[3em] a general model--based polarimetric decomposition...

A General Model–Based Polarimetric DecompositionScheme for Vegetated Areas

Maxim Neumann, Laurent Ferro-Famil, Eric Pottier

Motivation Volume Component Interpretation Experimental Results Conclusion

Motivation

T = fsTs + fdTd + fvTv

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Volume Component

• Layer of randomly distributed scatterers

• Single particle characteristics:position r uniformorientation ψ orientation distributiontilt Θ

particle scattering anisotropyshape Ξsize Dpermittivity εr

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Particle Scattering Anisotropy

Average backscattering matrix in eigenpolarizations:

〈F〉 =

[a 00 b

]=

a + b

2

[1 + δ∗ 0

0 1− δ∗]

Particle scattering anisotropy:

δ =

(a− b

a + b

)∗Depends on: shape, size, permittivity, tilt angle

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Particle Scattering AnisotropyParticle scattering anisotropy:

δ =

(a− b

a + b

)∗Effective particle shapes:

• |δ| → 0 =⇒ sphere, disk

• |δ| → 1 =⇒ dipole

Directly related to

• Alpha angle α ∈ [0, 14π] (Cloude and Pottier, 1997):

|δ| = tanα

• Shape parameter ρ ∈ [ 13 , 1] (Freeman, 2007):

|δ| = 21− ρ1 + ρ

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Orientation Distribution

• Mean particle polarization orientation: ψ̃

• Degree of orientation randomness: τ ∈ [0, 1]

• τ → 0 =⇒ aligned (no randomness)

• τ → 1 =⇒ complete random orientations

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Orientation DistributionCentral limit theorem =⇒Unimodal circular normal von Mises distribution:

p(ψ) =eκ cos(2(ψ− eψ))

πI0(κ)

κ: degree of concentration, I0: modified Bessel function of order 0.

Definition of τ :

τ =

∫pψ(ψ − ψ̃)dψ

πmax pψ(ψ)=

1

πpψ(ψ̃)=⇒ τ = I0(κ)e−κ

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Different Orientation Distributions

Truncated uniform distribution:

pU(ψ| eψ,Ψ)

=

{1Ψ |2(ψ − ψ̃)| ≤ Ψ

0 otherwise

Truncated Gaussian distribution:

pG(ψ| eψ,σ)

=

p

G(ψ| eψ,σ)R π/2−π/2

pG(ψ′| eψ,σ)

dψ′−π

2 ≤ ψ ≤π2

0 otherwise

Circular Normal (von Mises) distribution:

pC (ψ|ψ̃, κ) =eκ cos(2(ψ− eψ))

πI0(κ) 7 / 22


Polarimetric Volume Component

• Integration over the orientation angledistribution:

Tv =

∫ π/2

−π/2p(ψ)RT (2ψ)T

′vR

TT (2ψ) dψ

= RT (2 eψ)

[1 gcδ 0

gcδ∗(1+g)

2|δ|2 0

0 0 (1−g)2|δ|2

]RT

T (2 eψ)

with g = I2(κ)I0(κ) , gc = I1(κ)

I0(κ) , τ = I0(κ)e−κ

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Polarimetric Volume ComponentParameter space

Tv =

1 gc(τ)δ 0

gc(τ)δ∗ (1+g(τ))

2 |δ|2 0

0 0(1−g(τ))

2 |δ|2

Coherency matrix elements in dependence of |δ| and τ :

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Generalization

Tv =

1 gc(τ)δ 0

gc(τ)δ∗ 1+g(τ)2 |δ|2 0

0 0 1−g(τ)2 |δ|2

• For layers with a single dominant scattering mechanism type

• Ideal canonical scattering mechanisms:surface, sphere dipole dihedral

|δ| 0 1 ∞α 0 1

4π 1

2π

ρ 1 13

0

• Includes first–order forms as well as random forms

• Related to other volume and surface scattering models: Cloude, 1999, Hajnsek,

2001, Schuler, Lee, 2002, Yamaguchi, 2005.

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Parameter RetrievalScattering Anisotropy

Tv =

1 gc(τ)δ 0

gc(τ)δ∗ 1+g(τ)2 |δ|2 0

0 0 1−g(τ)2 |δ|2

• Magnitude:

|δ| =√

tv22 + tv33 =

√〈|Shh − Svv |2〉〈|Shv |2〉〈|Shh + Svv |2〉

• Phase:

arg δ = arg tv12 = arg(Shh + Svv )(Shh − Svv )∗

• How about the orientation randomness?11 / 22


Understanding the Orientation RandomnessLinear Approximation

Tv =

1 gc(τ)δ 0

gc(τ)δ∗ 1+g(τ)2 |δ|2 0

0 0 1−g(τ)2 |δ|2

Circular normal distribution:

g =I2(κ)

I0(κ), gc =

I1(κ)

I0(κ)

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Tv =

1 gc(τ)δ 0

gc(τ)δ∗ 1+g(τ)2 |δ|2 0

0 0 1−g(τ)2 |δ|2

Linear approximation:

g =

{1− 2τ τ ≤ 1

2

0 τ > 12

, gc = 1− τ12 / 22



Low orientation randomness τ ≤ 12

Tδ/τ =

1 (1− τ)δ 0(1− τ)δ∗ (1− τ)|δ|2 0

0 0 τ |δ|2

High orientation randomness τ > 1

2 :

Tδ/τ =

1 (1− τ)δ 0(1− τ)δ∗ 1

2 |δ|2 0

0 0 12 |δ|

2

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Test Site Traunstein

• Traunstein test site, 11.11.2003

• E–SAR L–band

• Incidence angles: 25◦– 56◦

• Resolution: 1.5m (slant range) x 95cm(azimuth)

• Data:

• Track 1: 09:00• Track 2: 08:50• Track 3: 08:40• Track 4: 08:00

• Used baselines:

• 1–2: 10min, 5m baseline• 1–3: 20min, 10m baseline• 1–4: 1h, 0m baseline

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Test Site Traunstein

S T Species Avg. hv ±hv

1 G c Fi-Ta-Bu 12.46m ±2.3m2 G c Fi 13.00m ±0m3 G d Ah-Bu-Fi 13.05m ±0m4 R c Fi-Bu 18.66m ±3.6m5 G c Fi-Ta-Bu 19.68m ±8.6m6 G c Fi-Bu-Ei 26.3m ±1.5m7 G c Fi-Bu-Ei 26.93m ±2.3m8 G d Bu-Ah-Es 27.20m ±3.1m9 G d Bu-Fi-La 27.32m ±0m

10 M c Fi-Bu-La 27.43m ±2.2m11 G c Fi-Bu-La 27.62m ±2.2m12 M c Fi-Bu-Ta 28.43m ±1.6m13 M c Fi-Bi-Bu 30.13m ±2.7m14 M c Fi-Bu-Es 32.49m ±1.6m15 R c Fi-Bu-Ta 33.14m ±2.7m16 R c Fi-Bu-Ah 34.34m ±1.8m17 M c Fi-Bu-Ei 34.59m ±2.5m18 R c Fi-Ta-Ki 34.66m ±1.4m19 R c Fi-Ta-Bu 35.23m ±1.2m20 R c Fi-Bu 36.10m ±1.8m

G: growth, M: mature, R: regenerating.c: coniferous, d: deciduous.Fi: north spruce, Ki: Scots pine, Ta: white fir, La: Eur. larch,Bu: Eur. beech, Es: ash, Ah: maple, Ei: oak, Bi: birch.

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Experimental Results: Degree of Orientation Randomness

m = {fg ,Reβ, Imβ, β22, β33, fv ,Re δ, Im δ, τ , hv , rh, σ, γsys , γtemp, φ0}

• Low SDEV.• Dependent on incidence angle.• Discriminates between different forest stands.

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Experimental Results: Degree of Orientation Randomness

m = {fg ,Reβ, Imβ, β22, β33, fv ,Re δ, Im δ, τ , hv , rh, σ, γsys , γtemp, φ0}

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Experimental Results: Particle Anisotropy

m = {fg ,Reβ, Imβ, β22, β33, fv ,Re δ, Im δ, τ, hv , rh, σ, γsys , γtemp, φ0}

• Anisotropy magnitude > 1 =⇒ possibly multiple scattering.

• Phase as an indicator for forest species type?

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Example of Direct DecompositionPauli–basis τ |δ| arg δ

• Forested region: high τ , medium to high |δ|, arg δ ≈ 0.

• Urban area: low τ , different |δ| and arg δ.

• Bare surfaces: low τ , low |δ|.• Crops: varying τ , |δ|, and arg δ =⇒ excellent for crops

classification.

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Conclusion

• Volume: particle characteristics + orientation randomness

• Polarization angle distributions

• Generalization to arbitrary scattering mechanism

• Linear approximation for intuitive interpretation

• Quantifying orientation effects in forest canopy

Using a–priori information, or approximations, or multi–angular, ormulti–temporal acquisitions:

=⇒ 3D particle orientations, shapes, permittivities.

=⇒ Vegetation monitoring

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Thank You!Questions?



Truncated uniform distribution:

pU(ψ| eψ,Ψ)

=

{1Ψ |2(ψ − ψ̃)| ≤ Ψ

0 otherwise

Truncated Gaussian distribution:

pG(ψ| eψ,σ)

=

p

G(ψ| eψ,σ)R π/2−π/2

pG(ψ′| eψ,σ)

dψ′−π

2 ≤ ψ ≤π2

0 otherwise

Circular Normal (von Mises) distribution:

pC (ψ|ψ̃, κ) =eκ cos(2(ψ− eψ))

πI0(κ) 22 / 22



Tv =

1 gcδ 0

gcδ∗ (1+g)

2 |δ|2 0

0 0 (1−g)2 |δ|2

22 / 22

[3em] a general model--based polarimetric decomposition...

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