Fusion and Inversion of SAR Data to Obtain a Superresolution ImageAli MOHAMMAD-DJAFARI 1, Franck DAOUT 2 and Philippe FARGETTE 3

1 Laboratoire de signaux et systemes (L2S), UMR 8506 CNRS-SUPELEC-Univ Paris Sud 11, Gif-sur-Yvette, France2 SATIE, ENS Cachan, Universite Paris 10, France

3 DEMR, ONERA, Palaiseau, France'

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AbsractThe Synthetic Aperture Radar (SAR) data obtained from a single emitter and a single receiver gives information in the Fourier domain of the sceneover a line segment whose width is related to the bandwidth of the emitted signal. The mathematical problem of image reconstruction in SAR thenbecomes a Fourier Synthesis (FS) inverse problem. When there are more than one emitter and/or receiver looking the same scene, the problembecomes fusion and inversion. In this paper we report on a Bayesian inversion framework to obtain a Super Resolution (SR) image doing jointlydata fusion and inversion. We applied the proposed method on some synthetic data to compare its performances to other classical methods and onexperimental data obtained at ONERA.

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Synthetic Aperture Radar (SAR)imaging

s(t, u) =

∫∫f (x, y) p(t − τ (x, y , u)) dx dy

k =

[kx

ky

]=

[k cos(θ)k sin(θ)

]|k| = k = ω/c

s(ω, u) =

∫∫f (x, y) exp [−jωτ (x, y, θ(u))] dx dy

=

∫∫f (x, y) exp [−j(kxx + kyy)] dx dy

u (rad/m)

v (r

ad/m

)

S(u,v)

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S(u,v)

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Monostatic Bistatic & Multistatic{kx = k cos(θ)ky = k sin(θ)

{kx = k (cos(θtc) + cos(θcr)ky = k (sin(θtc) + sin(θcr)

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Forward model: Fourier Synthesis

f (x, y) → G(kx, ky), M(kx, ky)

Inverse problem:G(kx, ky), M(kx, ky) → f (x, y)

g(ui, vi) =

∫∫f (x, y) exp [−j(uix + viy)] dxdy

g = Hf + ǫ'

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Bayesian Estimation Approach◮ Forward model M : g = Hf + ǫ

◮ Likelihood: p(g|f ; M) = pǫ(g − Hf)

◮ A priori information: p(f |M)

◮ Bayes :

p(f |g; M) =p(g|f ; M) p(f |M)

p(g|M)

◮ Estimators:◮ Mode (Maximum A Posteriori)◮ Mean (Posterior Mean)◮ Marginal modes

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Proposed Bayesian Approach

f = arg minf

{J(f) = − ln p(g|f) − ln p(f)}

p(g|f) ∝ exp[

1

2σǫ2‖g − Hf‖2

]

◮ Generalized Gaussian:p(f) ∝ exp

[γ

∑j |fj|

2]

◮ Cauchy:p(f) ∝ exp

[γ

∑j ln(1 − |fj|

2]

◮ Generalized Gauss-Markov

p(f) ∝ exp

γ∑

j

|fj − fj−1|β

◮ Gauss-Markov-Potts'

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Comparison with classical methodsfh(x,y)

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Gerchberg-Papoulis Least Squaresfh(x,y)

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Quad. Reg. Proposed MAP'

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Multistatic data fusion methodsMethod 1:Data Fusion followed by inversion

G1(u, v)M1(u, v)

−

|G2(u, v)M2(u, v)

−

→G(kx, ky)M(u, v)

→ Inversion → f (x, y)

withG(u, v) =

G1(u,v)+G2(u,v)2 (u, v) ∈ M1(u, v) ∩ G2(u, v)

G1(u, v) (u, v) ∈ M1(u, v)G2(u, v) (u, v) ∈ M2(u, v)

and M(kx, ky) = M1(u, v) ∪ M2(u, v)

Method 2:Separte inversion followed by image fusion

G1(u, v)M1(u, v)

− Inversion − f1(x, y)

|

G2(u, v)M2(u, v)

− Inversion − f2(x, y)

→ Fusion→f (x, y)

◮ Image fusion◮ Coherent addition

f (x, y) = (f1(x, y) + f2(x, y))/2◮ Incoherent addition

f (x, y) = (|f1(x, y)| + |f2(x, y)|)/2

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Bayesian SimultaneousData Fusion and Inversion

G1(u, v)M1(u, v)

−

|G2(u, v)M2(u, v)

−

→Joint Fusion

andInversion

→ f (x, y)

{g1 = H1f + ǫ1

g2 = H2f + ǫ2

p(f |g1, g2) ∝ p(g1|f) p(g2|f) p(f)

MAP :

f = arg maxf

{p(f |g1, g2)} = arg minf

{J(f)}

withJ(f) = − ln p(g1|f) − ln p(g2|f) − ln p(f)

=‖g1 − H1f‖2

2σ2ǫ1

+‖g2 − H2f‖2

2σ2ǫ2

+ γ∑

j

[Df ]j|β

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Simulated datafh(x,y)

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fh(x,y)

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Data Fusion followed Joint Fusionby Inversion and Inversion

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Experimental data (Vv polarisation)

BF1 band x (m)

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Reconstruction by backpropagation

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ReferencesA. Mohammad-Djafari, F. Daout & Ph. Fargette, Fusion

et inversion des signaux SAR pour obtenir une image

super resolue, GRETSI 2009, 7-10 Sept., Dijon, France.A. Mohammad-Djafari, Sh. Zhu, F. Daout & Ph. Far-

gette, Fusion of Multistatic Synthetic Aperture Radar

Data to obtain a Superresolution Image, WIO 2009, 20-

24 July, Paris, France.