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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
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
◮ 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 )
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
◮ 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)
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
Proposed TOP LS approach (example with Legendre polynomials)
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)
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
Proposed TOP LS approach (example with Legendre polynomials)
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)
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
Proposed TOP LS approach (example with Legendre polynomials)
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)
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
TOP LS estimator : application to real data
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
TOP LS estimator : application to real data
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
L. 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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
◮ . . .
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
◮ . . .
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)
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
◮ . . .
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
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
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
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
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
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]
L. 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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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
L. Ferro-Famil, Y. Huang, A. Reigber and M. Neumann POLTOMSAR, IGARSS 2011, Vancouver, 27/08/2011
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