Bayesian X-ray Computed Tomography using a Three-level Hierarchical Prior Model

Transcription

1 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 1/26 Bayesian X-ray Computed Tomography using a Three-level Hierarchical Prior Model Li Wang, Ali Mohammad-Djafari, Nicolas Gac Laboratoire des Signaux et Systèmes (L2S) UMR8506 CNRS-CentraleSupélec-UNIV PARIS SUD SUPELEC, Gif-sur-Yvette, France li.wang@lss.supelec.fr djafari@lss.supelec.fr nicolas.gac@lss.supelec.fr

2 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 2/26 General presentation Context: X ray Computed Tomography (CT) Classical methods: Analytic(FBP), Algebraic and iterative(ls,qr,l1,reg,tv,etc.) Proposed Bayesian methods: Sparsity enforcing unsupervised three-level hierarchical model 1 Simulation performances Conclusion and perspectives [1], Computed tomography reconstruction based on a hierarchical model and variational Bayesian method. L.Wang, A.Mohamma-Djafari, N.Gac, M.Dumitru. ICASSP 16.

3 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 3/26 Context: X-ray Computed Tomography (CT) (I) Low dose Reduce intensity of rays Reduce number of projections Figure: X-ray CT scanner.

4 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 4/26 Context: X-ray Computed Tomography (CT) (II) Non Destructive Testing Limited number of projections Limited angles of projections Figure: X-ray CT scanner. left: right:

5 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 5/26 Context: Radon Transformation g(r, ) I0 l(r, ) f(x,y) Figure: Parallel beam tomography projection interpretation. y r Hij fj x projection i g(r, φ) = l r,φ f (x, y)dl Without noise: g [i] = j H [i, j] f [j] Accounting for errors: g [i] = j H [i, j] f [j] + ɛ[i] Notations: f = {f 1, f 2,, f N } The quantified object pixel value; ɛ = {ɛ 1, ɛ 2,, ɛ N } additive noise; H = (h mn ) 1<n<N,1<m<M the projection operator; g = {g 1, g 2,, g N } measured projection data;

6 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 6/26 Operators and back-projections g = Hf + ɛ (1) Forward operator: g = Hf Adjoint operator: f = H g Back-projection: f = H g Filtered back-projection: f = H ( HH ) 1 g H huge dimensional In general not accessible directly

7 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 7/26 An example of the direct model original image Projection Back-Projection f ( ) g (64 128) fbp Filtered Back-Projection Filtered Back-Projection Filtered Back-Projection f FBP (64 in [0, π]) ffbp (32 in [0, π]) ffbp (64 in [0, 1 2 π]) Figure: An example of projection.

8 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 8/26 Outline Classical methods Proposed methods: The Hierarchical Haar Transformation Based Method (HHBM) Conclusion and Perspectives

9 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 9/26 Context: Classical methods f = arg minf g Hf λr(f ) Least Square R(f ) = 0 l 2 : Quadratic Regularization(QR) R(f ) = j f j 2 = f 2 l 2 : General QR R(f ) = 2 j [Df ] j = Df 2 l 1 : LASSO R(f ) = j f j = f 1 Total Variation(1D) R(f ) = Df 1 Total Variation(2D) R(f ) = D x f 1 + D y f 1 Main difficulties: How to choose λ? (Cross Validation, L-Curve) How to quantify uncertainties?

10 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 10/26 Bayesian inference Supervised method: Unsupervised method: p (f g) = p (g f ) p (f ) p (g) (2) p (f, θ g) = p (g f, θ) p (f θ) p (θ) p (g) (3) where θ the hyper-parameters. Main steps: Choose prior models: p(f θ) and p(θ); Computation: JMAP, VBA...

11 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 11/26 Outline Classical methods Proposed methods: The Hierarchical Haar Transformation Based Method (HHBM) Conclusion and Perspectives

12 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 12/26 Sparsity in a transform domain Sparsity Specific: volume domain, f is sparse Generally: transformation domain, Df is sparse D: Fourier, Wavelet, Dictionary, etc Sparsity enforcing distribution: Generalized Gaussian distributions (Laplace, double exponential, etc) Gaussian Mixture distributions Heavy tailed distributions (Student-t, Cauchy, etc)

13 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 13/26 Multilevel Haar Transformation (a) (b) (c) (a) Original image; (b) Level 1 Haar transformation; (c) Level 2 Haar transformation. f = Dz + ξ, where z and ξ sparse.

14 (Generalized) Student-t and Sparsity coefficient Pdf Normal, St t and St g Normal St t St g Figure: Pdfs of Normal, Student-t and Generalized Student-t distribution St(f ν) = 0 St g (f α, β) = N (f 0, z) IG(z ν 2, ν 2 ) dz = Γ ( ) ν+1 2 ( νπ Γ ν ) (1 + f 2 ν 2 0 ) ν+1 2 N (f 0, v)ig(v α, β) dv = Γ(α + 1 ) ( f ) 2 (α+ 2 1 ) 2βπ Γ(α) 2β [1] A generalization of Student-t based on Infinite Gaussian Scaled Mixture model and its use as a sparsity enforcing in Bayesian signal processing. M.Dumitru, L.Wang, A.Mohammad-Djafari. L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 14/26

15 Proposed three-level hierarchical model f H g ɛ Direct model { g = Hf + ɛ L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 15/26

16 Proposed three-level hierarchical model ξ z D f ɛ H g Direct model { g = Hf + ɛ f = Dz + ξ D: Haar transformation operator. z: Haar transformation coefficient of f. L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 16/26

17 Proposed three-level hierarchical model α z0, β z0 v z ξ z D f ɛ H g Direct model { g = Hf + ɛ f = Dz + ξ D: Haar transformation operator. z: Haar transformation coefficient of f. L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 17/26

18 Proposed three-level hierarchical model α ξ0, β ξ0 α z0, β z0 Direct model v ξ v z v ɛ ξ α ɛ0, β ɛ0 z D f ɛ H (Details: Appendix I) g { g = Hf + ɛ f = Dz + ξ D: Haar transformation operator. z: Haar transformation coefficient of f. Likelihood: p(g f, v ɛ ). Prior p(f z, v ξ ), p(z v z ), p(v ɛ ), p(v ξ ) and p(v z ). Posterior p(f, z, v ɛ, v ξ, v z g) L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 18/26

19 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 19/26 Estimation Algorithms Bayesian Inference p(f, z, v ɛ, v ξ, v z g) p(g f, v ɛ ) p(f z, v ξ )p(z v z )p(v ɛ )p(v ξ )p(v z ) }{{}}{{} Likelihood Prior Joint Maximum A Posterior (JMAP) ( f, ẑ, v ɛ, v ξ, v z ) = arg max f,z,v ɛ,v ξ,v z {p(f, z, v ɛ, v ξ, v z g)} Posterior Mean (PM) via Variational Bayesian Approach (VBA) p(f, z, v ɛ, v ξ, v z g) q(f, z, v ɛ, v ξ, v z ) = q 1 (f )q 2 (z)q 3 (v ɛ )q 4 (v ξ )q 5 (v z ) by minimizing the Kullback-Leibler divergence: ( ) KL q(θ) : p(θ g) = q(θ) ln q(θ) p(θ g) dθ

20 Results(I) Original images and Reconstructed results using 128 projections with SNR=dB. δ f = f f 2 f Original Image FBP (δ f = 0.32) QR (δ f = 0.056) LASSO (δ f = ) TV (δ f = ) HHBM (δ f = 0.038) L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. /26

21 f Results(II) Zone of profiles of Original and reconstructed Shepp-Logan image. Results are obtained with 128 projections and noise of db. Profile case 128 projection and noise db Original FBP QR LASSO L1Haar TV HHBM x L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 21/26

22 Results(III) δ f δ f with 128 projections and noise db FBP QR LASSO TV HHBM δ f iteration δ f while using 128 projections with SNR=dB δ f with 64 projections and noise db FBP QR LASSO TV HHBM iteration δ f while using 64 projections with SNR=dB L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 22/26

23 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 23/26 Outline Classical methods Proposed methods: The Hierarchical Haar Transformation Based Method (HHBM) Conclusion and Perspectives

24 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 24/26 Conclusion and Perspectives Conclusions: Using sparsity of Haar transformation coefficient Joint MAP estimations Method realizable in 3D case Perspectives: Reconstruction of object and contours simultaneously Using our methods and ASTRA toolbox to solve large scale data problems Find ways to decrease the iterative methods cost

25 Thank you very much! L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 25/26

26 L. Wang, A. Mohammad-Djafari, N. Gac, MaxEnt 16, Ghent, Belgium. 26/26 Appendix (I) The three-level hierarchical haar transformation based model: p(g f, v ɛ ) = N (g Hf, V ɛ ) V ɛ 1 2 exp [ 1 2 (g Hf ) V 1 ɛ (g Hf ) ] [ ] p(f z, v ξ ) = N (f Dz, v ξ ) v ξ 1 2 exp 1 2 (f Dz) V 1 ξ (f Dz) p(z v z ) = N (z 0, V z ) V z 1 2 exp [ ] 1 2 z V 1 z z p(v z α z0, β z0 ) = N j IG(v zj α z0, β z0 ) [ ] N j v (αz 0 +1) z j exp β z0 v 1 z j p(v ɛ α ɛ0, β ɛ0 ) = M i IG(v ɛi α ɛ0, β ɛ0 ) M i v (αɛ 0 +1) ɛ i exp [ ] β ɛ0 v 1 ɛ i p(v ξ α ξ0, β ξ0 ) = N j IG(v ξj α ξ0, β ξ0 ) [ N j v (α ξ 0 +1) exp β ξ0 v 1 ξ j ξ j ]