Fusion and Inversion of SAR Data to Obtain a Superresolution Image Ali MOHAMMAD-DJAFARI 1 , Franck DAOUT 2 and Philippe FARGETTE 3 1 Laboratoire de signaux et syst ` emes (L2S), UMR 8506 CNRS-SUPELEC-Univ Paris Sud 11, Gif-sur-Yvette, France 2 SATIE, ENS Cachan, Universit´ e Paris 10, France 3 DEMR, ONERA, Palaiseau, France ✬ ✫ ✩ ✪ 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. ✬ ✫ ✩ ✪ Synthetic Aperture Radar (SAR) imaging s (t , u )= f (x , y ) p (t − τ (x , y , u )) dx dy k = k x k y = k cos(θ ) k sin(θ ) |k | = k = ω/c s (ω, u )= f (x , y ) exp [−j ωτ (x , y ,θ (u ))] dx dy = f (x , y ) exp [−j (k x x + k y y )] dx dy u (rad/m) v (rad/m) S(u,v) 15 20 25 30 35 40 45 50 55 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 u (rad/m) v (rad/m) S(u,v) 10 15 20 25 30 35 40 45 50 55 -70 -60 -50 -40 -30 -20 -10 0 10 Monostatic Bistatic & Multistatic k x = k cos(θ ) k y = k sin(θ ) k x = k (cos(θ tc )+ cos(θ cr ) k y = k (sin(θ tc )+ sin(θ cr ) ✬ ✫ ✩ ✪ Forward model: Fourier Synthesis f (x , y ) → G (k x , k y ), M (k x , k y ) Inverse problem: G (k x , k y ), M (k x , k y ) → f (x , y ) g (u i , v i ) = f (x , y ) exp [−j (u i x + v i y )] dx dy g = Hf + ǫ ✬ ✫ ✩ ✪ Bayesian Estimation Approach ◮ Forward model M : g = H f + ǫ ◮ Likelihood: p (g |f ; M)= p ǫ (g − H f ) ◮ 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 ✬ ✫ ✩ ✪ Proposed Bayesian Approach f = arg min f {J (f )= − ln p (g |f ) − ln p (f )} p (g |f ) ∝ exp 1 2σ ǫ 2 ‖g − Hf ‖ 2 ◮ Generalized Gaussian: p (f ) ∝ exp γ ∑ j |f j | 2 ◮ Cauchy: p (f ) ∝ exp γ ∑ j ln(1 −|f j | 2 ◮ Generalized Gauss-Markov p (f ) ∝ exp γ j |f j − f j −1 | β ◮ Gauss-Markov-Potts ✬ ✫ ✩ ✪ Comparison with classical methods fh(x,y) 0 50 100 150 200 250 0 50 100 150 200 250 fh(x,y) 0 50 100 150 200 250 0 50 100 150 200 250 Gerchberg-Papoulis Least Squares fh(x,y) 0 50 100 150 200 250 0 50 100 150 200 250 fh(x,y) 0 50 100 150 200 250 0 50 100 150 200 250 Quad. Reg. Proposed MAP ✬ ✫ ✩ ✪ Multistatic data fusion methods Method 1: Data Fusion followed by inversion G 1 (u , v ) M 1 (u , v ) − | G 2 (u , v ) M 2 (u , v ) − → G (k x , k y ) M (u , v ) → Inversion → f (x , y ) with G (u , v ) = G 1 (u ,v )+G 2 (u ,v ) 2 (u , v ) ∈ M 1 (u , v ) ∩ G 2 (u , v ) G 1 (u , v ) (u , v ) ∈ M 1 (u , v ) G 2 (u , v ) (u , v ) ∈ M 2 (u , v ) and M (k x , k y ) = M 1 (u , v ) ∪ M 2 (u , v ) Method 2: Separte inversion followed by image fusion G 1 (u , v ) M 1 (u , v ) − Inversion − f 1 (x , y ) | G 2 (u , v ) M 2 (u , v ) − Inversion − f 2 (x , y ) → Fusion → f (x , y ) ◮ Image fusion ◮ Coherent addition f (x , y ) =( f 1 (x , y ) + f 2 (x , y ))/2 ◮ Incoherent addition f (x , y ) =(| f 1 (x , y )| + | f 2 (x , y )|)/2 ✬ ✫ ✩ ✪ Bayesian Simultaneous Data Fusion and Inversion G 1 (u , v ) M 1 (u , v ) − | G 2 (u , v ) M 2 (u , v ) − → Joint Fusion and Inversion → f (x , y ) g 1 = H 1 f + ǫ 1 g 2 = H 2 f + ǫ 2 p (f |g 1 , g 2 ) ∝ p (g 1 |f ) p (g 2 |f ) p (f ) MAP : f = arg max f {p (f |g 1 , g 2 )} = arg min f {J (f )} with J (f )= − ln p (g 1 |f ) − ln p (g 2 |f ) − ln p (f ) = ‖g 1 − H 1 f ‖ 2 2σ 2 ǫ 1 + ‖g 2 − H 2 f ‖ 2 2σ 2 ǫ 2 + γ j [D f ] j | β ✬ ✫ ✩ ✪ Simulated data fh(x,y) 0 50 100 150 200 250 0 50 100 150 200 250 fh(x,y) 0 50 100 150 200 250 0 50 100 150 200 250 Data Fusion followed Joint Fusion by Inversion and Inversion ✬ ✫ ✩ ✪ Experimental data (Vv polarisation) BF1 band x (m) y (m) Reconstruction by backpropagation -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x (m) y (m) Reconstruction by backpropagation -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x (m) y (m) Reconstruction by backpropagation -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 BF2 band x (m) y (m) fh(x,y) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x (m) y (m) fh(x,y) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x (m) y (m) fh(x,y) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 BF1 & BF2 x (m) y (m) fh(x,y) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x (m) y (m) fh(x,y) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x (m) y (m) fh(x,y) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 ✬ ✫ ✩ ✪ References A. Mohammad-Djafari, F. Daout & Ph. Fargette, Fusion et inversion des signaux SAR pour obtenir une image super r ´ esolue, 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.