. Bayesian Discrete Tomography from a few number of projections Ali Mohammad-Djafari Laboratoire des Signaux et Syst`emes (L2S) UMR8506 CNRS-CentraleSup´elec-UNIV PARIS SUD SUPELEC, 91192 Gif-sur-Yvette, France http://lss.centralesupelec.fr Email: [email protected] http://djafari.free.fr http://publicationslist.org/djafari Invited talk at Workshop on Discrete Tomography, Polytechnico de Milan, Italy

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Contents 1. Limited angle Tomography 2. Basic Bayesian approach 3. Two main steps: I I

Choosing appropriate Prior model Do the computational efficiently

4. Hierarchical prior modelling I I

Sparsity enforcing models through Student-t and IGSM Gauss-Markov-Potts models

5. Computational tools: JMAP, Gibbs Sampling MCMC, VBA 6. Case study: Image Reconstruction with only two projections 7. Implementation issues I

I

Main GPU implementation steps: Forward and Back Projections Multi-Resolution implementation

8. Conclusions

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Limited angle Tomography: Limitations of analytical methods

Original

Data

Backprojection Filtered Backprojection

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 3

Algebraic methods: Discretization y 6

S•

Hij

r 

@ @

Q Q

f1 Q

@ @ @ f (x, y )@ @@

@
@ φ @ @ HH @ H

QQ fjQ Q Q Q Qg -

x

P f b (x, y ) j j j 1 if (x, y ) ∈ pixel j bj (x, y ) = 0 else f (x, y ) =

@ @ •D

@ @

g (r , φ) Z
g (r , φ) =

f (x, y ) dl L

i

fN

gi =

N X

Hij fj + i

j=1

gk = Hk f + k , k = 1, · · · , K −→ g = Hf +  gk projection at angle φk , g all the projections.

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 4

Algebraic methods 

   g1 H1 X  ..   ..  Hk f+k = Hf+ g =  .  , H =  .  → gk = Hk f+k → g = k gK HK I I I I I I

I

H is huge dimensional: 2D: 106 × 106 , 3D: 109 × 109 . Hf corresponds to forward projection Ht g corresponds to Back projection (BP) H may not be invertible and even not square H is, in general, ill-conditioned In limited angle tomography H is under determined, si the problem has infinite number of solutions Minimum Norm Solution X bf = Ht (HHt )−1 g = Htk (Hk Htk )−1 gk k

can be interpreted as the Filtered Back Projection solution.

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 5

Algebraic methods I

I

I

Minimum Norm Solution: minimize kfk22 s.t. Hf = g −→ bf = Ht (HHt )−1 g Least square Solution: bf = arg min J(f) = kg − Hfk2 → bf = (Ht H)−1 Ht g f Quadratic Regularization: J(f) = kg − Hfk2 + λkfk22 −→ bf = (Ht H + λI)−1 Ht g

I I I

L1 Regularization: J(f) = kg − Hfk2 + λkfk1 Lpq Regularization: J(f) = kg − Hfkpp + λkDfkqq More general Regularization: X X J(f) = φ(g i − [Hf]i ) + λ ψ((Df]j ) i

j

J(f) = ∆1 (g, Hf) + λ∆2 (f, f 0 ) with ∆1 and ∆2 any distances (L2, L2, ..) or divergence (KL)

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 6

y

6 Computed Tomography with only two projections

Hij
f (x, y )


Q Q

f1 Q QQ fjQ Q Q Q Qg

i

fN

@ @ -

x

HH @

Z g (r , φ) =

f (x, y ) dl PL f (x, y ) = j fj bj (x, y )  1 if (x, y ) ∈ pixel j bj (x, y ) = 0 else N X gi = Hij fj + i j=1

g = Hf + 

Case study: Reconstruction from 2 projections R g1 (x) = R f (x, y ) dy , g2 (y ) = f (x, y ) dx Very ill-posed inverse problem f (x, y ) = g1 (x) g2 (y ) Ω(x, y ) RΩ(x, y ) is a Copula: R Ω(x, y ) dx = 1 Ω(x, y ) dy = 1

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 7

Simple example

   

1 2 3 g1 g2 g3 g4

3 4 ? 4 6 ? 7 3   1   0 =   1 0

I

? 4 f1 f3 g3 1 -1 ? 6 f2 f4 g4 -1 1 7 g1 g2 0 0   f1 f4 f1 1 0 0  f2  f2 f5 0 1 1       f3 f6 f3 0 1 0 g1 g2 f4 1 0 1 Hf = g −→ bf = H−1 g if H invertible.

0 0

-1 1 0

1 0 -1 0 0

I

H is rank deficient: rank(H) = 3

I

Problem has infinite number of solutions.

I

How to find all those solutions ?

I

Which one is the good one? Needs prior information.

I

To find an unique solution, one needs either more data or prior information.

f7 g4 f8 g5 f9 g6 g3

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 8

Prior information or constraints fj > 0 or fj ∈ IR+

I

Positivity:

I

Boundedness:

I

Smoothness: fj depends on the neighborhoods.

I

Sparsity: many fj are zeros.

I

Sparsity in a transform domain: f = Dz and many zj are zeros.

I

Discrete valued (DV):

I

Binary valued (BV):

I

Compactness: f (r) is non zero in one or few non-overlapping compact regions

I

Combination of the above mentioned constraints Main mathematical questions:

I

I I

1 > fj ) > 0 or fj ∈ [0, 1]

fj ∈ {0, 1, ..., K } fj ∈ {0, 1}

Which combination results to unique solution ? How to apply them ?

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 9

Deterministic approaches I

I I I

I

Iterative methods: SIRT, ART, Quadratic or L1 regularization, Bloc Coordinate Descent, Multiplicative ART,... Criteria: J(f) = kg − Hfk2 + λkDfk2 Gradient based algorithms: ∇J(f) = −2Ht (g − Hf) + 2λDt Df Simplest algorithm: h i bf (k+1) = bf (k) + α(k) Ht (g − Hbf (k) ) + 2λDt Dbf (k) More criteria: P P J(f) = i φ(g i − [Hf]i ) + λ j ψ((Df]j ) with φ(t) and ψ(t) = {t 2 , |t|, |t|p , ...} or J(f) = ∆1 (g, Hf) + λ∆2 (f, f 0 )

I I

Imposing constraints in each iteration (example: DART) Mathematical studies of uniqueness and convergence of these algorithms are necessary

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Bayesian estimation approach M: I

I I I

g = Hf + 

Observation model M + Hypothesis on the noise  −→ p(g|f; M) = p (g − Hf) A priori information p(f|M) p(g|f; M) p(f|M) Bayes : p(f|g; M) = p(g|M) Maximum A Posteriori (MAP) : bf = arg max {p(f|g)} = arg max {p(g|f) p(f)} f f = arg min {J(f) = − ln p(g|f) − ln p(f)} f

I

Link with Regularization: bf = arg min {J(f) = ∆1 (g, Hf) + λR(f)} f with ∆1 (g, Hf) = − ln p(g|f)

and

λR(f) = − ln p(f)

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Case of linear models and Gaussian priors I

g = Hf +  Prior knowledge on the noise:

I

I

I

I



 1 2  ∼ N (0, v I) → p(g|f) ∝ exp − 2 kg − Hfk 2v Prior knowledge on f:   1 2 2 0 −1 f ∼ N (0, vf (D D) ) → p(f) ∝ exp − 2 kDfk 2vf A posteriori:   1 1 2 2 p(f|g) ∝ exp − 2 kg − Hfk − 2 kDfk 2v 2vf MAP : bf = arg maxf {p(f|g)} = arg minf {J(f)} 2 with J(f) = kg − Hfk2 + λkDfk2 , λ = vv2 f Advantage : characterization of the solution b p(f|g) = N (bf, Σ) with

bf = H0 H + λD0 D −1 H0 g, Σ b = v H0 H + λD0 D −1 2

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

MAP estimation with other priors: bf = arg min {J(f)} with J(f) = 1 kg − Hfk2 + Ω(f) v f Separable priors: I

Gaussian:   P p(fj ) ∝ exp −α|fj |2 −→ Ω(f) = α j |fj |2 = kfk22

I

Generalized Gaussian: P p(fj ) ∝ exp [−α|fj |p ] , 1 < p < 2 → Ω(f) = α j |fj |p = kfkpp P Gamma: p(fj ) ∝ fjα exp [−βfj ] −→ Ω(f) = α j ln fj + βfj

I I

Beta: P P p(fj ) ∝ fjα (1 − fj )β −→ Ω(f) = α j ln fj + β j ln(1 − fj )

Markovian models:  p(fj |f) ∝ exp −α

 X i∈Nj

φ(fj , fi ) −→

Ω(f) = α

XX j

φ(fj , fi ),

i∈Nj

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Sparsity enforcing models I

I

I

3 classes of models: 1- Generalized Gaussian, 2- Mixture models and 3- Heavy tailed (Cauchy and Student-t) Student-t model  
ν+1 2 log 1 + f /ν St(f |ν) ∝ exp − 2 Infinite Gausian Scaled Mixture (IGSM) equivalence Z ∞ St(f |ν) ∝ N (f |, 0, 1/z) G(z|α, β) dz, with α = β = ν/2 0

  p(f|z)      p(z|α, β)       p(f, z|α, β)

i h 1P 2 N (f |0, 1/z ) ∝ exp − z f j j j j j j 2 Q Q (α−1) = j G(z hPj |α, β) ∝ j z j iexp [−βz j ] ∝ exp (α − 1) ln z j − βz j h jP i ∝ exp − 21 j z j fj2 + (α − 1) ln z j − βz j =

Q

j p(fj |z j ) =

Q

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Non stationary noise and sparsity enforcing model – Non stationary noise: g = Hf+, i ∼ N (i |0, vi ) →  ∼ N (|0, V = diag [v1 , · · · , vM ]) – Student-t prior model and its equivalent IGSM : f j |vfj ∼ N (f j |0, vfj ) and vfj ∼ IG(vfj |αf0 , βf0 ) → f j ∼ St(f j |αf0 , βf0 ) 

p(g|f, v ) = N (g|Hf, V ), V = diag [v ] p(f|vf ) = N (g|0, Vf ), Vf = diag [vf ]  Q ?  ?  p(v ) = Qi IG(vi |α0 , β0 ) vf v   p(vf ) = i IG(vfj |αf0 , βf0 ) ?  ?  p(f, v , vf |g) ∝ p(g|f, v ) p(f|vf ) p(v ) p(vf )  f  – JMAP: (b f, vˆ , vˆf ) = arg max(f ,v ,vf ) {p(f, v , vf |g)} H ?  – VBA: Approximate p(f, v , vf |g) g by q1 (f) q2 (v ) q3 (vf ) 

αf0 , βf0 α0 , β0

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Sparse model in a Transform domain 1 g = Hf + , f = Dz, z sparse  p(g|z, v ) = N (g|HDf, v I) Vz = diag [vz ] p(z|vz ) = N (z|0, Vz ), p(v ) = IG(v αz0 , βz0 Q  |α0 , β0 ) p(v ) = z i IG(vz j |αz0 , βz0 ) ?  p(z, v , vz , v ξ |g) ∝p(g|z, v ) p(z|vz ) p(v ) p(vz ) p(v ξ ) vz α , β 0 0 – JMAP: ?  ?  (b z, vˆ , b vz ) = arg max {p(z, v , vz |g)} v z (z,v ,vz )   D ?  Alternate optimization: ?    b z = arg minz {J(z)} with: f     −1/2 2 1  zk J(z) = 2vˆ kg − HDzk2 + kVz H 2 βz0 +b zj ?  vbzj = αz +1/2  g  0    vˆ = β0 +kg−HDzbk2  α0 +M/2 – VBA: Approximate p(z, v , vz , v ξ |g) by q1 (z) q2 (v ) q3 (vz ) Alternate optimization.

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Sparse model in a Transform domain 2 g = Hf + , f = Dz + ξ, z sparse  p(g|f, v ) = N (g|Hf, v I) p(f|z) = N (f|Dz, v ξ I), αξ0 , βξ0 αz0 , βz0  p(z|v Vz = diag [vz ] z ) = N (z|0, Vz ),  ?  ?  vξ vz α , β p(v ) = IG(v Q  |α0 , β0 )  0 0 p(vz ) = i IG(vz j |αz0 , βz0 ) ?  ?  ? p(v ) = IG(v |α , β )  ξ0 ξ ξ ξ0 v z ξ  p(f, z, v , vz , v ξ |g) ∝p(g|f, v ) p(f|zf ) p(z|vz )    D ?  @  ? p(v ) p(vz ) p(v ξ ) R f @  – JMAP:   (bf, b z, vˆ , b vz , vbξ ) = arg max {p(f, z, v , vz , v ξ |g)} H ?  (f ,z,v ,vz ,v ξ ) g Alternate optimization.  – VBA: Approximate p(f, z, v , vz , v ξ |g) by q1 (f) q2 (z) q3 (v ) q4 (vz ) q5 (v ξ ) Alternate optimization.

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Gauss-Markov-Potts prior models for images

f (r)

z(r)

c(r) = 1 − δ(z(r) − z(r0 ))

p(f (r)|z(r) = k, mk , vk ) = N (mk , vk ) X p(f (r)) = P(z(r) = k) N (mk , vk ) Mixture of Gaussians I I

k Q Separable iid hidden variables: p(z) = r p(z(r)) Markovian hidden variables:  p(z) Potts-Markov:  X p(z(r)|z(r0 ), r0 ∈ V(r)) ∝ exp γ δ(z(r) − z(r0 ))   r0 ∈V(r) X X p(z) ∝ exp γ δ(z(r) − z(r0 )) r∈R r0 ∈V(r)

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Four different cases To each pixel of the image is associated 2 variables f (r) and z(r) I

f|z Gaussian iid, z iid : Mixture of Gaussians

I

f|z Gauss-Markov, z iid : Mixture of Gauss-Markov

I

f|z Gaussian iid, z Potts-Markov : Mixture of Independent Gaussians (MIG with Hidden Potts)

I

f|z Markov, z Potts-Markov : Mixture of Gauss-Markov (MGM with hidden Potts)

f (r)

z(r)

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 1

Gauss-Markov-Potts prior models for images

f (r) z(r) c(r) = 1 − δ(z(r) − z(r0 ))  a0 g = Hf +  m 0 , v0 γ α0 , β0 p(g|f, v ) = N (g|Hf, v I) α0 , β0 p(v ) = IG(v |α0 , β0 ) ?  ?  ?   p(f = k,Q mk , vk ) = N (f (r)|mk , vk )  (r)|z(r) P   v z θ  p(f|z, θ) =  k r∈Rk ak N (f (r)|mk , v k ),     θ = {(a , m , k k v k ), k = 1, · · · , K } @  ?  ?  R f @ p(θ) = D(a|a )N  0 , v 0)IG(v|α0 , β0 )   h0 P(a|m i  P    0 p(z|γ) ∝ exp γ δ(z(r) − z(r )) Potts MRF 0 r r ∈N (r) H ?  p(f, z, θ|g) ∝ p(g|f, v ) p(f|z, θ) p(z|γ) g MCMC: Gibbs Sampling  VBA: Alternate optimization.

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Bayesian Computation and Algorithms I

Joint posterior probability law of all the unknowns f, z, θ p(f, z, θ|g) ∝ p(g|f, θ 1 ) p(f|z, θ 2 ) p(z|θ 3 ) p(θ)

I

Often, the expression of p(f, z, θ|g) is complex.

I

Its optimization (for Joint MAP) or its marginalization or integration (for Marginal MAP or PM) is not easy

I

Two main techniques: I

I

MCMC: Needs the expressions of the conditionals p(f|z, θ, g), p(z|f, θ, g), and p(θ|f, z, g) VBA: Approximate p(f, z, θ|g) by a separable one q(f, z, θ|g) = q1 (f) q2 (z) q3 (θ) and do any computations with these separable ones.

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

MCMC based algorithm p(f, z, θ|g) ∝ p(g|f, z, θ 1 ) p(f|z, θ 2 ) p(z|θ 3 ) p(θ) General Gibbs sampling scheme: bf ∼ p(f|b b g) −→ b b g) −→ θ b ∼ (θ|bf, b z, θ, z ∼ p(z|bf, θ, z, g) I

b g) ∝ p(g|f, θ) p(f|b b Generate samples f using p(f|b z, θ, z, θ) When Gaussian, can be done via optimization of a quadratic criterion.

I

b g) ∝ p(g|bf, b b p(z) Generate samples z using p(z|bf, θ, z, θ) Often needs sampling (hidden discrete variable)

I

Generate samples θ using p(θ|bf, b z, g) ∝ p(g|bf, σ2 I) p(bf|b z, (mk , vk )) p(θ) Use of Conjugate priors −→ analytical expressions.

I

After convergence use samples to compute means and variances.

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Application in CT: Reconstruction from 2 projections

g|f g = Hf +  g|f ∼ N (Hf, σ2 I) Gaussian

f|z iid Gaussian or Gauss-Markov

z iid or Potts

c q(r) ∈ {0, 1} 1 − δ(z(r) − z(r0 )) binary

p(f, z, θ|g) ∝ p(g|f, θ 1 ) p(f|z, θ 2 ) p(z|θ 3 ) p(θ)

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Proposed algorithms p(f, z, θ|g) ∝ p(g|f, θ 1 ) p(f|z, θ 2 ) p(z|θ 3 ) p(θ) • MCMC based general scheme: bf ∼ p(f|b b g) −→ b b g) −→ θ b ∼ (θ|bf, b z, θ, z ∼ p(z|bf, θ, z, g) Iterative algorithme: I

I

I

b g) ∝ p(g|f, θ) p(f|b b Estimate f using p(f|b z, θ, z, θ) Needs optimization of a quadratic criterion. b g) ∝ p(g|bf, b b p(z) Estimate z using p(z|bf, θ, z, θ) Needs sampling of a Potts Markov field. Estimate θ using p(θ|bf, b z, g) ∝ p(g|bf, σ2 I) p(bf|b z, (mk , vk )) p(θ) Conjugate priors −→ analytical expressions.

• Variational Bayesian Approximation I

Approximate p(f, z, θ|g) by q1 (f) q2 (z) q3 (θ)

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Results with two projections

Original

Backprojection

Filtered BP

Gauss-Markov+pos

GM+Line process

GM+Label process

c

LS

z

c

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Implementation issues I

In almost all the algorithms, the step of computation of bf needs an optimization algorithm.

I

The criterion to optimize is often in the form of J(f) = kg − Hfk2 + λkDfk2

I

Very often, we use the gradient based algorithms which need to compute ∇J(f) = −2Ht (g − Hf) + 2λDt Df

I

So, for the simplest case, in each step, we have h i bf (k+1) = bf (k) + α(k) Ht (g − Hbf (k) ) + 2λDt Dbf (k)

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Gradient based algorithms h

 i  bf (k+1) = bf (k) + α H0 g − Hbf (k) − λD0 Dbf (k) b = Hbf (Forward projection) 1. Compute g b (Error or residual) 2. Compute δg = g − g 0 3. Compute δf 1 = H δg (Backprojection of error) 4. Compute δf 2 = −D0 Dbf (Correction due to regularization) 5. Update

bf (k+1) = bf (k) + [δf 1 + δf 2 ]

projections of Initial estimated Forward guess −→ image −→ projection −→ estimated image −→ H b g = Hf (k) f (0) f (k) ↑ update ↑ correction term in image space δf = H0 δg − λD0 Df (k)

I

–

Measured ← projections g

↓ compare ↓ ←−

Backprojection ←− H0

correction term in projection space δg = g − b g

Steps 1 and 3 need great computational cost and have been implemented on GPU.

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Multi-Resolution Implementation Sacle 1: black g(1) = H(1) f (1) ( N × N ) Sacle 2: green g(2) = H(2) f (2) (N/2 × N/2) Sacle 3: red g(3) = H(3) f (3) (N/4 × N/4)

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Results with 4 projection

Original

Projections

Initialization

Final result

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 2

Conclusions I I I I I I I I I

I

Limited angle Computed Tomography is a very ill-posed Inverse problem Analytical methods have many limitations Algebraic methods push further these limitations Deterministic Regularization methods push still further the limitations of ill-conditioning. Probabilistic and in particular the Bayesian approach has many potentials Hierarchical prior model with hidden variables are very powerful tools for Bayesian approach to inverse problems. Gauss-Markov-Potts models for images incorporating hidden regions and contours Main Bayesian computation tools: JMAP, MCMC and VBA Application in different imaging system (X ray CT, Microwaves, PET, Ultrasound, Optical Diffusion Tomography (ODT), Acoustic source localization,...) Current Projects: Efficient implementation in 2D and 3D cases

A. Mohammad-Djafari, Bayesian Discrete Tomography from a few number of projections, Mars 21-23, 2016, Polytechnico de Milan, Italy. 3