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Bayesian Approach for Inverse Problem in Microwave Tomography
Hacheme Ayasso
Ali Mohammad-Djafari Bernard Duchene
Laboratoire des Signaux et Systemes, UMRS 08506 (CNRS-SUPELEC-UNIV PARIS SUD 11),3 rue Joliot-Curie, F-91192 Gif-sur-Yvette cedex, France
Journees Problemes Inverses et Optimisation de Forme17 - 18 decembre 2008
Introduction
Microwave Tomography: Reconstruction of unknown object (electricalpermetivity and conductivity) from several measurement around it.
Interaction betweenincident field andobject → Scatteredfield measurementaround the object
Several sourcepositions
Several excitationfrequencies
Einc
y
z
x
D
S
Application: Medical imaging (Breast Cancer), burried object detection, NDT
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 2 / 23
General Context
Forward Model
~E scat = S (χ)
~E scat : scatter field measurement around the object (data)
χ : unknown object (contrast function) under test
S : system transfere function (Forward Model).
Difficulties
1 Calculating χ given ~E scat & S → An ill-posed inverse problem (Hadamard).
2 S is non-linear.
3 High computational cost.
Proposed Method1 Bayesian inference framework with Gauss-Markov Potts priors
2 Joint source contrast estimation
3 Careful choice of calculation methods & parallel programing
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 3 / 23
Outline
1 Forward ModelFormulationDifficultiesGradiant-FFT methodsValidation
2 Bayesian Inversion ApproachFormulationPrior ModelIterative LinearizationJoint estimation
3 Conclusion
4 References
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 4 / 23
1. Forward Model
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 5 / 23
Forward Formulation(Continuous)
Using the integral equation representation for electrical field
~E scat(~r) =
(1 +
1
k20
~∇~r ~∇~r)∫
D
G(~r ,~r ′)χ(~r ′)~E (~r ′)d~r ′, ~r ∈ S
~E (~r) = ~E inc(~r) +
(1 +
1
k20
~∇~r ~∇~r)∫
D
G(~r ,~r ′)χ(~r ′)~E (~r ′)d~r ′, ~r ∈ D
χ(~r) = ω2ε0µ0εr (r) + iωµ0σ(r)~E (~r),~r ∈ D: total field in thedomain of interest D~E scat ,~r ∈ S : scatter field inmeasuement domain SFoward problem: χ, ~E inc → ~E scat ,Inverse problem: ~E scat , ~E inc → χ Einc
y
z
x
D
S
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 6 / 23
Forward Formulation(Discretization)
Two derivatives → G (particular consideration for non-integrable singularity)
~E scat(~r) =
∫D
(G(~r ,~r ′) +
1
k20
~∇~r ~∇~rG(~r ,~r ′)
)χ(~r ′)~E (~r ′)d~r ′
Method of Moments (MoM), Cubical voxels~E , χ constants inside the voxelGreen Dyadic function is calculated on a sphere with the same size of a voxel
Discret model, εc & εo → Measurement, model, and discretization errors
E scat = G o × (X ◦ E ) + εo (Observation)
E = E inc + G c × (X ◦ E ) + εc (Coupling)
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 7 / 23
Difficulties
1 Nonlinear forward model → calculate E
E scat = G o ×(
X ◦(I − G c • XT
)−1
× Einc
)+ ε
2 A =(I − G c • XT
)Large full complex matrix
Memory problem.E = A−1E inc High computational cost
Solution: “Gradiant-FFT” Method
Cost Min required(One frequency!)voxel per axe n 64X 2× n3 × Nf ≈ 0.5Me = 1.5MB
E ou E inc N = 6× n3 × Nf ≈ 1.5Me = 12MBG c ou A 18× n6 × Nf ≈ 1.2Te = 10TBGE (Mem/Cal) O(N2)/O(N3) 10TB/0.5EmCG (Mem/Cal) O(N)/O(K × N2) 12MB/K × 10TmCG-FFT O(N)/O(K × N log(N)) 12MB/10Mm
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 8 / 23
Gradiant-FFT methods
Solve large complex linear system
E inc =(E − G c × (X ◦ E )
)= A (E )
G c × (X ◦ E )⇔ Discret Convolution product → G o is diagonal in Fourierdomain (Need for zero padding ”circularity”)
Forward operator
A (E ) = E − FT−1(FT (G c)× FT (X ◦ E )
)Adjoint operator
A† (E ) = E − X ∗ ◦ FT−1(FT (G c)× FT (E )
)BiCGSTAB-FFT [Xu2002] High convergence speed compared to classsicalCG-FFTs method
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 9 / 23
Validation
Comparaison with scattered field real data.
Amplitude
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 10 / 23
Validation
Comparaison with scattered field real data.
Phase
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 11 / 23
2.Bayesian Inversion Approach
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 12 / 23
Formulation
Bayes Formula
p(X |E scat ;M
)=
p(E scat |X ;M
)p (X |M)
p(E scat |M
)Errors Model p(ε)→ Likelihood p(E scat |X ;M)
Prior information over the contrast→ p(X |M)
Model Evidence p(E scat |M)
Estimation
MAP : X = argmaxX
(p(X |E ;M))
MP : X = E (X )p(X |E scat ;M)
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 13 / 23
Prior ModelPrior information: objects are composed of a finite number of homogeneousmaterials
Hidden field representing materials z
Homogeneity within a class → p(X |z) ∼ N (µz , vz)
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 14 / 23
Prior ModelPrior information: objects are composed of a finite number of homogeneousmaterials
Hidden field representing materials z
Spatial dependency between hidden field sites → Potts prior p(z)
Independent prior for X |z → Mixture of Independent Gaussians (MIG)
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 15 / 23
Prior ModelPrior information: objects are composed of a finite number of homogeneousmaterials
Hidden field representing materials z
Spatial dependency between hidden field sites → Potts prior p(z)
Gauss markov prior for X |z → Mixture of Gauss-Markov (MGM)
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 16 / 23
Hierarchical Model
Partialy unsupervised approach → Estimation of most of the model hyperparameterObject
MIG : p(X (r)|z(r) = k,mk , vk) = N (mk , vk)MGM : p(X (r)|z(r),mk , vk , f (r ′), z(r ′), r ′ ∈ V(r)) = N (µz(r), vz(r))
Hidden field
p(z |γ) ∝ exphP
r∈R Φ(z(r))
+ 12γ
Pr∈R
Pr′∈V(r) δ(z(r)− z(r ′))
iHyper-parameters:
p(mk |m0, v0) = N (m0, v0), ∀kp(v−1
k |a0, b0) = G(a0, b0), ∀kp(α|α0) = D(α0, · · · , α0)p(v−1
ε |ae0 , be0 ) = G(ae0 , be0 )
Constants(Hyper-hyperparameter):m0, v0, a0, b0, α0, ae0 , be0 , γ
X
zk
αk
mkvk
m0 , v0
α0
S + +
0v
a0 ,b0 ae0 ,be0
Escat
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 17 / 23
Iterative Linearization
Nonlinear Forward model → Estimate the total field E in alternance with thecontrast X and other model parameters.
Algorithm
1 Initial guess for total field E = E inc (Born approximation) and Modelparameter
2 E itr−1, z itr−1, θitr−1 -MAP
X itr (A linear inverse problem)
p(X itr |E itr−1, Escat, z itr−1, θitr−1) ∝ p(E scat |X itr , E itr−1, z itr−1, θitr−1)p(X |E itr−1, z itr−1, θitr−1)
3 X itr -Forward Model
E itr
4 X itr -MAP
z itr , θitr (segmentation problem)
p(z itr , θitr |X itr ) ∝ p(X itr |z itr , θitr )p(z itr , θitr )
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 18 / 23
Alternated Estimation (Application 1)
MIG prior with independent hidden field sites:Sphere
(a) BIM (Real Part) (b) BIM (imaginary Part)
(c) Bayesian(Real Part) (d) Bayesian (Imaginary Part)
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 19 / 23
Alternated Estimation (Application 2)
Two spheres:
(a) BIM (Real Part) (b) BIM (imaginary Part)
(c) Bayesian(Real Part) (d) Bayesian (Imaginary Part)
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 20 / 23
Perspective: Joint Estimation
Iterative estimation → Low contrast, Local minimum solution
Nonlinear forward model (but bilinear) → Joint estimation of current densityJ = X ◦ E and Contrast X
E scat = G oJ + εo Observation
J = X ◦ E inc + X ◦ G cJ + εo Contrast-Coupling
Joint posterior of currents J, contrast X , hidden field z , and hyperparameters θ
p(J,X , z , θ|E scat) ∝ p(E scat |J,X )p(J|X )p(X |z , θ)p(z |θ)p(θ)
”Observation” → p(E scat |J,X ) , ”Contrast-Coupling” → p(J|X )Prior model → p(X |zu), p(z |θ), p(θ)
Estimator → Stochastic sampling or Variational Bayes technique
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 21 / 23
Conclusion
Microwave Tomography: Nonlinear ill-posed inverse problem with highcomputational cost.
BiCGSTAB-FFT method for forward problem.
Bayesian framework for inverse problem.
Hierarchical mixture model to account for the knowmedge of number ofmaterials and piecewise homogeneity prior information.
Iterative estimation: better results than conventional methods
Perspective:
Joint estimation of current density, contrast, hidden field, and hyperparametersApplication of Variational Bayes approximation technique for the posterior
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 22 / 23
References
H. Ayasso, B. Duchene, A. Mohammad-Djafari,“A Bayesian approach to microwave
imaging in a 3-D configuration”,in OIPE, 2008.
O. Feron, B. Duchene, A. Mohammad-Djafari, ”Microwave imaging of
inhomogeneous objects made of a finite number of dielectric and conductive
materials from experimental data”, Inverse Problems, vol. 21, no. 6, 2005, pp.
S95–S115.
O. Feron, B. Duchene, A. Mohammad-Djafari, ”Microwave imaging of piecewise
constant objects in a 2D-TE configuration”, Int. J. Appl. Electromagn. Mechan.,
vol. 26, 2007, pp. 167–174.
X. Xu, Q.H. Liu, et Z.W. Zhang, “The stabilized biconjugate gradient fast Fourier
transform method for electromagnetic scattering,” Antennas and Propagation
Society International Symposium, 2002. IEEE, vol. 2, 2002.
H. Ayasso, A. Mohammad-Djafari, ”Variational Bayes With Gauss-Markov-Potts
Prior Models For Joint Image Restoration And Segmentation”, in VISSAP, 2008.
H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, 17-18 Decemeber 2008 23 / 23
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