +1 , [G]ij = gij
γ
Compact LS solution : ˆ = arg mina c = [ˆ0 , a T ]T
a a ˆ
1
ˆ0 = 0.5 ⇒
a f (z)dz = 1, ˆ = (X + XT )−1 (v + v∗ )
a
−1
G = [g0 Gr ], X = G† Gr , v = G† (γ − 0.5go )
r r
Selecting polynomial order
M MB-inSAR images : NoPCT = 2(M − 1) DoF
No = NoPCT : PCT ”inversion”. No < NoPCT : LS fitting
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

10. Proposed TOP LS approach (example with Legendre polynomials)
Least-Square TOP compact solution (generalized order PCT)
LS criterion at arbitrary order No : c = ||ˆ − Ga||2 , a ∈ RNo +1 , [G]ij = gij
γ
Compact LS solution : ˆ = arg mina c = [ˆ0 , a T ]T
a a ˆ
1
ˆ0 = 0.5 ⇒
a f (z)dz = 1, ˆ = (X + XT )−1 (v + v∗ )
a
−1
G = [g0 Gr ], X = G† Gr , v = G† (γ − 0.5go )
r r
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
No
γ (kzj ) =
ˆ ai gij + ˜(kzj ) + γn
r ˆ
i =0
High No value (PCT inversion) sensitive to mismodeling
Volume limits, z0 , z0 + hv need to be estimated
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

11. TOP LS estimator : influence of the selected order No
1
No= 1 0
10
0.8 No= 2
No= 3
0.6 No= 4
No= 5 −1
10
0.4 No= 6
N =7 rel rmse
o
0.2
N =8
o
z N =9
0 o −2
10
No=10
−0.2 f’(z)
−0.4
−3
10
−0.6
−0.8 direct fit
−4
stabilized fi
−1 10
−1 −0.5 0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 9 10
estimated f’(z) No
Configuration Behavior w.r.t. No
M = 6 images No < Ntrue : underfitting
NoPCT = 10 NoTRUE ≤ No ≤ NoPCT − 2 : correct fit
NoTRUE = 3 NoPCT − 1 ≤ No ≤ NoPCT : severe instability
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

12. TOP LS estimator : influence of orthogonal terms
0.7
0.6
0.5
M = 3 images, NoPCT = 4 0.4
rmse
ˆ
NoTRUE = 4 → estimate a 0.3
Add terms at orders 5 and 6 → ˆnew
a 0.2
RMSE for lower order terms 0.1
0
1 2 3 4
No
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

13. TOP LS estimator : influence of orthogonal terms
0.7
0.6
0.5
M = 3 images, NoPCT = 4 0.4
rmse
ˆ
NoTRUE = 4 → estimate a 0.3
Add terms at orders 5 and 6 → ˆnew
a 0.2
RMSE for lower order terms 0.1
0
1 2 3 4
No
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

14. TOP LS estimator : influence of volume limit selection
30
25
20
z
15
M = 6, dkz = 0.2 10 f’(x)
case a, N =5
TOP solutions for various No o
case b, N =5
o
5 case a, N =N
Adaptive solution o
case b, N =N
o
oPCT
oPCT
ˆ, z0 , hv = arg min ||ˆ − Ga||2
adaptive, N =5
a γ 0
o
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2
reflectivity
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

15. TOP LS estimator : application to real data
TROPISAR Campaign over French
Guyana
ONERA/SETHI MB-POLinSAR data
at P band
Resolutions : δr = 1m, δaz = 1.245m
M = 6 images
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

16. TOP LS estimator : application to real data
TROPISAR Campaign over French
Guyana
ONERA/SETHI MB-POLinSAR data
at P band
Resolutions : δr = 1m, δaz = 1.245m
M = 6 images
60
40
Estimation of volume limits
20
Capon HH tomogram
SB-PolinSAR volume bounds z
0
hv may be underestimated (tunable)
z0 overestimated −20
POLTOMSAR bounds
see Huang et al. IGARSS 2011 −40
POLinSAR bounds
LIDAR bounds
POLTOMSAR bounds
−60
50 100 150 200 250 300 350
range bins
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

17. TOP LS estimator : application to real data
TOP LS, HH Capon tomogram
60 60
40 40
20 20
z z
0 0
−20 −20
−40 −40
−60 −60
50 100 150 200 250 300 350 50 100 150 200 250 300 350
range bins range bins
M=6 Similar global features
No = 6 NoP CT Similar z-resolution (color coding)
Fixed bounds, −40m ≤ z ≤ 40m Legendre polynomials
⇒ high intensity at estimation bounds
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

18. TOP LS estimator : application to real data
60 60 60
40 40 40
20 20 20
z z z
0 0 0
−20 −20 −20
−40 −40 −40
−60 −60 −60
50 100 150 200 250 300 350 50 100 150 200 250 300 350 50 100 150 200 250 300 350
range bins range bins range bins
No = 6 No = 6 No = 4
−40m ≤ z ≤ 40m −60m ≤ z ≤ 60m POLTOMSAR bounds
High sensitivity to the volume bounds, possibly unstable
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

19. Outline
Tomography using Orthogonal Polynomials (TOP)
Fully Polarimetric TOP
Perspectives for Adaptive TOP
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

20. Full rank POLTOM : FR-P-CAPON
Coherence tomography and spectral estimators
z0 +hv z0 +hv 1 k z hv
jkzj z jkzj zm j
γ(kzj ) = f (z) e dz/ f (z)dz = e f (z) ej 2
z
dz
z0 z0 −1
Equivalent to a normalized, or ”whitened” covariance matrix
f (z) = aδ(z − z1 ), optimal estimation by BF
f (z) = i
ai δ(z − zi ) , optimal estimation by CAPON
...
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

21. Full rank POLTOM : FR-P-CAPON
Coherence tomography and spectral estimators
z0 +hv z0 +hv 1 k z hv
jkzj z jkzj zm j
γ(kzj ) = f (z) e dz/ f (z)dz = e f (z) ej 2
z
dz
z0 z0 −1
Equivalent to a normalized, or ”whitened” covariance matrix
f (z) = aδ(z − z1 ), optimal estimation by BF
f (z) = i
ai δ(z − zi ) , optimal estimation by CAPON
...
Classical CAPON estimators
ˆ
SP-CAPON : f (z) = (a(z)† R−1 a(z))−1
ˆ(z), ω opt (z) : limited rank-1 POLSAR information (H = 0)
P-CAPON : f ˆ
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

22. Full rank POLTOM : FR-P-CAPON
Coherence tomography and spectral estimators
z0 +hv z0 +hv 1 k z hv
jkzj z jkzj zm j
γ(kzj ) = f (z) e dz/ f (z)dz = e f (z) ej 2
z
dz
z0 z0 −1
Equivalent to a normalized, or ”whitened” covariance matrix
f (z) = aδ(z − z1 ), optimal estimation by BF
f (z) = i
ai δ(z − zi ) , optimal estimation by CAPON
...
Classical CAPON estimators
ˆ
SP-CAPON : f (z) = (a(z)† R−1 a(z))−1
ˆ(z), ω opt (z) : limited rank-1 POLSAR information (H = 0)
P-CAPON : f ˆ
Full-rank CAPON estimator
ˆ
Full rank POLSAR coherency matrix : T(z) (H = 0)
ˆ
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

23. FR-P-CAPON over the Paracou tropical forest at P band
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

24. FR-P-CAPON over the Paracou tropical forest at P band
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

25. FR-P-CAPON over the Dornstetten temperate forest at L band
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

26. FR-P-CAPON over the Dornstetten temperate forest at L band
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

27. FR-P-CAPON over the Dornstetten temperate forest at L band
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

28. FR-P-CAPON over the Dornstetten temperate forest at L band
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

29. 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,
ω 1 , ω 2 , ω 3 = arg max ω1 ⊥ω 2 ⊥ω3 i ,j
|γ(ω i , kzj ))|2
Estimate fω 1 (z), fω2 (z), fω 3 (z)
Compute fpq (z) from fω i , i = 1 . . . 3, using U3 = [ω 1 ω 2 ω 3 ]
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

30. Outline
Tomography using Orthogonal Polynomials (TOP)
Fully Polarimetric TOP
Perspectives for Adaptive TOP
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

31. Adaptive TOP
z0 +hv z0 +hv 1 k z hv
j
γ(kzj ) = f (z) ejkzj z dz/ f (z)dz = ejkzj zm f (z) ej 2
z
dz
z0 z0 −1
Principle of adaptivity
Classical spectral estimator tracks zm , OP set characterizes the structure
of the volume
Joint OP-spectral cost function : fast semi-analytical solutions
z0 , hv and No are adaptively estimated
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011

32. Conclusion
Tomography using Orthogonal Polynomials (TOP)
LS fit using any polynomial order No : analytical compact solution
Stability and robustness assessment :
Order selection, Volume bounds
No NoPCT
Fully Polarimetric TOP
Full Rank P-CAPON : T(az, rg, z)
3-D polarimetry, under-cover soil moisture . . .
FP-TOP
Perspectives for Adaptive TOP
OP based spectral estimators
Adaptive order and volume bound selection
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011