Bayesian Approach for Inverse Problem in Microwave Tomography

Transcription

1 Bayesian Approach for Inverse Problem in Microwave Tomography Hacheme Ayasso Ali Mohammad-Djafari Bernard Duchêne Laboratoire des Signaux et Systèmes, UMRS (CNRS-SUPELEC-UNIV PARIS SUD 11), 3 rue Joliot-Curie, F Gif-sur-Yvette cedex, France Journées Problèmes Inverses et Optimisation de Forme décembre 2008

2 Introduction Microwave Tomography: Reconstruction of unknown object (electrical permetivity and conductivity) from several measurement around it. S Interaction between incident field and object Scattered field measurement around the object y z x Several source positions Several excitation frequencies E inc D Application: Medical imaging (Breast Cancer), burried object detection, NDT H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

3 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 Method 1 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, Decemeber / 23

4 Outline 1 Forward Model Formulation Difficulties Gradiant-FFT methods Validation 2 Bayesian Inversion Approach Formulation Prior Model Iterative Linearization Joint estimation 3 Conclusion 4 References H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

5 1. Forward Model H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

6 Forward Formulation(Continuous) Using the integral equation representation for electrical field ( E scat ( r) = ) r r G( r, r )χ( r ) E( r )d r, k 2 0 ( E( r) = E inc ( r) ) k0 2 r r G( r, r )χ( r ) E( r )d r, D D r S r D S χ( r) = ω 2 ε 0 µ 0 ε r (r) + iωµ 0 σ(r) E( r), r D: total field in the domain of interest D E scat, r S: scatter field in measuement domain S Foward problem: χ, E inc E scat, Inverse problem: E scat, E inc χ E inc y z D x H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

7 Forward Formulation(Discretization) Two derivatives G (particular consideration for non-integrable singularity) ( E scat ( r) = G( r, r ) + 1 ) D k0 2 r r G( r, r ) χ( r ) E( r )d r Method of Moments (MoM), Cubical voxels E, χ constants inside the voxel Green Dyadic function is calculated on a sphere with the same size of a voxel Discret model, ɛ c & ɛ o Measurement, model, and discretization errors E scat = G o (X E) + ɛ o (Observation) E = E inc + G c (X E) + ɛ c (Coupling) H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

8 Difficulties 1 Nonlinear forward model calculate E ( ( ) ) E scat = G o X I G c X T 1 E inc + ɛ 2 A = ( I G c X T ) Large full complex matrix Memory problem. E = A 1 E inc High computational cost Solution: Gradiant-FFT Method Cost Min required(one frequency!) voxel per axe n 64 X 2 n 3 N f 0.5Me = 1.5MB E ou E inc N = 6 n 3 N f 1.5Me = 12MB G c ou A 18 n 6 N f 1.2Te = 10TB GE (Mem/Cal) O(N 2 )/O(N 3 ) 10TB/0.5Em CG (Mem/Cal) O(N)/O(K N 2 ) 12MB/K 10Tm CG-FFT O(N)/O(K N log(n)) 12MB/10Mm H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

9 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 Fourier domain (Need for zero padding circularity ) Forward operator Adjoint operator A (E) = E FT 1 ( FT (G c ) FT (X E) ) A (E) = E X FT 1 ( FT (G c ) FT (E) ) BiCGSTAB-FFT [Xu2002] High convergence speed compared to classsical CG-FFTs method H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

10 Validation Comparaison with scattered field real data. Amplitude H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

11 Validation Comparaison with scattered field real data. Phase H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

12 2.Bayesian Inversion Approach H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

13 Formulation Bayes Formula p ( X E scat ; M ) = p ( E scat X ; M ) p (X M) p ( E scat M ) Errors Model p(ɛ) Likelihood p(e scat X ; M) Prior information over the contrast p(x M) Model Evidence p(e scat M) Estimation MAP : ˆX = argmax(p(x E; M)) X MP : ˆX = E(X ) p(x E scat ;M) H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

14 Prior Model Prior information: objects are composed of a finite number of homogeneous materials Hidden field representing materials z Homogeneity within a class p(x z) N (µ z, v z ) H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

15 Prior Model Prior information: objects are composed of a finite number of homogeneous materials 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, Decemeber / 23

16 Prior Model Prior information: objects are composed of a finite number of homogeneous materials 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, Decemeber / 23

17 Hierarchical Model Partialy unsupervised approach Estimation of most of the model hyperparameter Object MIG : p(x (r) z(r) = k, m k, v k ) = N (m k, v k ) MGM : p(x (r) z(r), m k, v k, f (r ), z(r ), r V(r)) = N (µ z(r), v z(r)) Hidden field m 0, v 0 a 0,b 0 a e0,b e0 h P p(z γ) exp r R Φ(z(r)) + 1 γ P P i 2 r R r V(r) δ(z(r) z(r )) m k v k v 0 Hyper-parameters: p(m k m 0, v 0) = N (m 0, v 0), k p(v 1 k a 0, b 0) = G(a 0, b 0), k p(α α 0) = D(α 0,, α 0) p(vɛ 1 a e0, b e0 ) = G(a e0, b e0 ) α k X z k S + + E scat Constants(Hyper-hyperparameter): m 0, v 0, a 0, b 0, α 0, a e0, b e0, γ α 0 H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

18 Iterative Linearization Nonlinear Forward model Estimate the total field E in alternance with the contrast X and other model parameters. Algorithm 1 Initial guess for total field E = E inc (Born approximation) and Model parameter 2 E itr 1, z itr 1, θ itr 1 MAP X itr (A linear inverse problem) p(x itr E itr 1, E scat, z itr 1, θ itr 1 ) p(e scat X itr, E itr 1, z itr 1, θ itr 1 )p(x E itr 1, z itr 1, θ itr 1 ) 3 X itr Forward Model 4 X itr MAP E itr z itr, θ itr (segmentation problem) p(z itr, θ itr X itr ) p(x itr z itr, θ itr )p(z itr, θ itr ) H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

19 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, Decemeber / 23

20 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, Decemeber / 23

21 Perspective: Joint Estimation Iterative estimation Low contrast, Local minimum solution Nonlinear forward model (but bilinear) Joint estimation of current density J = X E and Contrast X E scat = G o J + ɛ o Observation J = X E inc + X G c J + ɛ o 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, Decemeber / 23

22 Conclusion Microwave Tomography: Nonlinear ill-posed inverse problem with high computational cost. BiCGSTAB-FFT method for forward problem. Bayesian framework for inverse problem. Hierarchical mixture model to account for the knowmedge of number of materials and piecewise homogeneity prior information. Iterative estimation: better results than conventional methods Perspective: Joint estimation of current density, contrast, hidden field, and hyperparameters Application of Variational Bayes approximation technique for the posterior H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23

23 References H. Ayasso, B. Duchêne, A. Mohammad-Djafari, A Bayesian approach to microwave imaging in a 3-D configuration,in OIPE, O. Féron, B. Duchêne, 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. Féron, B. Duchêne, A. Mohammad-Djafari, Microwave imaging of piecewise constant objects in a 2D-TE configuration, Int. J. Appl. Electromagn. Mechan., vol. 26, 2007, pp 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, IEEE, vol. 2, H. Ayasso, A. Mohammad-Djafari, Variational Bayes With Gauss-Markov-Potts Prior Models For Joint Image Restoration And Segmentation, in VISSAP, H.AYASSO & al. (LSS(CNRS-SUPELEC-UPS)) Bayesian Microwave Tomography Reconstruction Nantes, Decemeber / 23