Inverse problems arising in different synthetic

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). Sh. Zhu, A. Mohammad-Djafari et al.,. SPIE, Electronic Imaging, San ...
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Inverse problems arising in different synthetic aperture radar imaging systems and a general Bayesian approach for them Sha Zhua,b , Ali Mohammad-Djafaria , Xiang Lib and Junjie Maoc a

Laboratoire des Signaux et Syst`emes, UMR8506 CNRS-SUPELEC-Univ Paris Sud, Gif-sur-Yvette, France; b Research Institute of Space Information Technology, School of Electronic Science and Engineering, c Research Center of Microwave Technology, School of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, China;

Presenter: Ali Mohammad-Djafari [email protected] SPIE, Electronic Imaging, San Francisco, January 23-27, 2011 Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Summary ◮

Introduction to SAR imaging



Monostatic, Bistatic and Multistatic SAR imaging



Forward modeling as a Fourier Synthesis inverse problem



Classical inversion methods ◮ ◮

Inverse Fourier Transform Least square and deterministic regularization



Bayesian estimation approach



Proposed method of joint data fusion and super-resolution reconstruction



Simulation and experimental data results



Conclusions and Discussions

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Synthetic Aperture Radar (SAR) imaging

s(t, u) = s(t, u) =

XX

ZZ

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f (m, n) p(t − τm,n (u))

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f (x, y) p(t − τ (x, y, u)) dx dy

ZZ s(ω, θ(u)) = P (ω) f (x, y) exp {−jωτ (x, y, θ(u))} dx dy     k cos(θ) kx = k= |k| = k = ω/c ky k sin(θ) Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Monostatic, Bistatic and Multstatic cases ◮

Mono-static case (same transmitter-receivers) ZZ s(ω, θ(u)) = P (ω) f (x, y) exp {−jωτ (x, y, θ(u))} dx dy τ (x, y, u(θ)) =

2 2p 2 x + (y − u)2 = (kx x + ky (y − u)) c ω S(u,v) −70 −65 −60

kx = k cos(θ)

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ky = k sin(θ)

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s(ω, θ(u)) = P (ω) = P (ω) Sh. Zhu, A. Mohammad-Djafari et al.,

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f (x, y) exp {−jωτ (x, y, θ(u))} dx dy f (x, y) exp {−j(kx x + ky y)} dx dy

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Bistatic and Multstatic cases ◮ ◮

Bistatic case (one transmitter, many receivers) Multistatic case (one transmitter, many receivers) ZZ s(t, u(θ)) = f (x, y) p(t − τtc (x, y) − τrc (x, y, u(θ))) dx dy τtc + τcr =

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kx = k (cos(θtc ) + cos(θcr ) ky = k (sin(θtc ) + sin(θcr ) s(ω, θ(u)) = P (ω) Sh. Zhu, A. Mohammad-Djafari et al.,

ZZ

f (x, y) exp {−j(kx x + ky y)} dx dy

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Monostatic, Bistatic and Multstatic cases s(ω, θ(u)) = P (ω)

ZZ

f (x, y) exp {−j(kx x + ky y)} dx dy

|P (ω)| = 1

ω ∈ [ωmin , ωmax ]

S(u,v)

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 Monostatic  Bistatic & Multistatic kx = k cos(θ) kx = k (cos(θtc ) + cos(θcr ) ky = k sin(θ) ky = k (sin(θtc ) + sin(θcr ) For each position of transmitter/receiver we get information on the Fourier domain of the scene on a ligne segment which length is proportional to the bandwidth of the transmitted signal and its orientation depends on the relative positions of Transmitter-Scene-Receiver

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Forward modeling as a Fourier Synthesis inverse problem G(kx , ky ) = M (kx , ky )F (kx , ky ) orginal target f(x,y)

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Forward model: f (x, y) −→ G(kx , ky ), M (kx , ky ) Inverse problem: G(kx , ky ), M (kx , ky ) −→ f (x, y) Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Fourier synthesis inverse problem g(ui , vi ) =

g(si ) =

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Z Z

f (x, y) exp {−j(ui x + vi y)} dxdy g = Hf + ǫ

f (r) h(si , r) dr −→ Discretization: f (r) =

X

fj bj (r)

j

g = Hf + ǫ



H Forward FT matrix



Hf : Fourier transform of f



H t g: IFT of g assuming the missing data are equal to zero.



Remark: HH t = I but H t H 6= I.

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Classical analytical methods s(ω, θ(u)) = P (ω) = P (ω)

ZZ

ZZ

f (x, y) exp {−jωτ (x, y, θ(u))} dx dy f (x, y) exp {−j(kx x + ky y)} dx dy

Assuming |P (ω)| = 1, we can write ZZ f (x, y) = s(ω, θ(u)) exp {+j(kx x + ky y)} dkx dky plan kx ky 20

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Interpolation: s(ω, θ(u)) −→ F (kx , ky )

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Inverse Fourier Transform: F (kx , ky ) −→ IF T −→ f (x, y) All the unknown values of F (kx , ky ) are assumed to be equal to zero.

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Bayesian estimation approach M:

g = Hf + ǫ



Observation model M + Hypothesis on the noise ǫ −→ 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)

Link with regularization : Maximum A Posteriori (MAP) : b = arg max {p(f |g)} = arg max {p(g|f ) p(f )} f f f = arg min {− ln p(g|f ) − ln p(f )} f with Q(g, Hf ) = − ln p(g|f ) and λΩ(f ) = − ln p(f ) But, Bayesian inference is not only limited to MAP Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Proposed method: Bayesian with different a priori b = arg min {J(f ) = − ln p(g|f ) − ln p(f )} f f   1 2 kg − Hf k p(g|f ) ∝ exp 2σǫ2 ◮ ◮ ◮ ◮

o n P p(f ) ∝ exp γ j |fj |2 o n P Cauchy p(f ) ∝ exp γ j ln(1 − |fj |2 nP o 2 Sparse Gaussian p(f ) ∝ exp γ |f | j j j Generalized Gaussian

Generalized Gauss-Markov      X   X  p(f ) ∝ exp γ |fj − fj−1 |β ∝ exp γ |[Df ]j |β     j



j

Gauss-Markov-Potts model

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Sparse Gaussian prior model ◮

Sparse Gaussian model:

    X p(f ) ∝ exp − γj |fj |2   j



Full Bayesian:

p(f , θ|g) ∝ p(g|f , θ 1 ) p(f |θ 2 ) p(θ) θ 1 = {σǫ2 },

◮ ◮ ◮

θ 2 = {γj }

Inverse Gamma priors for θ b , θ) b = arg max Joint MAP: (f Marginalizing

p(θ|g) =

Z

f ,θ {p(f , θ|g)} or

p(f , θ|g) df

b = arg max {p(θ|g)} −→ f b = arg max {p(f |θ; g)} θ θ f Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Gauss-Markov-Potts prior models for images ”In many imaging applications, the objects are, in general, composed of a finite number of materials, and the pixels/voxels corresponding to each materials are grouped in compact regions”

How to model this prior information?

f (r) z(r) ∈ {1, ..., K} p(f (r)|z(r) = k) = N (mk , vk )     X p(z(r)|z(r ′ ), r ′ ∈ V(r)) ∝ exp γ δ(z(r) − z(r ′ ))  ′  r ∈V(r ) p(f , z, θ|g) ∝ p(g|f , θ 1 ) p(f |z, θ 2 ) p(z|γ) p(θ) Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Comparison of different inversion methods on data set 1 fh(x,y)

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Multistatic data fusion methods Method 1: Data Fusion followed by inversion

G1 (u, v) − M1 (u, v) G(kx , ky ) b v) −→ Inversion −→ fb(x, y) −→ G(u, |−→ M (kx , ky ) G2 (u, v) − M2 (u, v) with   (G1 (u, v) + G2 (u, v))/2 (u, v) ∈ M1 (u, v) ∩ G2 (u, v) G (u, v) G(kx , ky ) = (u, v) ∈ M1 (u, v)  1 G2 (u, v) (u, v) ∈ M2 (u, v) and

M (kx , ky ) = M1 (u, v) ∪ M2 (u, v) Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Multistatic data fusion methods Method 1: Data Fusion followed by inversion

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Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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G1 (u, v) − Inversion − fb1 (x, y) M1 (u, v) b v) |−→ Fusion −→ fb(x, y) −→ G(u, G2 (u, v) − Inversion − fb2 (x, y) M2 (u, v) ◮

Image fusion ◮ ◮



Coherent addition Incoherent addition

fb(x, y) = (fb1 (x, y) + fb2 (x, y))/2 b f (x, y) = (|fb1 (x, y)| + |fb2 (x, y)|)/2

May need image registration

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Multistatic data fusion methods Method 3: Simultaneous Data Fusion and Inversion

G1 (u, v) − M1 (u, v) | −→ G2 (u, v) − M2 (u, v)

Fusion et Inversion 

b 1 (u, v) −G b −→ f (x, y) −→ | b 2 (u, v) −G

g 1 = H 1 f + ǫ1 g 2 = H 2 f + ǫ2

Regularization: J(f ) = kg 1 − H 1 f k2 + kg 2 − H 2 f k2 + λkDf k2

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Bayesian Approach for Simultaneous Data Fusion and Inversion ◮

Joint MAP with sparse Gaussian model: p(f , θ|g 1 , g 2 ) ∝ p(g 1 |f , σǫ21 ) p(g 2 |f , σǫ22 ) p(f |{γj }) p(θ) θ = {σǫ21 , σǫ22 , {γj }}



b , θ) b = arg max {p(f , θ|g 1 , g 2 )} (f (f ,θ ) Gibbs sampling with Gauss-Markov-Potts model: p(f , z, θ|g 1 , g 2 ) ∝ p(g 1 |f , σǫ21 ) p(g 2 |f , σǫ22 ) p(f |z, {mk , vk }) p(θ) θ = {σǫ21 , σǫ22 , {mk , vk }}



Note that, in both cases, the estimation of f , we optimize: J(f ) = − ln p(g 1 |f ) − ln p(g 2 |f ) − ln p(f ) X 1 1 2 2 = f k + f k + γj fj2 − H − H kg kg 1 1 2 2 2σǫ21 2σǫ22 j

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Bayesian Approach for Simultaneous Data Fusion and Inversion Masque

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SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Comparison of data fusion methods

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SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Results on experimental data (Vv polarisation)

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SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Results on experimental data (2 bands fusion) Reconstruction by backpropagation

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SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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Conclusions and Perspectives ◮

Bayesian estimation framework is an appropriate one for handeling inverse problems and in particular Fusion and inversion of SAR imaging data



Proposed methods show good results both on simulated and experimental data



For experimental data, we still need to account for polarisation information



Present forward modeling assumes a scene with non interacting real point sources



More accurate forward models are needed for accounting for real scenes: Complexe valued, interacting, polarisation, multiple trajectories, ...

Sh. Zhu, A. Mohammad-Djafari et al.,

SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.

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