Bayesian Microwave Breast Imaging

Transcription

1 Bayesian Microwave Breast Imaging Leila Gharsalli 1 Hacheme Ayasso 2 Bernard Duchêne 1 Ali Mohammad-Djafari 1 1 Laboratoire des Signaux et Systèmes (L2S) (UMR8506: CNRS-SUPELEC-Univ Paris-Sud) Plateau de Moulon, Gif-sur-Yvette, FRANCE. 2 GIPSA-LAB, Département Image Signal (CNRS-Univ Grenoble) BP , Saint Martin d Hères, France.

2 Summary 1 Introduction 2 Forward problem 3 Inversion 4 Inversion 5 Application and results 6 Conclusions and prospects 7 References 2/18

3 Motivation Breast cancer Most commonly diagnosed cancer Second cause of cancer death for women Need to early stage cancer detection Detection techniques X-ray mammography: Ionizing radiation, costly, high false-positive rate, low sensitivity Ultrasound and MRI: High spatial resolution but luck of functional information Microwave imaging: non negligible dielectric contrast between cancerous and normal healthy breast tissues, non-ionizing nature, easy accessibility Promising alternative for breast tumor detection 3/18

4 Microwave imaging Different source positions Several receivers and frequencies in the microwave range Retrieve unknown target (electrical permittivity and conductivity) from measurements of the scattered field, the incident wave being known Non-linear ill-posed inverse problem 4/18

5 Domain integral representation E(r): total field in the object Observation equation y(r) = k1 2 G(r, r ) χ(r )E(r) dr D Coupling equation E(r) = E inc (r)+k1 2 G(r, r ) χ(r )E(r) dr χ(r) = (k(r) 2 k1 2)/k2 1 : normalized contrast function k(r) 2 = ω 2 ɛ 0 µ 0 ɛ r (r) + iωµ 0 σ(r), k 1 the outside medium G(r, r ): Green s function D = propagation constant of 5/18

6 Discretized model Test domain D partitioned into N D elementary pixels Coupling equation rewritten in terms of contrast sources: w(r ) = χ(r )E(r ) Method of moments [Harrington, 1993] bilinear discret model: { y = G o w + ɛ (observation) w = χ E inc + χ G c w + ξ (coupling) ɛ, ξ: model and measurements errors 6/18

7 Bayesian framework { y = G o w + ɛ (observation) w = χ E inc + χ G c w + ξ (coupling) Let consider the first equation: y = G o w + ɛ, p(ɛ) = N (ɛ 0, σ 2 ɛi) Bayes rule: p(y w) = p(y w)p(w) p(y) { p(y w) = N (y G o w, σɛi) 2 p(w) = N (w 0, σw 2 p(w y) = N (w ŵ, I) Σ) { ŵ = [G ot G o + λi] 1 G ot y Σ = [G o G o + λi] 1 We can do the same with the second equation. 7/18

8 Bayesian framework p(χ, w, Θ y; M) = p(y w, Θ; M) p(w χ, Θ; M) p(χ Θ; M) p(θ M) p(y M) Θ: hyper-parameters p(y w, Θ; M) (observation equation) p(w χ, Θ; M) (coupling equation) p(χ Θ; M): a priori on the object p(θ M): a priori information on the hyperparameters p(χ, w, Θ y; M): a posteriori 8/18

9 Mixture of independant Gaussian laws (MGI): pixels of the same class are assumed independant Gauss-Markov mixture model (MGM): a high correlation between pixels of the same class by means of a Markov field 9/18 Gauss-Markov-Potts prior Objects composed of a finite number K of materials distributed into homogenous and compact regions

10 Gauss-Markov-Potts prior Homogeneity of a class: p(χ(r) z(r)) = N (m k (r), v k (r)), k = 1,..., K, - v k (r) = v k r R k m k if C(r) = 1 - m k (r) = 1 χ(r ) if C(r) = 0, N V r V(r) C(r) = 1 r V(r) δ(z(r ) z(r)), V(r) is a neighborhood of r made of the four nearest pixels and N V = card(v). Compactness of the regions (Potts): p(z λ) exp r D ( Φ(z(r)) + λ 2 r V(r) δ (z(r) z(r ))) 10/18

11 Probabilistic modeling Model and measurement errors: ɛ N (0, v ɛ I), ξ N (0, v ξ I) Hyperparameters: p(m k ) = N (µ, τ), p(v k ) = IG(η, φ) p(v ɛ ) = IG(η ɛ, φ ɛ ), p(v ξ ) = IG(η ξ, φ ξ ) A posteriori: p (χ, w, z, Θ y) p (y w, v ɛ ) p (w χ, v ξ ) p (χ z, m, v) p (z λ) p (m µ, τ) p (v η, φ) p (v ɛ η ɛ, φ ɛ ) p (v ξ η ξ, φ ξ ) with Θ = {m, v, v ɛ, v ξ }. 11/18

12 Bayesian inversion techniques p (χ, w, z, Θ y) p (y w, v ɛ ) p (w χ, v ξ ) p (χ z, m, v) p (z λ) p (m µ, τ) p (v η, φ) p (v ɛ η ɛ, φ ɛ ) p (v ξ η ξ, φ ξ ) Joint Maximum A Posteriori (JMAP) (intractable solutions in non-linear cases) [Gharsalli et al., 2014] needs to approximate the joint a posteriori distribution Monte-Carlo Markov Chain sampling (MCMC) (costly) [Féron, 2006] Variational Bayesian Approach (VBA) (less costly alternative) [Ayasso et al., 2012] 12/18

13 Variational Bayesian Approach (VBA) [Mackay, 2003] Aim: Approximating p(χ, w, z, Θ y; M) by a separable law: q(χ, w, z, Θ) = q 1 (χ)q 2 (w)q 3 (z)q 4 (Θ) Criterion: KL(q : p) = q ln q qp p q = = F(q) + ln p(y M) Free negative energy: ( ) F(q) = ln p u,y M ( ), u = {χ, w, z, Θ} q u q Minimize KL(q : p) Maximize F(q) Find: q opt = arg maxq F(q) { } q(u i ) exp log(p(u, y)) q(u j i j ) 13/18

14 Measurement configuration: application to breast cancer detection S source 7.5 cm D₁ receivers skin ( D₂ ) tumor ( D₄ ) 2 cm 10 cm breast ( D₃ ) D Breast model built up from MRI scan [Zastrow and Hagness, 2008] 64 sources 64 receivers 6 frequencies in the band GHz 12.2 cm D = N x N y D1 (ɛ 1, σ 1 (S/m)) D2 (ɛ 2, σ 2 (S/m)) D3 (ɛ 3, σ 3 (S/m)) D4 (ɛ 3, σ 3 (S/m)) (10, 0.5) (6.12, 0.11) ([2.46, 60.6], [0.01, 2.28]) (55.3, 1.57) 14/18

15 Reconstruction CSI: Contrast Source Inversion, VBA: Variational Bayesian Approach, MGI: Independent Gaussian mixture, MGM: Gauss-Markov mixture 15/18

16 Reconstruction Original VBA MGI VBA MGM Hidden field and 3029 pixels have undergone a change of class during the iterative process Re(m k ) Im(m k ) Class 1 Class 2 Class 3 Class Iterations Iterations 16/18

17 Conclusions : - Bayesian approach to microwave imaging: complexity of breast imaging compared to previous applications [Ayasso, 2010] - Better resolution with VBA than with CSI - Better identication of heterogeneous areas with MGM than with MGI prior - Risk of hiding small details of interest Prospects - Accelerate the convergence by introducing a gradient step - Account for breast tissue frequency dispersivity by means of Debye models - Test on laboratory controlled experimental data involving breast phantoms - Study the false alarm rate of the method need of a reference database 17/18

18 , Ayasso, H. (2010). PhD thesis, Université Paris sud 11,. Ayasso, H., Duchêne, B., and Mohammad-Djafari, A. (2012). Inverse Problems in Science and Engineering, 20(1): Féron, O. (2006). PhD thesis, Université Paris sud 11. Gharsalli, L., Ayasso, H., Duchêne, B., and Mohammad-Djafari, A. (2014). Inverse scattering in a bayesian framework: application to microwave imaging for breast cancer detection. Inverse Problems, 30(11): Harrington, R. F. (1993). Field Computation by Moment Methods. Wiley-IEEE Press. Mackay, D. J. C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press. Zastrow, E., D. S. L. M. K. F. V. V. B. and Hagness, S. (2008). Database of 3d grid-based numerical breast phantoms for use in computational electromagnetics simulations. 18/18

19 Thanks! Questions? 18/18