. Bayesian Computed Tomography Ali Mohammad-Djafari Groupe Probl` emes Inverses Laboratoire des Signaux et Syst` emes UMR 8506 CNRS - SUPELEC - Univ Paris Sud 11 Sup´ elec, Plateau de Moulon, 91192 Gif-sur-Yvette, FRANCE.
[email protected] http://djafari.free.fr http://www.lss.supelec.fr NIMI, Gent, Belgium, Mai 2010
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
1/41
Content ◮
Image reconstruction in Computed Tomography: An ill posed invers problem
◮
Two main steps in Bayesian approach: Prior modeling and Bayesian computation Prior models for images:
◮
◮ ◮ ◮
◮
◮
Separable Gaussian, GG, ... Gauss-Markov, General one layer Markovian models Hierarchical Markovian models with hidden variables (contours and regions) Gauss-Markov-Potts
Bayesian computation ◮ ◮
MCMC Variational and Mean Field approximations (VBA, MFA)
◮
Application: Computed Tomography in NDT
◮
Conclusions and Work in Progress
◮
Questions and Discussion
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
2/41
Computed Tomography: Making an image of the interior of a body ◮ ◮ ◮
f (x, y ) a section of a real 3D body f (x, y , z) gφ (r ) a line of observed radiographe gφ (r , z) Forward model: Line integrals or Radon Transform Z gφ (r ) = f (x, y ) dl + ǫφ (r ) L
ZZ r,φ f (x, y ) δ(r − x cos φ − y sin φ) dx dy + ǫφ (r ) =
◮
Inverse problem: Image reconstruction Given the forward model H (Radon Transform) and a set of data gφi (r ), i = 1, · · · , M find f (x, y )
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
3/41
2D and 3D Computed Tomography 3D
2D Projections
80
60 f(x,y)
y 40
20
0 x −20
−40
−60
−80 −80
gφ (r1 , r2 ) =
Z
f (x, y , z) dl
−60
gφ (r ) =
Lr1 ,r2 ,φ
−40
Z
−20
0
20
40
60
f (x, y ) dl Lr,φ
Forward probelm: f (x, y ) or f (x, y , z) −→ gφ (r ) or gφ (r1 , r2 ) Inverse problem: gφ (r ) or gφ (r1 , r2 ) −→ f (x, y ) or f (x, y , z) A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
4/41
80
CT as a linear inverse problem Fan beam X−ray Tomography −1
−0.5
0
0.5
1
Source positions
−1
g (si ) =
Z
−0.5
Detector positions
0
0.5
1
f (r) dli + ǫ(si ) −→ Discretization −→ g = Hf + ǫ
Li
◮
g, f and H are huge dimensional
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
5/41
Inversion: Deterministic methods Data matching ◮
Observation model gi = hi (f ) + ǫi , i = 1, . . . , M −→ g = H(f ) + ǫ
◮
Misatch between data and output of the model ∆(g, H(f )) fb = arg min {∆(g, H(f ))} f
◮
Examples:
– LS
∆(g, H(f )) = kg − H(f )k2 =
X
|gi − hi (f )|2
i
– Lp – KL
p
∆(g, H(f )) = kg − H(f )k = ∆(g, H(f )) =
X i
◮
X
|gi − hi (f )|p , 1 < p < 2
i
gi gi ln hi (f )
In general, does not give satisfactory results for inverse problems.
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
6/41
Inversion: Regularization theory Inverse problems = Ill posed problems −→ Need for prior information Functional space (Tikhonov): g = H(f ) + ǫ −→ J(f ) = ||g − H(f )||22 + λ||Df ||22 Finite dimensional space (Philips & Towmey): g = H(f ) + ǫ • Minimum norme LS (MNLS): J(f ) = ||g − H(f )||2 + λ||f ||2 • Classical regularization: J(f ) = ||g − H(f )||2 + λ||Df ||2 • More general regularization: or
J(f ) = Q(g − H(f )) + λΩ(Df )
J(f ) = ∆1 (g, H(f )) + λ∆2 (f , f 0 ) Limitations: • Errors are considered implicitly white and Gaussian • Limited prior information on the solution • Lack of tools for the determination of the hyperparameters A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
7/41
Bayesian estimation approach M:
g = Hf + ǫ
◮
Observation model M + Hypothesis on the noise ǫ −→ p(g|f ; M) = pǫ(g − Hf )
◮
A priori information
p(f |M)
◮
Bayes :
p(f |g; M) =
p(g|f ; M) p(f |M) p(g|M)
Link with regularization : Maximum A Posteriori (MAP) : fb = arg max {p(f |g)} = arg max {p(g|f ) p(f )} f
f
= arg min {− ln p(g|f ) − ln p(f )} f
with Q(g, Hf ) = − ln p(g|f ) and λΩ(f ) = − ln p(f ) But, Bayesian inference is not only limited to MAP A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
8/41
Case of linear models and Gaussian priors g = Hf + ǫ ◮
◮
◮
◮
Hypothesis on the noise: ǫ ∼ N (0, σǫ2 I) 1 p(g|f ) ∝ exp − 2 kg − Hf k2 2σǫ Hypothesis on f : f ∼ N (0, σf2 I) 1 2 p(f ) ∝ exp − 2 kf k 2σf A posteriori: 1 1 2 2 p(f |g) ∝ exp − 2 kg − Hf k − 2 kf k 2σǫ 2σf MAP : fb = arg maxf {p(f |g)} = arg minf {J(f )} with
◮
J(f ) = kg − Hf k2 + λkf k2 ,
λ=
Advantage : characterization of the solution f |g ∼ N (fb, Pb ) with fb = Pb H t g,
A. Mohammad-Djafari,
σǫ2 σf2
Pb = H t H + λI
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
−1
9/41
MAP estimation with other priors: fb = arg min {J(f )} with J(f ) = kg − Hf k2 + λΩ(f ) f
Separable priors: ◮
◮ ◮
◮
Gaussian: P p(fj ) ∝ exp −α|fj |2 −→ Ω(f ) = kf k2 = α j |fj |2 P Gamma: p(fj ) ∝ fjα exp {−βfj } −→ Ω(f ) = α j ln fj + βfj
Beta: P P p(fj ) ∝ fjα (1 − fj )β −→ Ω(f ) = α j ln fj + β j ln(1 − fj ) Generalized Gaussian: p(fj ) ∝ exp {−α|fj |p } ,
1 < p < 2 −→
Markovian models: X p(fj |f ) ∝ exp −α φ(fj , fi ) −→
Ω(f ) = α
i ∈Nj
A. Mohammad-Djafari,
Ω(f ) = α
NIMI Workshop on medical imaging, Gent, Belgium,
P
XX j
j
|fj |p ,
φ(fj , fi ),
i ∈Nj
May 29, 2010.
10/41
Main advantages of the Bayesian approach ◮
MAP = Regularization
◮
Posterior mean ? Marginal MAP ?
◮
More information in the posterior law than only its mode or its mean
◮
Meaning and tools for estimating hyper parameters
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Meaning and tools for model selection
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More specific and specialized priors, particularly through the hidden variables More computational tools:
◮
◮
◮ ◮
◮
Expectation-Maximization for computing the maximum likelihood parameters MCMC for posterior exploration Variational Bayes for analytical computation of the posterior marginals ...
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
11/41
Full Bayesian approach M:
g = Hf + ǫ
◮
Forward & errors model: −→ p(g|f , θ 1 ; M)
◮
Prior models −→ p(f |θ 2 ; M)
◮
Hyperparameters θ = (θ 1 , θ 2 ) −→ p(θ|M)
◮
Bayes: −→ p(f , θ|g; M) =
◮
Joint MAP:
◮
◮
◮
p(g|f,θ;M) p(f|θ;M) p(θ|M) p(g|M)
b = arg max {p(f , θ|g; M)} (fb, θ) (f,θ) R p(f |g; M) = R p(f , θ|g; M) df Marginalization: p(θ|g; M) = p(f , θ|g; M) dθ ( R fb = f p(f , θ|g; M) df dθ R Posterior means: b = θ p(f , θ|g; M) df dθ θ
Evidence of the model: ZZ p(g|M) = p(g|f , θ; M)p(f |θ; M)p(θ|M) df dθ
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
12/41
MAP estimation with different prior models
g = Hf + ǫ g = Hf + ǫ = f = Cf + z with z ∼ N (0, σf2 I) p(f ) = N 0, σf2 (D t D)−1 Df = z with D = (I − C) p(f |g) = N (fb, Pbf ) with fb = Pbf H t g,
Pbf = H t H + λD t D
J(f ) = − ln p(f |g) = kg − Hf k2 + λkDf k2
−1
——————————————————————————– g = Hf + ǫ = g = Hf + ǫ p(f ) = N 0, σf2 (W W t ) f = W z with z ∼ N (0, σf2 I) p(z|g) = N (b z , Pbz ) with zb = Pbz W t H t g, Pbz = W t H t HW + λI J(z) = − ln p(z|g) = kg − HW zk2 + λkzk2 −→ fb = W zb
z decomposition coefficients A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
13/41
−1
MAP estimation and Compressed Sensing
g = Hf + ǫ f = Wz
◮
W a code book matrix, z coefficients
◮
Gaussian:
◮
o n P p(z) = N (0, σz2 I) ∝ exp − 2σ1 2 j |z j |2 z P J(z) = − ln p(z|g) = kg − HW zk2 + λ j |z j |2
Generalized Gaussian (sparsity, β = 1): o n P p(z) ∝ exp −λ j |z j |β
J(z) = − ln p(z|g) = kg − HW zk2 + λ
◮
z = arg minz {J(z)} −→ fb = W zb
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
P
j
|z j |β
May 29, 2010.
14/41
Two main steps in the Bayesian approach ◮
Prior modeling ◮
◮ ◮
◮
Separable: Gaussian, Generalized Gaussian, Gamma, mixture of Gaussians, mixture of Gammas, ... Markovian: Gauss-Markov, GGM, ... Separable or Markovian with hidden variables (contours, region labels)
Choice of the estimator and computational aspects ◮ ◮ ◮ ◮ ◮
MAP, Posterior mean, Marginal MAP MAP needs optimization algorithms Posterior mean needs integration methods Marginal MAP needs integration and optimization Approximations: ◮ ◮ ◮
A. Mohammad-Djafari,
Gaussian approximation (Laplace) Numerical exploration MCMC Variational Bayes (Separable approximation)
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
15/41
Which images I am looking for? 50 100 150 200 250 300 350 400 450 50
A. Mohammad-Djafari,
100
150
200
250
300
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
16/41
Which image I am looking for?
Gaussian p(fj |fj−1 ) ∝ exp −α|fj − fj−1 |2
Generalized Gaussian p(fj |fj−1 ) ∝ exp {−α|fj − fj−1 |p }
Piecewize Gaussian p(fj |qj , fj−1 ) = N (1 − qj )fj−1 , σf2
Mixture of GM p(fj |zj = k) = N mk , σk2
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
17/41
Gauss-Markov-Potts prior models for images ”In NDT applications of CT, the objects are, in general, composed of a finite number of materials, and the voxels corresponding to each materials are grouped in compact regions”
How to model this prior information?
f (r) z(r) ∈ {1, ..., K } p(f (r)|z(r) = k, mk , vk ) = N (mk , vk ) X p(f (r)) = P(z(r) = k) N (m k , vk ) Mixture of Gaussians k X p(z(r)|z(r ′ ), r ′ ∈ V(r)) ∝ exp γ δ(z(r) − z(r ′ )) ′ r ∈V(r)
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
18/41
Four different cases To each pixel of the image is associated 2 variables f (r) and z(r) ◮
f |z Gaussian iid, z iid : Mixture of Gaussians
◮
f |z Gauss-Markov, z iid : Mixture of Gauss-Markov
◮
f |z Gaussian iid, z Potts-Markov : Mixture of Independent Gaussians (MIG with Hidden Potts)
◮
f |z Markov, z Potts-Markov : Mixture of Gauss-Markov (MGM with hidden Potts)
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
f (r)
z(r) May 29, 2010.
19/41
Four different cases
Case 1: Mixture of Gaussians
Case 2: Mixture of Gauss-Markov
Case 3: MIG with Hidden Potts
Case 4: MGM with hidden Potts
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
20/41
Four different cases f (r)|z(r) z(r) f (r)|z(r) ′
z(r)|z(r ) ′
′
f (r)|f (r ), z(r), z(r ) z(r) 0
0
0
1
0
0
1
0
1
0
q(r) = {0, 1} ′
′
f (r)|f (r ), z(r), z(r ) ′
z(r)|z(r ) 0
0
0
A. Mohammad-Djafari,
1
0
0
1
0
1
NIMI Workshop on medical imaging, Gent, Belgium,
0
q(r) = {0, 1}
May 29, 2010.
21/41
f |z Gaussian iid,
Case 1:
z iid
Independent Mixture of Independent Gaussiens (IMIG): p(f (r)|z(r) = k) = N (mk , vk ), p(f (r)) = Q
p(z) = Noting
PK
k=1
∀r ∈ R P
αk N (mk , vk ), with
r p(z(r)
= k) =
Q
r αk
Rk = {r : z(r) = k},
k
=
Q
k
αk = 1.
αnkk
R = ∪k Rk ,
mz (r) = mk , vz (r) = vk , αz (r) = αk , ∀r ∈ Rk we have: p(f |z) =
Y
N (mz (r), vz (r))
r∈R
p(z) =
Y r
A. Mohammad-Djafari,
αz (r) =
Y
P
αk
r∈R
δ(z(r)−k)
k
NIMI Workshop on medical imaging, Gent, Belgium,
=
Y
αnkk
k
May 29, 2010.
22/41
Case 2:
f |z Gauss-Markov,
z iid
Independent Mixture of Gauss-Markov (IMGM): p(f (r)|z(r), z(r ′ ), f (r ′ ), r ′ ∈ V(r)) = N (µz (r), Pvz (r)), ∀r∗ ∈′ R 1 µz (r) = |V(r)| r′ ∈V(r) µz (r ) µ∗z (r ′ ) = δ(z(r ′ ) − z(r)) f (r ′ ) + (1 − δ(z(r ′ ) − z(r)) mz (r ′ ) = (1 − c(r ′ )) f (r ′ ) + c(r ′ ) mz (r ′ ) Q Q p(f |z) ∝ Qr N (µz (r), vz (r)) ∝ Qk αk N (mk 1, Σk ) p(z) = r vz (r) = k αnkk
with 1k = 1, ∀r ∈ Rk and Σk a covariance matrix (nk × nk ).
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
23/41
Case 3: f |z Gauss iid, z Potts Gauss iid as in Case 1: Q p(f |z) = Qr∈R Q N (mz (r), vz (r)) = k r∈Rk N (mk , vk )
Potts-Markov:
X p(z(r)|z(r ′ ), r ′ ∈ V(r)) ∝ exp γ δ(z(r) − z(r ′ )) ′ r ∈V(r)
X X p(z) ∝ exp γ δ(z(r) − z(r ′ )) ′ r∈R r ∈V(r)
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
24/41
Case 4: f |z Gauss-Markov, z Potts Gauss-Markov as in Case 2: p(f (r)|z(r), z(r ′ ), f (r ′ ), r ′ ∈ V(r)) = N (µz (r), vz (r)), ∀r ∈ R µz (r) µ∗z (r ′ )
1 P ∗ ′ = |V(r)| r′ ∈V(r) µz (r ) = δ(z(r ′ ) − z(r)) f (r ′ ) + (1 − δ(z(r ′ ) − z(r)) mz (r ′ )
p(f |z) ∝
Q
r N (µz (r), vz (r))
∝
Q
k
αk N (mk 1, Σk )
Potts-Markov as in Case 3: X X p(z) ∝ exp γ δ(z(r) − z(r ′ )) ′ r∈R r ∈V(r)
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
25/41
Summary of the two proposed models
f |z Gaussian iid z Potts-Markov
f |z Markov z Potts-Markov
(MIG with Hidden Potts)
(MGM with hidden Potts)
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
26/41
Bayesian Computation p(f , z, θ|g) ∝ p(g|f , z, vǫ ) p(f |z, m, v) p(z|γ, α) p(θ) θ = {vǫ , (αk , mk , vk ), k = 1, ·, K }
p(θ) Conjugate priors
◮
Direct computation and use of p(f , z, θ|g; M) is too complex
◮
Possible approximations : ◮ ◮ ◮
◮
Gauss-Laplace (Gaussian approximation) Exploration (Sampling) using MCMC methods Separable approximation (Variational techniques)
Main idea in Variational Bayesian methods: Approximate p(f , z, θ|g; M) by q(f , z, θ) = q1 (f ) q2 (z) q3 (θ) ◮ ◮
Choice of approximation criterion : KL(q : p) Choice of appropriate families of probability laws for q1 (f ), q2 (z) and q3 (θ)
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
27/41
MCMC based algorithm p(f , z, θ|g) ∝ p(g|f , z, θ) p(f |z, θ) p(z) p(θ) General scheme:
◮
◮
◮
b g) −→ zb ∼ p(z|fb, θ, b g) −→ θ b ∼ (θ|fb, zb, g) fb ∼ p(f |b z , θ, b g) ∝ p(g|f , θ) p(f |b b Sample f from p(f |b z , θ, z , θ) Needs optimisation of a quadratic criterion. b g) ∝ p(g|fb, zb, θ) b p(z) Sample z from p(z|fb, θ, Needs sampling of a Potts Markov field.
Sample θ from p(θ|fb, zb, g) ∝ p(g|fb, σǫ2 I) p(fb|b z , (mk , vk )) p(θ) Conjugate priors −→ analytical expressions.
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
28/41
Application of CT in NDT Reconstruction from only 2 projections
◮
◮
g1 (x) =
Z
f (x, y ) dy
g2 (y ) =
Z
f (x, y ) dx
Given the marginals g1 (x) and g2 (y ) find the joint distribution f (x, y ). Infinite number of solutions : f (x, y ) = g1 (x) g2 (y ) Ω(x, y ) Ω(x, y ) is a Copula: Z Z Ω(x, y ) dx = 1 and Ω(x, y ) dy = 1
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
29/41
Application in CT 20
40
60
80
100
120 20
40
60
80
100
120
g|f f |z z q g = Hf + ǫ iid Gaussian iid q(r) ∈ {0, 1} g|f ∼ N (Hf , σǫ2 I) or or 1 − δ(z(r) − z(r ′ )) Gaussian Gauss-Markov Potts binary Forward model Gauss-Markov-Potts Prior Model Auxilary Unsupervised Bayesian estimation: p(f , z, θ|g) ∝ p(g|f , z, θ) p(f |z, θ) p(θ) A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
30/41
Results: 2D case
Original
Backprojection
Gauss-Markov+pos
Filtered BP
GM+Line process
GM+Label process
20
20
20
40
40
40
60
60
60
80
80
80
100
100
100
120
120 20
A. Mohammad-Djafari,
LS
40
60
80
100
120
c
120 20
40
60
80
100
NIMI Workshop on medical imaging, Gent, Belgium,
120
z
May 29, 2010.
20
40
60
80
31/41
100
120
c
Some results in 3D case (Results obtained with collaboration with CEA)
A. Mohammad-Djafari,
M. Defrise
Phantom
FeldKamp
Proposed method
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
32/41
Some results in 3D case
FeldKamp
A. Mohammad-Djafari,
Proposed method
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
33/41
Some results in 3D case Experimental setup
A photograpy of metalique esponge
Reconstruction by proposed method
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
34/41
Application: liquid evaporation in metalic esponge
Time 0
A. Mohammad-Djafari,
Time 1
Time 2
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
35/41
Conclusions ◮
Gauss-Markov-Potts are useful prior models for images incorporating regions and contours
◮
Bayesian computation needs often pproximations (Laplace, MCMC, Variational Bayes)
◮
Application in different CT systems (X ray, Ultrasound, Microwave, PET, SPECT) as well as other inverse problems
Work in Progress and Perspectives : ◮
Efficient implementation in 2D and 3D cases using GPU
◮
Evaluation of performances and comparison with MCMC methods
◮
Application to other linear and non linear inverse problems: (PET, SPECT or ultrasound and microwave imaging)
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
36/41
Some references ◮ ◮ ◮ ◮ ◮ ◮ ◮ ◮ ◮
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A. Mohammad-Djafari (Ed.) Probl` emes inverses en imagerie et en vision (Vol. 1 et 2), Hermes-Lavoisier, Trait´ e Signal et Image, IC2, 2009, A. Mohammad-Djafari (Ed.) Inverse Problems in Vision and 3D Tomography, ISTE, Wiley and sons, ISBN: 9781848211728, December 2009, Hardback, 480 pp. H. Ayasso and Ali Mohammad-Djafari Joint NDT Image Restoration and Segmentation using Gauss-Markov-Potts Prior Models and Variational Bayesian Computation, To appear in IEEE Trans. on Image Processing, TIP-04815-2009.R2, 2010. H. Ayasso, B. Duchene and A. Mohammad-Djafari, Bayesian Inversion for Optical Diffraction Tomography Journal of Modern Optics, 2008. A. Mohammad-Djafari, Gauss-Markov-Potts Priors for Images in Computer Tomography Resulting to Joint Optimal Reconstruction and segmentation, International Journal of Tomography & Statistics 11: W09. 76-92, 2008. A Mohammad-Djafari, Super-Resolution : A short review, a new method based on hidden Markov modeling of HR image and future challenges, The Computer Journal doi:10,1093/comjnl/bxn005:, 2008. O. F´ eron, B. Duch` ene and A. Mohammad-Djafari, Microwave imaging of inhomogeneous objects made of a finite number of dielectric and conductive materials from experimental data, Inverse Problems, 21(6):95-115, Dec 2005. M. Ichir and A. Mohammad-Djafari, Hidden markov models for blind source separation, IEEE Trans. on Signal Processing, 15(7):1887-1899, Jul 2006. F. Humblot and A. Mohammad-Djafari, Super-Resolution using Hidden Markov Model and Bayesian Detection Estimation Framework, EURASIP Journal on Applied Signal Processing, Special number on Super-Resolution Imaging: Analysis, Algorithms, and Applications:ID 36971, 16 pages, 2006. O. F´ eron and A. Mohammad-Djafari, Image fusion and joint segmentation using an MCMC algorithm, Journal of Electronic Imaging, 14(2):paper no. 023014, Apr 2005. H. Snoussi and A. Mohammad-Djafari, Fast joint separation and segmentation of mixed images, Journal of Electronic Imaging, 13(2):349-361, April 2004. A. Mohammad-Djafari, J.F. Giovannelli, G. Demoment and J. Idier, Regularization, maximum entropy and probabilistic methods in mass spectrometry data processing problems, Int. Journal of Mass Spectrometry, 215(1-3):175-193, April 2002.
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
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Thanks, Questions and Discussions Thanks to:
My graduated PhD students:
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H. Snoussi, M. Ichir, (Sources separation) F. Humblot (Super-resolution) H. Carfantan, O. F´ eron (Microwave Tomography) S. F´ ekih-Salem (3D X ray Tomography)
My present PhD students:
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H. Ayasso (Optical Tomography, Variational Bayes) D. Pougaza (Tomography and Copula) —————– Sh. Zhu (SAR Imaging) D. Fall (Emission Positon Tomography, Non Parametric Bayesian)
My colleages in GPI (L2S) & collaborators in other instituts:
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B. Duchˆ ene & A. Joisel (Inverse scattering and Microwave Imaging) N. Gac & A. Rabanal (GPU Implementation) Th. Rodet (Tomography) —————– A. Vabre & S. Legoupil (CEA-LIST), (3D X ray Tomography) E. Barat (CEA-LIST) (Positon Emission Tomography, Non Parametric Bayesian) C. Comtat (SHFJ, CEA)(PET, Spatio-Temporal Brain activity)
Questions and Discussions A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
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Continuous Signals and Images Simple Gauss-Markov model g = Hf + ǫ f (r) Continuous Image : Gauss-Markov p(f ) = N (0, Σf ) p(fj |fi , i 6= j) = N (βf , σ2 ) j−1 P f p(f (r)|f (s)) = N β s∈V(r) f (s), σf2
MAP : fb = arg maxf {p(f |g)} = arg minf {J(f )} J(f ) = kg − Hf k2 +
P
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(f j − βf j−1 )2
J(f ) = kg − Hf k2 2 P P + r f (r) − β s∈V(r) f (s)
A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
39/41
Piecewise continuous signals and images Compound Gauss-Markov with contours hidden variable model f (r) piecewise continuous image : Compound Intensity-Contours MRF Hidden contour variable: q(r) p(fj |qj , fi , i 6= j) = N (β(1 − qj )fj−1 , σf2 ) p(f (r)|q(r), f (s)) P = N β(1 − q(r)) s∈V(r) f (s), σf2 b = arg maxf,q {p(f , q|g)} MAP : (fb, q)
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fb = arg max {p(f |g, q)} = arg min {J(f )} f
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NIMI Workshop on medical imaging, Gent, Belgium,
May 29, 2010.
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Objects composed of a finite number of homogeneous materials f : Compound intensity-regions MRF Introduction of a class label variable z(r) z(r) = k, k = 1, · · · , K Rk = {r : z(r) = k}, R = ∪k Rk p(f (r)|z (r) = k) = N (f (r)|mk , σk2 ) z = {z(r), r ∈ R} a segmented image Potts MRF: X X p(z) ∝ exp α δ(z(r) − z(s))
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p(f , z, θ|g) ∝ p(g|f , θ 1 ) p(f |z, θ 2 ) p(z|θ 2 ) p(θ) b b g) f ∼ p(f |b z , θ, b g) zb ∼ p(z|fb, θ, b b θ ∼ p(θ|f , zb, g) A. Mohammad-Djafari,
NIMI Workshop on medical imaging, Gent, Belgium,
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