Fusion and Inversion of SAR Data to Obtain a Superresolution Image 1
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Ali MOHAMMAD-DJAFARI , Franck DAOUT and Philippe FARGETTE 1
` Laboratoire de signaux et systemes (L2S), UMR 8506 CNRS-SUPELEC-Univ Paris Sud 11, Gif-sur-Yvette, France 2 SATIE, ENS Cachan, Universite´ Paris 10, France 3 DEMR, ONERA, Palaiseau, France $
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Absract The Synthetic Aperture Radar (SAR) data obtained from a single emitter and a single receiver gives information in the Fourier domain of the scene over a line segment whose width is related to the bandwidth of the emitted signal. The mathematical problem of image reconstruction in SAR then becomes a Fourier Synthesis (FS) inverse problem. When there are more than one emitter and/or receiver looking the same scene, the problem becomes fusion and inversion. In this paper we report on a Bayesian inversion framework to obtain a Super Resolution (SR) image doing jointly data fusion and inversion. We applied the proposed method on some synthetic data to compare its performances to other classical methods and on experimental data obtained at ONERA.
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Synthetic Aperture Radar (SAR) Proposed Bayesian Approach imaging
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G1(u, v ) − M1(u, v ) Joint Fusion | → and f (x, y) →b Inversion G2(u, v ) − M2(u, v ) g 1 = H1 f + ǫ 1 g 2 = H2 f + ǫ 2 p(f |g 1, g 2) ∝ p(g 1|f ) p(g 2|f ) p(f ) MAP : fb = arg max {p(f |g 1, g 2)} = arg min {J(f )}
Generalized Gaussian:h P i p(f ) ∝ exp γ j |fj |2
Cauchy: h P i p(f ) ∝ exp γ j ln(1 − |fj |2 Generalized Gauss-Markov X p(f ) ∝ exp γ |fj − fj−1|β
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f (x, y) exp [−j(kx x + ky y)] dx dy
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J(f ) = − ln p(g 1|f ) − ln p(g 2|f ) − ln p(f ) X kg 1 − H1f k2 kg 2 − H2f k2 β ] | = + + γ [Df j 2 2 2σǫ1 2σǫ2 j
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fb = arg min {J(f ) = − ln p(g|f ) − ln p(f )} f 1 2 kg − Hf k p(g|f ) ∝ exp 2σǫ2 ◮
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Bistatic & Multistatic Monostatic Method 1: kx = k cos(θ) kx = k (cos(θtc ) + cos(θcr ) Data Fusion followed by inversion ky = k sin(θ) ky = k (sin(θtc ) + sin(θcr )
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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(g|f ; M) p(f |M) p(f |g; M) = p(g|M) ◮ Estimators: ◮
Mode (Maximum A Posteriori) ◮ Mean (Posterior Mean) ◮ Marginal modes
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G2(u, v) f2(x, y) − Inversion − b M2(u, v) ◮
Coherent addition b f (x, y) = (b f1(x, y) + b f2(x, y))/2 ◮ Incoherent addition b f1(x, y)| + |b f2(x, y)|)/2 f (x, y) = (|b ◮
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G1(u, v) − M1(u, v) G(kx , ky ) |→ f (x, y) → Inversion → b M(u, v) G2(u, v) − M2(u, v) with 2 (u,v ) G1(u,v )+G (u, v) ∈ M1(u, v) ∩ G2(u, v) 2 G(u, v) = G1(u, v) (u, v) ∈ M1(u, v) G2(u, v) (u, v) ∈ M2(u, v)
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References A. 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. Fargette, Fusion of Multistatic Synthetic Aperture Radar Data to obtain a Superresolution Image, WIO 2009, 2024 July, Paris, France.
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