Proximal tools for image reconstruction in dynamic Positron Emission Tomography

Transcription

1 Proximal tools for image reconstruction in dynamic Positron Emission Tomography Nelly Pustelnik 1 joint work with Caroline Chaux 2, Jean-Christophe Pesquet 3, and Claude Comtat 4 1 Laboratoire de Physique, ENS Lyon CNRS UMR LATP, Université d Aix-Marseille 1 CNRS UMR LIGM, Université Paris-Est CNRS UMR SHFJ, CEA Séminaire DROITE September, 17th 2012

2 Positron Emission Tomography (PET) Image reconstruction for dynamic PET 2/32 Measures Reconstructed Benzodiazepine images receptors density Difficulties: Degradation: projection operator and noise. Large data size.

3 Outline 3/32 1. Degradation model and state-of-the-art Variational approach, sparsity, frames 2. Proximal algorithms Proximity operators, properties, DR, PPXA, Space+time PET reconstruction PPXA, hybrid regularization, simulated and preclinical data 4. Conclusion and future works

4 4/32 Degradation model A P α Original image Ay Sinogram y R N z R M Degradation model z = P α (Ay) A R M N : matrix associated with the projection operator. P α : Poisson noise with scaling parameter α > 0. GOAL: Find a vector ŷ the closest to y from the observations z

5 Variational approach: State-of-the-art Quadratic regularization - Wiener filtering [Wiener,1949]: ŷ = argmin Ay z λ y 2 2 where λ > 0 y R N 5/32

6 Variational approach: State-of-the-art Quadratic regularization - Wiener filtering [Wiener,1949]: ŷ = argmin Ay z λ y 2 2 where λ > 0 y R N 5/32 Use of frames [Haar,1910] [Mallat,2009] - Soft-thresholding: ŷ = argmin Ay z λ Fy 1 where λ > 0 y R N

7 Variational approach: State-of-the-art 6/32 F } {{ } y R N F }{{} x = Fy R K with K N F R K N : matrix associated with the analysis frame operator. F R N K : matrix associated with the synthesis frame operator. Tight frame condition: F F = µid pour µ > 0.

8 Variational approach: State-of-the-art Quadratic regularization - Wiener filtering [Wiener,1949]: ŷ = argmin Ay z λ y 2 2 where λ > 0 y R N Use of frames [Haar,1910] [Mallat,2009] - Soft-thresholding: ŷ = argmin y R N Ay z λ Fy 1 2 interpretations 7/32

9 Variational approach: Bayesian formulation 8/32 u = Ay = (u (i) ) 1 i N : realization of a random vector U. z: realization of a random vector Z. x = Fy = (x (k) ) 1 k K : realization of a random vector X = (X (k) ) 1 k K having independent components. MAP estimator (Maximum A Posteriori) max y P(U = Ay Z = z) max P(Z = z U = AF x) P(X = x) x K min lnp(z = z U = AF x) ln p x }{{} X (k)(x (k) ) k=1 Data fidelity }{{} A priori

10 Variational approach: Bayesian formulation 9/32 F } {{ } y R N F }{{} x = Fy R K with K N

11 Variational approach: Bayesian formulation K min lnp(z = z U = AF x) ln p x }{{} X (k)(x (k) ) k=1 Fidelity term }{{} A priori 10/32 where and P(Z = z U = AF x) = 1 { (2πα) exp N/2 p X (k)(x (k) ) = 1 C k exp{ λ k x (k) } AF x z 2 } 2 2α Consequently min x 1 K 2α AF x z λ k x (k) }{{} k=1 }{{} Fidelity term A priori

12 11/32 Variational approach: Compressed sensing Sparsity assumption y 0 = argmin y R N Fy 0 s.t. Ay z 2 2 ε where ( y = (y i ) 1 i N R N ) y 0 #{i : y i 0} and ε > 0 Convex relaxation : ŷ = arg min y R N Fy 1 s.t. Ay z 2 2 ε where ( y = (y i ) 1 i N R N ) y 1 N i=1 y i Lagragian formulation : ŷ = arg min y R N Ay z λ Fy 1 where λ > 0 Compressed sensing [Donoho,2004][Candès,2008]

13 Variational approach: Criterion choice Minimization problem Find ŷ Argmin y R N J f j (y) where (f j ) 1 j J belong to the class of convex functions, l.s.c., and proper from R N to ], + ]. j=1 12/32

14 Variational approach: Criterion choice 12/32 Minimization problem Find ŷ Argmin y R N J f j (y) j=1 f j may model the data fidelity term y R N, f j (y) = 1 2α Ay z 2 for a Gaussian noise y R N, f j (y) = D KL (αay, z) for a Poisson noise

15 Variational approach: Criterion choice 12/32 Minimization problem Find ŷ Argmin y R N J f j (y) j=1 f j may model the data fidelity term y R N, f j (y) = 1 2α Ay z 2 for a Gaussian noise y R N, f j (y) = D KL (αay, z) for a Poisson noise Poisson distribution: P(Z = z U = Ay) = Poisson anti-log likelihood: N exp ( ) α(ay) i (α(ty) i ) zi z i! i=1 log P(Z = z U = Ay) = N z i log(α(ay) i ) + α(ay) i i=1

16 Variational approach: Criterion choice 12/32 Minimization problem Find ŷ Argmin y R N J f j (y) j=1 f j may model the data fidelity term y R N, f j (y) = 1 2α Ay z 2 for a Gaussian noise y R N, f j (y) = D KL (αay, z) for a Poisson noise f j may model a prior y R N, f j (y) = λ y 2 2 : Tikhonov regularization y R N, f j (y) = λtv(y): total variation y R N, f j (y) = λ Fy 1 : model sparsity

17 Variational approach: Criterion choice 12/32 Minimization problem Find ŷ Argmin y R N J f j (y) j=1 f j may model the data fidelity term y R N, f j (y) = 1 2α Ay z 2 for a Gaussian noise y R N, f j (y) = D KL (αay, z) for a Poisson noise f j may model a prior y R N, f j (y) = λ y 2 2 : Tikhonov regularization y R N, f j (y) = λtv(y): total variation y R N, f j (y) = λ Fy 1 : model sparsity f j may model constraints y R N, f j (y) = ι C (y) (C = [0, 255] N ): data range constraint

18 Variational approach: Algorithm choice 13/32 Minimization problem Find ŷ Argmin y R N J f j (y) j=1 Properties of the involved functions smooth functions gradient-based methods (Newton, Quasi-Newton,...) constraints projection based methods (POCS, SIRT,...) non-smooth functions proximal algorithms (FB, DR, PPXA,...)

19 Proximity operator Definition [Moreau (1965)] Let ϕ: H R be a convex, l.s.c., and proper function. The proximity operator of ϕ at point u H is the unique point denoted by prox ϕ u such that 14/32 ( u H), prox ϕ u = arg min ϕ(v) + 1 u v 2 v H 2 Examples: If ϕ = ι C prox ιc = P C where P C denotes the projection onto the convex set C. If ϕ = χ with χ > 0 prox ϕ is the soft-thresholding with a fixed threshold χ. If ϕ = D KL (α, ζ) with ζ, α > 0 prox ϕ = 1 2 ( α + α 2 + 4ζ).

20 Proximity operator Proposition [Combettes,Pesquet,2007] H 1 and H 2 : real separable Hilbert spaces, ϕ: convex, l.s.c., and proper function from H 2 to R, L: H 1 H 2 : bounded linear operator such that L L = σid where σ > 0. Then, prox ϕ L = Id + L σ (prox σϕ Id) L. 15/32

21 Proximity operator Proposition [Combettes,Pesquet,2007] H 1 and H 2 : real separable Hilbert spaces, ϕ: convex, l.s.c., and proper function from H 2 to R, L: H 1 H 2 : bounded linear operator such that L L = σid where σ > 0. Then, prox ϕ L = Id + L σ (prox σϕ Id) L. 15/32 If L denotes a tight frame, i.e. L = F where F F = µid Proposition can be used (example: prox DKL(F,z) )

22 Proximity operator Proposition [Combettes,Pesquet,2007] H 1 and H 2 : real separable Hilbert spaces, ϕ: convex, l.s.c., and proper function from H 2 to R, L: H 1 H 2 : bounded linear operator such that L L = σid where σ > 0. Then, prox ϕ L = Id + L σ (prox σϕ Id) L. 15/32 If L denotes a tight frame, i.e. L = F where F F = µid Proposition can be used (example: prox DKL(F,z) ) Extension to the case of a separable ϕ and a linear operator such that L L = D where D denotes a diagonal matrix [Pustelnik et al, 2011].

23 16/32 Proximal algorithm: Forward-backward Forward-backward algorithm Let u 0 H. The algorithm constructs a sequence (u l ) l 1 by the iterations u l+1 = u l + λ l ( proxγl f 1 (u l γ l f 2 (u l )) u l )

24 16/32 Proximal algorithm: Forward-backward Forward-backward algorithm Let u 0 H. The algorithm constructs a sequence (u l ) l 1 by the iterations u l+1 = u l + λ l ( proxγl f 1 (u l γ l f 2 (u l )) u l ) Forward-backward convergence [Combettes and Wajs, 2005] f 2 is β-lipschitz differentiable on H with β > 0 γ l ]0, 2/β[ : algorithm step-size λ l ]0, 1] : relaxation parameter Under these assumptions, (u l ) l N converges to a solution of minf 1 (u) + f 2 (u).

25 Proximal algorithm: PPXA 17/32 Parallel ProXimal Algorithm [Combettes and Pesquet, 2008] γ ]0, + [, (ωj ) 1 j J ]0,1] J such that P J j=1 ω j = 1, (vj,0 ) 1 j J (R N ) J and u 0 = P J j=1 ω j v j,0, The sequence (x l ) l 1 is generated by the iterations: 8 For j = 1,..., J p j,l = prox γ/ωj f j v j,l compute every proximity operators >< p l = P J j=1 ω jp j,l λ l ]0, 2[ For j = 1,..., J v j,l+1 = v j,l + λ l (2 p l u l p j,l ) >: u l+1 = u l + λ l (p l u l ) Convergence [Combettes and Pesquet, 2008] Under some assumptions, the sequence (u l ) l 1 converges to a solution of min P J j=1 f j(u)

26 Proximal algorithms: ŷ Argmin y R N J j=1 f j(l j y) Forward-Backward [Combettes-Wajs, 2005] J = 2, L i = Id and a gradient Lipschitz function. 18/32

27 Proximal algorithms: ŷ Argmin y R N J j=1 f j(l j y) Forward-Backward [Combettes-Wajs, 2005] J = 2, L i = Id and a gradient Lipschitz function. Douglas-Rachford [Combettes-Pesquet, 2007] J = 2 and L i = Id. 18/32

28 Proximal algorithms: ŷ Argmin y R N J j=1 f j(l j y) Forward-Backward [Combettes-Wajs, 2005] J = 2, L i = Id and a gradient Lipschitz function. Douglas-Rachford [Combettes-Pesquet, 2007] J = 2 and L i = Id. PPXA [Combettes-Pesquet, 2008] L i = Id. 18/32

29 Proximal algorithms: ŷ Argmin y R N J j=1 f j(l j y) Forward-Backward [Combettes-Wajs, 2005] J = 2, L i = Id and a gradient Lipschitz function. Douglas-Rachford [Combettes-Pesquet, 2007] J = 2 and L i = Id. PPXA [Combettes-Pesquet, 2008] L i = Id. PPXA + [Pesquet-Pustelnik, 2012] i L i L i invertible. 18/32

30 Proximal algorithms: ŷ Argmin y R N J j=1 f j(l j y) Forward-Backward [Combettes-Wajs, 2005] J = 2, L i = Id and a gradient Lipschitz function. Douglas-Rachford [Combettes-Pesquet, 2007] J = 2 and L i = Id. PPXA [Combettes-Pesquet, 2008] L i = Id. PPXA + [Pesquet-Pustelnik, 2012] i L i L i invertible. M+SFBF [Combettes-Briceño, 2011] no required conditions. 18/32

31 Proximal algorithms: ŷ Argmin y R N J j=1 f j(l j y) Forward-Backward [Combettes-Wajs, 2005] J = 2, L i = Id and a gradient Lipschitz function. Douglas-Rachford [Combettes-Pesquet, 2007] J = 2 and L i = Id. PPXA [Combettes-Pesquet, 2008] L i = Id. PPXA + [Pesquet-Pustelnik, 2012] i L i L i invertible. M+SFBF [Combettes-Briceño, 2011] no required conditions. M+LFBF [Combettes-Pesquet, 2012] one function with a Lipschitz gradient. 18/32

32 PET reconstruction (dynamic) 19/32 Measures Reconstructed Benzodiazepine images receptors distribution

33 PET reconstruction (dynamic) 20/32 A P α Original image Ay t Sinogram y t R N z t R M Degradation model t {1,..., T }, z t = P α (Aȳ t ) A R M N : matrix associated with the projection operator. P α: Poisson noise with scaling parameter α > 0.

34 PET reconstruction: State-of-the-art Filter Back Projection: FBP EM-ML [Shepp, Vardi, 1982][Lange, Carson, 1984] Ordered Subsets EM: OSEM [Hudson, Larkin, 1994] Space-alternating Generalized EM: SAGE [Fessler, Hero, 1994] RAMLA [Browne, De Pierro, 1996] EM with stopping criteria [Veklerov, Llacer, 1987] SIEVES [Snyder, Miller, 1985] MAP [Herman et al., 1979], [Sauer, Bouman, 1993], [Fessler, 1995] Multiresolution [Turkheimer et al., 1999], [Alpert et al., 2006], [Zhou et al., 2007], Spatio-temporal assumptions [Nichols et al., 2002], [Kamasac et al., 2005], [Reader et al., 2006], [ Verhaeghe et al., 2008] 21/32

35 Proposed method 22/32 Criterion ŷ Argmin y R NT T t=1 D KL (αay t, z t ) D KL (αa, z): minus Poisson log-likelihood (data fidelity term)

36 Proposed method 22/32 Criterion ŷ Argmin y R NT T t=1 D KL (αay t, z t ) + κ Fy 1 D KL (αa, z): minus Poisson log-likelihood (data fidelity term) F R K NT : wavelet transform such that F F = FF = Id 1 : l 1 -norm

37 Proposed method Criterion ŷ Argmin y R NT T t=1 ( ) D KL (αay t, z t ) + ϑtv(y t ) + κ Fy 1 22/32 D KL (αa, z): minus Poisson log-likelihood (data fidelity term) F R K NT : wavelet transform such that F F = FF = Id 1 : l 1 -norm tv: total variation

38 Proposed method Criterion ŷ Argmin y R NT T t=1 ( ) D KL (αay t, z t ) + ϑtv(y t ) + κ Fy 1 + ι C (y) 22/32 D KL (αa, z): minus Poisson log-likelihood (data fidelity term) F R K NT : wavelet transform such that F F = FF = Id 1 : l 1 -norm tv: total variation C R NT : convex constraint

39 Proposed method Criterion ŷ Argmin y R NT T t=1 ( ) D KL (αay t, z t ) + ϑtv(y t ) + κ Fy 1 + ι C (y) 22/32 D KL (αa, z): minus Poisson log-likelihood (data fidelity term) F R K NT : wavelet transform such that F F = FF = Id 1 : l 1 -norm tv: total variation C R NT : convex constraint ϑ 0 and κ 0

40 Proposed method Criterion ŷ Argmin y R NT T t=1 ( ) D KL (αay t, z t ) + ϑtv(y t ) + κ Fy 1 + ι C (y) 22/32 D KL (αa, z): minus Poisson log-likelihood (data fidelity term) F R K NT : wavelet transform such that F F = FF = Id 1 : l 1 -norm non differentiable tv: total variation C R NT : convex constraint non differentiable ϑ 0 and κ 0

41 Proposed method Criterion ŷ Argmin y R NT T t=1 ( ) D KL (αay t, z t ) + ϑtv(y t ) + κ Fy 1 + ι C (y) 22/32 D KL (αa, z): minus Poisson log-likelihood (data fidelity term) F R K NT : wavelet transform such that F F = FF = Id 1 : l 1 -norm non differentiable tv: total variation C R NT : convex constraint non differentiable ϑ 0 and κ 0

42 Proposed method Criterion ŷ Argmin x R K T t=1 ( ) D KL (αay t, z t ) + ϑtv(y t ) + κ Fy 1 + ι C (y) 23/32 tv: anisotropic total variation 1 : l 1 -norm C R NT : convex constraint D KL (αa, z): Poisson anti-log likelihood

43 Proposed method Criterion ŷ Argmin x R K T t=1 ( ) D KL (αay t, z t ) + ϑtv(y t ) + κ Fy 1 + ι C (y) 23/32 tv: anisotropic total variation Closed form for prox tv after splitting 1 : l 1 -norm Closed form for prox 1 C R NT : convex constraint Closed form for prox ιc D KL (αa, z): Poisson anti-log likelihood

44 Proposed method Criterion ŷ Argmin x R K T t=1 ( ) D KL (αay t, z t ) + ϑtv(y t ) + κ Fy 1 + ι C (y) 23/32 tv: anisotropic total variation Closed form for prox tv after splitting 1 : l 1 -norm Closed form for prox 1 C R NT : convex constraint Closed form for prox ιc D KL (αa, z): Poisson anti-log likelihood closed form for prox DKL (α, z), NO closed form for prox because A DKL(αA,z) A D (where D denote a diagonal matrix)

45 24/32 Proposed method Tubes of response m How to compute prox DKL(αA,z)? Pixels n For every m {1,...,M}, Ψ m Γ 0 (R) We can write M D KL (αay, z) = Ψ m (αa m, y) m=1

46 24/32 Proposed method Pixels n Tubes of response m How to compute prox DKL(αA,z)? I r is a partition of {1,...,M} For every m {1,...,M}, Ψ m Γ 0 (R) We can write M D KL (αay, z) = Ψ m (αa m, y) m=1 R = Ψ(αA m, y) r=1 m I r Choice of I r

47 Proposed method Tubes of response m Pixels n Choice of I r : A (r) = (A m,n ) m Ir,1 n N? How to compute prox DKL(αA,z)? I r is a partition of {1,...,M} For every m {1,...,M}, Ψ m Γ 0 (R) We can write M D KL (αay, z) = Ψ m (αa m, y) m=1 R = Ψ(αA m, y) r=1 m I r R = Ψ (r) (αa (r) y) r=1 24/32

48 24/32 Proposed method Tubes of response m How to compute prox DKL(αA,z)? Pixels n For every m {1,...,M}, Ψ m Γ 0 (R) We can write M D KL (αay, z) = Ψ m (αa m, y) m=1 R = Ψ(αA m, y) r=1 m I r R = Ψ (r) (αa (r) y) Choice of I r : A (r) = (A m,n ) m Ir,1 n N? containing non-overlapping and thus orthogonal rows of A prox Ψ (r) A (r) takes a closed form (i.e. A (r) (A (r) ) D. r=1

49 Simulated data: Materials & methods Simulated data - Materials PET 2D+time cerebral data [ 18 F]-FDG HR+ scanner, based on Zubal cerebral phantom. 16 temporal frames, time: between 30 secondes and 5 minutes. Events number: between 48 and /32

50 Simulated data: Materials & methods Simulated data - Materials PET 2D+time cerebral data [ 18 F]-FDG HR+ scanner, based on Zubal cerebral phantom. 16 temporal frames, time: between 30 secondes and 5 minutes. Events number: between 48 and Method PPXA: 400 iterations 10 s each. System matrix split is R = 432 subsets. Wavelets: symmlet filters length 6 on 3 levels (space) and 2 levels of Daubechies-6 on interval (time). EM-ML with/without post-reconstruction smoothing, PPXA with/without temporal regularisation. 25/32

51 Simulated data: Results 26/32 EM-ML Sieves PPXA 2D PPXA 2D+t min.

52 Simulated data: Results 26/32 EM-ML Sieves PPXA 2D PPXA 2D+t min min.

53 Simulated data: MSE 27/32 Cortex Frame [min] EM-ML SIEVES PPXA 2D PPXA 2D+t Striatum Frame [min] EM-ML SIEVES PPXA 2D PPXA 2D+t

54 Simulated data: TAC 28/ Activity [Bq/cc] Original SIEVES 1 SIEVES 2 PPXA 1 PPXA Time [min] Cortex

55 Simulated data: TAC 28/ Activity [Bq/cc] Original SIEVES 1 SIEVES 2 PPXA 1 PPXA 2 Activity [Bq/cc] Time [min] Cortex Time [min] Striatum

56 Preclinical data: materials & methods 29/32 Baboon 2D brain PET [ 18 F]-FDG study on the HR+ scanner Prompt and delayeds stored separatly 128 time frames, from 10 to 30 second duration

57 Preclinical data: materials & methods 29/32 Baboon 2D brain PET [ 18 F]-FDG study on the HR+ scanner Prompt and delayeds stored separatly 128 time frames, from 10 to 30 second duration Sinograms rebinned into 16 time frames, from 80 to 240 second duration OP-PPXA: all corrections (except scatters) included in the model

58 Preclinical data: results FBP Sieves PPXA 2D+t 30/ min.

59 Preclinical data: results FBP Sieves PPXA 2D+t 30/ min min.

60 Conclusion 31/32 Efficient method to reconstruct dynamic PET data and obtain parametric map Taking into account time-frame characteristics by considering wavelets Using Parallel ProXimal Algorithm Possibility to deal with a large panel of regularization terms and constraints Results presented on simulated and preclinical data Future work: Impact of considering 16, 32, 64,... temporal frames for parametric map estimation Improve the convergence rate of the algorithm

61 Reference 32/32 J. Verhaegue, D. Van De Ville, I. Khalidov, Y. D Asseler, I. Lemahieu, and M. Unser. Dynamic PET reconstruction using wavelet regularization with adapted basis functions. IEEE TMI, 27(7): , F. C. Sureau, J.-C. Pesquet, C. Chaux, N. Pustelnik, A. J. Reader, C. Comtat, R. Trebossen, Temporal wavelet denoising of PET sinograms and images. IEEE NSS, 6 pp., Dresden, Germany, Oct N. Pustelnik, C. Chaux, J.-C. Pesquet, F. C. Sureau, E. Dusch and C. Comtat, Adapted Convex Optimization Algorithm for Wavelet-Based Dynamic PET Reconstruction, Fully 3D, 4 pp., Bejing, Chine, Sept N. Pustelnik, C. Chaux, J.-C. Pesquet, and C. Comtat, Parallel Algorithm and Hybrid Regularization for Dynamic PET Reconstruction, IEEE MIC, 4 pp., Knoxville, Tennessee, Oct Nov N. Pustelnik, C. Chaux and J.-C. Pesquet, Parallel ProXimal Algorithm for image restoration using hybrid regularization, IEEE TIP, 20(9): , Sep S. Anthoine, J.-F. Aujol, Y. Boursier, and C. Mélot, Some proximal methods for Poisson intensity CBCT and PET, Inverse Problems and Imaging, to appear, 2012.