Méthodes proximales en tomographie CBCT et TEP

Transcription

1 Méthodes proximales en tomographie CBCT et TEP Jean-François Aujol IMB, UMR CNRS 5251 Collaboration avec Sandrine Anthoine, Clothilde Melot (LATP, Marseille) Yannick Boursier, (CPPM, Marseille) Jeudi 20 octobre 2011 Jean-François Aujol Méthodes proximales en tomographie CBCT et TEP

2 Outline 1 CBCT and PET modeling 2 State of the art 3 Recalls in convex analysis 4 CBCT problem : solvers and results on synthetic data 5 PET problem : solvers and results on synthetic data 6 CBCT problem : results on real data 7 Conclusion 2/57 Jean-François Aujol Tomographie CBCT et TEP

3 Outline 1 CBCT and PET modeling 2 State of the art 3 Recalls in convex analysis 4 CBCT problem : solvers and results on synthetic data 5 PET problem : solvers and results on synthetic data 6 CBCT problem : results on real data 7 Conclusion 2/57 Jean-François Aujol Tomographie CBCT et TEP

4 Reminder on CBCT CBCT= Cone Beam Computerized Tomography CT : a medical imaging modality which provides anatomical information on contrast images. CBCT scan : X-ray source+xray camera, the imaged object in between, in Cone beam geometry. 3/57 Jean-François Aujol Tomographie CBCT et TEP

5 Reminder on CBCT The Beer-Lambert law claims I j = z j exp r j µ E (l)dl with µ : l µ E (l) the unknown absorption coefficient at point l on r j. z j a parameter proportional to the number of photons emitted by the source. 4/57 Jean-François Aujol Tomographie CBCT et TEP

6 Reminder on CBCT, in 3D sinogram 5/57 Jean-François Aujol Tomographie CBCT et TEP

7 Reminder on CBCT, in 3D sinogram with more angles 6/57 Jean-François Aujol Tomographie CBCT et TEP

8 Reminder on PET PET= Positron Emission Tomography PET : medical imaging modality that provides a measurement of the metabolic activity of an organ injection to the patient of a radiotracer attached to a molecule that will be absorbed by some organs, depending of their function radioactive decay emits a positron, which annihilates with an electron after a very short time, and this yields... two gamma rays radiation of 511 kev and opposite direction. Rings of detectors are supposed to detect them. parallel-beam geometry Possible absorption of photons when crossing the body 7/57 Jean-François Aujol Tomographie CBCT et TEP

9 simultaneous PET- CBCT Thus along a line L modeling the measurements : w L = L x(l)exp( µ 511(l)) l µ 511 (l) is supposed to be known. x is the unknown concentration of radioactive desintegration. Prototype developped by the CPPM : ClearPET/XPAD (ClearPET developped by EPFL+XPAD developped by CPPM) allows simultaneous PET/CT imaging based on hybrid pixels Hybrid pixels : a new generation of detectors which is in photons counting mode very low counting rate no charge integration : no dark noise with these detectors 8/57 Jean-François Aujol Tomographie CBCT et TEP

10 CBCT framework no additional gaussian noise in this setting! Let y R n the measurements µ R m the unknown to recover A M(R m, R n )thesystemmatrixwithn << m in general, and ill conditionned. pure Poisson noise : y j P(z j exp ( [Aµ] j )) with P(λ) the Poisson distribution of parameter λ. log likelihood yields the objective function with constraint µ 0 L CBCT (µ) = j y j [Aµ] j + z j exp ( [Aµ] j ) We consider the problem ˆµ =argmin µ 0 L CBCT (µ)+λj(µ) 9/57 Jean-François Aujol Tomographie CBCT et TEP

11 PET framework again no additional gaussian noise in this setting! Let y R n the measurements x R m the unknown to recover B M(R m, R n )thesystemmatrixwithn << m in general, and ill conditionned. pure Poisson noise : y j P([Bx] j )withp(λ) the Poisson distribution of parameter λ. log likelihood yields the objective function with constraint x 0 L PET (x) = j y j log([bx] j )+[Bx] j We consider the problem ˆx =argmin x 0 L PET (x)+λj(µ) 10/57 Jean-François Aujol Tomographie CBCT et TEP

12 Optimization strategy : remarks The CBCT data fidelity term is differentiable with our assumption. Several optimization schemes and penalization can be tested under the constraint that the result µ 0. the CBCT optimization problem should be less challenging than the PET one! Choice of a regularization term Total-variation J TV (u) = ( u) i,j 1 i,j N Regularized Total-Variation J reg TV = 1 i,j N 1 -norm inducing sparsity J 1,φ(u) = < u,φ λ > = R φ (u) 1 λ Λ α2 + u) i,j 2 11/57 Jean-François Aujol Tomographie CBCT et TEP

13 Outline 1 CBCT and PET modeling 2 State of the art 3 Recalls in convex analysis 4 CBCT problem : solvers and results on synthetic data 5 PET problem : solvers and results on synthetic data 6 CBCT problem : results on real data 7 Conclusion 12/57 Jean-François Aujol Tomographie CBCT et TEP

14 Quick non exhaustive review some algorithms to recover CBCT and PET images viewed as Poisson noisy data Filtered backprojection for Cone Beam geometry : FDK algorithm (Feldkamp and all ) EM algorithm and variants (Shepp and Vardi 1982, Lange and Carson 1984, Hudson and Larkin ) Regularization of EM type algorithms : quadratic surrogate functions (De Pierro 1994, Fessler and all ), Huber (Chlewicki and all ), TV (Harmany and all ) technics closed to the ones used in convex optimization 13/57 Jean-François Aujol Tomographie CBCT et TEP

15 Quick non exhaustive review Forward backward splitting (Combettes-Wajs 2005) after using an Anscombe transform to go back to Gaussian noise applied in the setting of Deconvolution problems with Poisson noisy data (Dupé et al 2009) Alternative Direction Method of Multipliers in the context of poissonian image reconstruction (Figueiredo 2010) PPXA algorithm applied in the context of dynamical PET (Pustelnik et al 2010) Primal dual algorithm using TV regularization in the context of blurred Poisson noisy data (Bonettini and Ruggiero 2010)... 14/57 Jean-François Aujol Tomographie CBCT et TEP

16 Outline 1 CBCT and PET modeling 2 State of the art 3 Recalls in convex analysis 4 CBCT problem : solvers and results on synthetic data 5 PET problem : solvers and results on synthetic data 6 CBCT problem : results on real data 7 Conclusion 15/57 Jean-François Aujol Tomographie CBCT et TEP

17 Proximal operator Let F be a convex proper function. We recall that the subgradient of F, which is denoted by F,isdefinedby F (x) ={p X such that F (y) F (x)+p, y x y} For any h > 0 the following problem always has a unique solution : min y This solution is given by : hf (y)+ 1 x y2 2 y =(I + h F ) 1 (x) =prox F h (x) The mapping (I + h F ) 1 is called the proximity operator. 16/57 Jean-François Aujol Tomographie CBCT et TEP

18 Examples min y hf (y)+ 1 x y2 2 When F is the indicator function of some closed convex set C, i.e. : 0ifx C F (x) = + otherwise then prox F h (x) is the orthogonal projection of x onto C. When F (x) =xḃ1,thenprox F 1,1 h (x) is the soft wavelet shrinkage of x with parameter h. When F (x) =J TV (x) thenprox F h (x) =x hp hk (x), with P hk orthogonal projection onto hk, and K = {div g / g i,j 1 i, j}. 17/57 Jean-François Aujol Tomographie CBCT et TEP

19 Forward-backward splitting min x F (x)+g(x) where F is a convex C 1,1 function, with F LLipschitz, and G a simple convex function (simple means that the proximity operator of G is easy to compute). The Forward-Backward algorithm reads in this case : x0 X x k+1 = (I + h G) 1 (x k h F (x k )) = prox G h (x k h F (x k )) This algorithm is known to converge provided h 1/L. In terms of objective functions, the convergence speed is of order 1/k. 18/57 Jean-François Aujol Tomographie CBCT et TEP

20 Acceleration It has been shown by Nesterov (2005) and by Beck-Teboule (2009) that the previous algorithm could be modified so that a convergence speed of order 1/k 2 is obtained. The FISTA algorithm proposed by Beck and Teboule is the following : x 0 X ; y 1 = x 0 ; t 1 = 1; x k = (I + h G) 1 (y k h F (y k )) t k+1 = y k+1 = tk 2 2 x k + t k 1 t k+1 (x k x k 1 ) This algorithm converges provided h 1/L. 19/57 Jean-François Aujol Tomographie CBCT et TEP

21 Projected gradient algorithm When G is the indicator function of some closed convex set C, i.e.: 0ifx C G(x) = + otherwise Then the forward-backward algorithm is nothing but the classical projected gradient method, and the FISTA algorithm an accelerated projected gradient method. Forward-backward algorithm reads : x0 X x k+1 = P C (x k h F (x k )) with P C is the orthogonal projection operator onto C. And FISTA reads : x 0 X y 1 = x 0 t 1 =1 x k = P C (y k h F (y k )) t k+1 = y k+1 = tk 2 2 x k + t k 1 t k+1 (x k x k 1 ) 20/57This algorithm converges Jean-François provided Aujol htomographie 1/L. CBCT et TEP

22 FISTA and constrained total variation Beck and Teboule have shown that FISTA could be used to solve the constrained total variation problem. with C a closed non empty convex set. Proposition : Let us set : min J TV (u)+ 1 u C 2λ f u2 (1) h(v) = H C (f λdiv v) 2 + f λdiv v 2 where H C (u) =u P C (u) andp C (u) is the orthogonal projection of u onto C. Letusdefine: ṽ = arg min v 1 h(v) Then the solution of problem (1) is given by : u = P C (f λdiv ṽ) 21/57 Jean-François Aujol Tomographie CBCT et TEP

23 FISTA and constrained regularization The previous result can be adapted to some general L 1 regularization : min Ku u C 2λ f u2 (1) with C a closed non empty convex set. K is a continuous linear operator from X to Y (two finite-dimensional real vector spaces). Proposition : Let us set : h K (v) = H C (f + λk v) 2 + f + λk v 2 where H C (u) =u P C (u) andp C (u) is the orthogonal projection of u onto C. Letusdefine: ṽ = arg min v 1 h K (v) Then the solution of problem (1) is given by : u = P C (f + λk ṽ) 22/57 Jean-François Aujol Tomographie CBCT et TEP

24 Chambolle-Pock algorithm X and Y are two finite-dimensional real vector spaces. K : X Y continuous linear operator. F and G convex functions. min x X (F (Kx)+G(x)) We remind the definition of the Legendre-Fenchel conjugate of F : F (y) =max x X The associated saddle point problem is : (x, y F (x)) (1) min max (Kx, y + G(x) F (y)) x X y Y = Arrow-Urwicz method (ascent in y, descentinx). 23/57 Jean-François Aujol Tomographie CBCT et TEP

25 Chambolle-Pock algorithm Theorem min max (Kx, y + G(x) F (y)) (1) x X y Y Initialization : Choose τ,σ > 0, (x 0, y 0 ) X Y ), and set x 0 = x 0. Iterations (n 0) :Update x n, y n, x n as follows : y n+1 =(I + σ F ) 1 (y n + σk x n ) x n+1 =(I + τ G) 1 (x n τk y n+1 ) x n+1 =2x n+1 x n (2) Let L = K, and assume problem (1) has a saddle point. Choose τσl 2 < 1, and let (x n, x n, y n ) be defined by (2). Then there exists a saddle point (x, y )suchthatx n x and y n y. Notice that both F and G can be non smooth. 24/57 Jean-François Aujol Tomographie CBCT et TEP

26 Remarks Forward-backward splitting : Lions-Mercier 1979, Combettes-Wajs TV L2 model in image processing : Rudin-Osher-Fatemi 1992 (algorithm : steepest gradient descent on a regularized functional) Theoretical proof of convergence : Chambolle-Lions 1995 (and remark that prox TV h (x) =x hp hk (x)) Solving the functional without regularization : Chambolle 2002 (fixed point algorithm), Chambolle 2005 (projected gradient algorithm), Combettes-Pesquet 2004 (subgradient algorithm) Accelerations : Nesterov 2005, Weiss et al 2009, Beck-Teboule 2009, Chambolle-Pock /57 Jean-François Aujol Tomographie CBCT et TEP

27 Outline 1 CBCT and PET modeling 2 State of the art 3 Recalls in convex analysis 4 CBCT problem : solvers and results on synthetic data 5 PET problem : solvers and results on synthetic data 6 CBCT problem : results on real data 7 Conclusion 26/57 Jean-François Aujol Tomographie CBCT et TEP

28 Solving the CBCT problem y j [Aµ] j + z j exp ( [Aµ] j )+χ {µ 0} + λj(µ) j belongs to the class of problems arg min x X F (x)+g(x) with F and G proper, convex, lower semi-continuous functions, and FLLipschitz differentiable recall the definition prox F (x) =argmin y X F (y)+ 1 2 x y 2 Forward-backward splitting iterations (Combettes-Wajs 2005, Daubechies-De Mol 2004) x k+1 = prox hg (x k h F (x k )). Converge if h 1 L. with G = λj TV + χ C (C = {x 0}) can be solved with the algorithm FISTA (Beck and Teboulle 2009) with G = λj 1,φ + χ C, can be solved using again FISTA 27/57 Jean-François Aujol Tomographie CBCT et TEP

29 Algorithms for CBCT [TVreg] using J reg TV, accelerated projected gradient descent. [FB-TV] using J TV, Forward-Backward algorithm combined with FISTA. [FB-wav] using J 1,φ, Forward-Backward algorithm combined with FISTA. tested against three algorithms implemented in the IRT toolbox [FBP] Filtered backprojection [MLEM] MLEM algorithm [MLEM-H] MLEM algorithm penalized by a Huber function. 28/57 Jean-François Aujol Tomographie CBCT et TEP

30 Results on simulated data Simulated phantoms to recover : Zubal Contrast Resolution Criteria, T being the true object and I the reconstructed image SNR(I, T ) = 10 log mean(i 2 ) 10 mean( I T 2 ) SSIM(I, T )=mean w var(iin )+var(i out ) CNR(I )= mean(i in) mean(i out ) (2mean(Iw )mean(t w )+a)(2 cov(i w,t w )+b) mean(i w ) 2 +mean(t 2 w +a)(var(i w )+var(t w )+b) 29/57 Jean-François Aujol Tomographie CBCT et TEP

31 CBCT Zubal z =1e3 photons 30/57 Jean-François Aujol Tomographie CBCT et TEP

32 CBCT Zubal z =1e2 photons 31/57 Jean-François Aujol Tomographie CBCT et TEP

33 CBCT z =1e3 andz =1e2 Photon count 1e3 1e2 Algorithm snr ssim λ nb. iter. time (s) TVreg FB-Wav FB-TV FBP MLEM MLEM-H e5 752 TVreg FB-Wav FB-TV FBP MLEM MLEM-H e /57 Jean-François Aujol Tomographie CBCT et TEP

34 CBCT Contrast for 60 projections 33/57 Jean-François Aujol Tomographie CBCT et TEP

35 CBCT Contrast for 60 projections photon count Algorithm cnr ssim snr 1e4 FB-Wav FB-TV MLEM-H e3 FB-Wav FB-TV MLEM-H e2 FB-Wav FB-TV MLEM-H /57 Jean-François Aujol Tomographie CBCT et TEP

36 Influence of the number of projections Photon count 1e3 1e3 Nb. angles Algorithm cnr ssim snr cnr ssim snr TVreg FB-Wav FB-TV FBP MLEM MLEM-H Nb. angles 30 TVreg FB-Wav FB-TV FBP MLEM MLEM-H /57 Jean-François Aujol Tomographie CBCT et TEP

37 CBCT Resolution for 60 projections 36/57 Jean-François Aujol Tomographie CBCT et TEP

38 Outline 1 CBCT and PET modeling 2 State of the art 3 Recalls in convex analysis 4 CBCT problem : solvers and results on synthetic data 5 PET problem : solvers and results on synthetic data 6 CBCT problem : results on real data 7 Conclusion 37/57 Jean-François Aujol Tomographie CBCT et TEP

39 Solving the PET problem y j log([bx] j )+[Bx] j + χ {x 0} + λj(x) j belongs to the class of problems arg min x X F (Kx)+G(x) with F and G proper, convex, lower semi-continuous functions, F and G non differentiable, K a continuous linear operator Primal-dual algorithm : Chambolle-Pock algorithm (2010) y n+1 = prox σf (y n + σk x n ) x n+1 = prox τg (x n τk y n+1 ) x n+1 =2x n+1 x n 38/57 Jean-François Aujol Tomographie CBCT et TEP