.
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.
1/25
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.
2/25
Synthetic Aperture Radar (SAR) imaging
s(t, u) = s(t, u) =
XX
ZZ
m
f (m, n) p(t − τm,n (u))
n
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.
3/25
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)
−55
v (rad/m)
−50 −45 −40
ky = k sin(θ)
−35 −30 −25 −20 15
s(ω, θ(u)) = P (ω) = P (ω) Sh. Zhu, A. Mohammad-Djafari et al.,
ZZ
ZZ
20
25
30
35 40 u (rad/m)
45
50
55
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.
4/25
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 =
2 (kx x + ky (y − u)) ω S(u,v) −70
−60
−50
v (rad/m)
−40
−30
−20
−10
0
10 10
15
20
25
30 35 u (rad/m)
40
45
50
55
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.
5/25
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)
S(u,v) −70
−70 −65
−60
3 −60 −50
−50
0
Tx
5
−40
v (rad/m)
−55
1 v (rad/m)
2
−45
−30
−40 −20
4 3
−35
2
−10 −30
1 0 5
−1
0
−25
4 −2
3
−20
10
2
−3 −4
1 0
15
20
25
30
35 40 u (rad/m)
45
50
55
10
15
20
25
30 35 u (rad/m)
40
45
50
55
T/T/R config.
◮
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.
6/25
Forward modeling as a Fourier Synthesis inverse problem G(kx , ky ) = M (kx , ky )F (kx , ky ) orginal target f(x,y)
log(1+abs(F(u,v))
0
100
150
200
250 50
100
150
200
250
f (x, y)
0
−0.4
−0.3
−0.3
−0.2
−0.2
−0.1
−0.1
0
0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0
D(u,v)
Masque
−0.4
50
50
100
150
200
250
0.5 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
F (kx , ky )
0.4
0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
M (kx , ky )
0.5
0
50
100
150
200
G(kx , ky )
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.
7/25
250
Fourier synthesis inverse problem g(ui , vi ) =
g(si ) =
Z
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.
8/25
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
10
0
−10
−20
◮
Interpolation: s(ω, θ(u)) −→ F (kx , ky )
−30
−40
−50
−60
−70
−80
◮ ◮
0
10
20
30
40
50
60
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.
9/25
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.
10/25
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.
11/25
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.
12/25
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.
13/25
Comparison of different inversion methods on data set 1 fh(x,y)
fh(x,y)
0
0
50
50
100
100
150
150
200
200
250
250 0
50
100
150
200
250
Gerchberg-Papoulis (GP)
0
50
100
150
200
250
Least Squares (LS)
fh(x,y)
fh(x,y)
0
0
50
50
100
100
150
150
200
200
250
250 0
50
100
150
200
250
Quad. Reg. (QR) Sh. Zhu, A. Mohammad-Djafari et al.,
0
50
100
150
200
250
Proposed method
SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.
14/25
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.
15/25
Multistatic data fusion methods Method 1: Data Fusion followed by inversion
Masque
D(u,v) 0
−0.4
−0.3
50
−0.2 100
−0.1
0 150
0.1
0.2 Muv
fh(x,y)
Guv
0.4 250
0.5 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0
50
100
150
200
−0.4
−0.3
−0.3
−0.2
−0.2
−0.1
−0.1
−0.4
50
0
0
0.1
0.1
0.2
0.2
−0.2 100
−0.1
150
0.1
0
0.2
D(u,v)
200
0
0.3
−0.4
0.3
0.3
0.4
−0.3
−0.3
250
M1 (u, v) G1 (u, v) Masque
−0.4
Gh(u,v)
0
200
0.3
0.4
0.4 250
50
0.5
0.5 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.5 0
50
100
150
200
250
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.2 100
−0.1
0 150
0.1
0.2
M (kx , ky )G(kx , ky )
b v) fb(x, y) G(u,
Data Fusion
Inversion
200
0.3
0.4 250
0.5 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0
50
100
150
200
250
M2 (u, v) G2 (u, v)
Sh. Zhu, A. Mohammad-Djafari et al.,
SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.
16/25
0.5
Multistatic data fusion methods Method 2: Separte inversion followed by image fusion
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.
17/25
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.
18/25
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.
19/25
Bayesian Approach for Simultaneous Data Fusion and Inversion Masque
D(u,v) 0
−0.4
−0.3
50
−0.2 100
−0.1
0 150
0.1
fh(x,y)
Gh(u,v)
0
0.2
−0.4
200
0.3
50
0.4
−0.3
−0.2
250
0.5 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0
M1 (u, v)
50
100
150
200
250
G1 (u, v)
Masque
100
−0.1
150
0.1
0
D(u,v) 0.2
0 200
−0.4
−0.3
0.3
50
0.4 250
−0.2
0.5 0
100
−0.1
150
0.2 200
0.3
100
150
fb(x, y)
0
0.1
50
0.4
200
250
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
b v) G(u,
250
0.5 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
M2 (u, v)
0.4
0.5
0
50
100
150
200
250
G2 (u, v) Joint Fusion & Inversion
Sh. Zhu, A. Mohammad-Djafari et al.,
SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.
20/25
0.4
0.5
Simulated target and the two data sets D(u,v)
Masque
0
−0.4
50
−0.3
−0.2
100
−0.1
f(x,y) 0 0
150
0.1
50 0.2
200
0.3
100 0.4
250
0.5 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
0.5
50
100
150
200
250
150
M1 (u, v)
200
G1 (u, v) D(u,v)
Masque
250 0
50
100
150
Original target f (x, y)
200
250
0
−0.4
50
−0.3
−0.2
100
−0.1
0
150
0.1
0.2
200
0.3
0.4
250
0.5 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
M2 (u, v)
Sh. Zhu, A. Mohammad-Djafari et al.,
0.4
0.5
0
50
100
150
200
250
G2 (u, v)
SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.
21/25
Comparison of data fusion methods
fh(x,y)
fh(x,y)
0
0
50
50
100
100
150
150
200
200
250
250 0
50
100
150
200
250
Data Fusion followed by Inversion
Sh. Zhu, A. Mohammad-Djafari et al.,
0
50
100
150
200
250
Joint Fusion and Inversion
SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.
22/25
Results on experimental data (Vv polarisation)
Reconstruction by backpropagation
Reconstruction by backpropagation −0.6
−0.4
−0.4
−0.4
−0.2
−0.2
−0.2
0
0.2
y (m)
−0.6
y (m)
y (m)
Reconstruction by backpropagation −0.6
0
0.2
0.4
0.4
0.6
0.4
0.6 −0.6
−0.4
−0.2
0 x (m)
0.2
Sh. Zhu, A. Mohammad-Djafari et al.,
0.4
0.6
0
0.2
0.6 −0.6
−0.4
−0.2
0 x (m)
0.2
0.4
0.6
−0.6
−0.4
−0.2
0 x (m)
0.2
0.4
0.6
SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.
23/25
Results on experimental data (2 bands fusion) Reconstruction by backpropagation
−0.4
−0.4
−0.4
−0.2
−0.2
−0.2
0
y (m)
−0.6
0.2
0
0.2
0.4
BF1 band
0.4
0.6 −0.6
−0.4
−0.2
0 x (m)
0.2
0.4
0.6
0.6 −0.6
−0.4
−0.2
0.2
0.4
0.6
−0.6
−0.6
−0.4
−0.4
−0.4
−0.2
−0.2
−0.2
0
y (m)
−0.6
0
0.2
0.4
−0.2
0 x (m)
0.2
0.4
0.6
−0.4
−0.2
0 x (m)
0.2
0.4
0.6
−0.6
−0.4
−0.2
−0.2
−0.2
y (m)
−0.6
−0.4
y (m)
−0.6
−0.4
0
0.2
Sh. Zhu, A. Mohammad-Djafari et al.,
−0.4
−0.2
0 x (m)
−0.2
0.2
0.4
0.6
0 x (m)
0.2
0.4
0.6
0.2
0.4
0.6
0
0.4
0.6 −0.6
−0.4
0.2
0.4
0.6
0.6
fh(x,y)
−0.6
0.4
0.4
0
fh(x,y)
0.2
0.2
0.6 −0.6
fh(x,y)
0
0 x (m)
0.4
0.6 −0.4
−0.2
0.2
0.4
0.6 −0.6
−0.4
fh(x,y)
−0.6
0.2
y (m)
0 x (m)
fh(x,y)
y (m)
y (m)
fh(x,y)
BF1 & BF2
0
0.2
0.4
0.6
BF2 band
Reconstruction by backpropagation
−0.6
y (m)
y (m)
Reconstruction by backpropagation −0.6
0.6 −0.6
−0.4
−0.2
0 x (m)
0.2
0.4
0.6
−0.6
−0.4
−0.2
0 x (m)
SPIE, Electronic Imaging, San Francisco, January 23-27, 2011.
24/25
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.
25/25