Image Reconstruction in Computed Tomography From deterministic to

Unknown quantity: f (r) = k2. 0 (n2(r) â 1). Intermediate quantity : Ï(r) Ïd (ri ) = â«â«D. Gm(ri ,râ²)Ï(râ²) f (r. â²) drâ², ri â S Ï(r) = Ï0(r) + â«â«D. Go(r,râ²)Ï(râ²)f ...
. Image Reconstruction in Computed Tomography From deterministic to probabilistic methods 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 European School of Medical Physics, Oct.-Nov. 2011

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Content

Seeing outside of a body: Image restoration

Seeing inside of a body: Image reconstruction

Computed Tomography: Different imaging systems

Common inverse problem formulation

Analytical Methods

Algebraic Deterministic Methods

Probabilistic Methods

Bayesian approach

Examples and case studies

Questions and Discussion

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Seeing outside of a body: Making an image with a camera, a microscope or a telescope ◮

f (x, y ) real scene

g (x, y ) observed image

Forward model: Convolution ZZ g (x, y ) = f (x ′ , y ′ ) h(x − x ′ , y − y ′ ) dx ′ dy ′ + ǫ(x, y ) h(x, y ): Point Spread Function (PSF) of the imaging system

Inverse problem: Image restoration Given the forward model H (PSF h(x, y ))) and a set of data g (xi , yi ), i = 1, · · · , M find f (x, y )

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Making an image with an unfocused camera Forward model: 2D Convolution ZZ g (x, y ) = f (x ′ , y ′ ) h(x − x ′ , y − y ′ ) dx ′ dy ′ + ǫ(x, y ) ǫ(x, y )

f (x, y ) - h(x, y )

? - + -g (x, y )

Inversion: Deconvolution ? ⇐=

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Different ways to see inside of a body Incident wave

6  object -

Y

object

-

Passive Imaging

Active Imaging Measurement

Incident wave -

Measurement Incident wave

object

-

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Reflection

Transmission

R

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Seeing inside of a body: Computed Tomography ◮

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 )

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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,

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80

Microwave or ultrasound imaging Mesaurs: diffracted wave by the object φd (ri ) Unknown quantity: f (r) = k02 (n2 (r) − 1) Intermediate quantity : φ(r)

y

Object

ZZ

r'

Gm (ri , r′ )φ(r′ ) f (r′ ) dr′ , ri ∈ S D ZZ Go (r, r′ )φ(r′ ) f (r′ ) dr′ , r ∈ D φ(r) = φ0 (r) + φd (ri ) =

Measurement

plane

Incident

plane Wave

x

D

Born approximation (φ(r′ ) ≃ φ0 (r′ )) ): ZZ Gm (ri , r′ )φ0 (r′ ) f (r′ ) dr′ , ri ∈ S φd (ri ) = D

z

-

φ0 Discretization :   φd = H(f) φd = Gm Fφ −→ with F = diag(f) φ = φ0 + Go Fφ  H(f) = Gm F(I − Go F)−1 φ0 A. Mohammad-Djafari,

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r

(φ, f ) g

Fourier Synthesis in X ray ZZ Tomography

f (x, y ) δ(r − x cos φ − y sin φ) dx dy

g (r , φ) =

G (Ω, φ) = F (ωx , ωy ) = F (ωx , ωy ) = P(Ω, φ) y 6 s I

Z

g (r , φ) exp {−jΩr } dr

ZZ

f (x, y ) exp {−jωx x, ωy y } dx dy

for

ωx = Ω cos φ and ωy = Ω sin φ ωy 6 α Ω

r



I

f (x, y ) φ



F (ωx , ωy )

-

x

p(r , φ)–FT–P(Ω, φ)

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φ

-

ωx

Fourier Synthesis in X ray tomography

F (ωx , ωy ) =

ZZ

f (x, y ) exp {−jωx x, ωy y } dx dy

v 50 100

u

? =⇒

150 200 250 300 350 400 450 50

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100

150

200

250

300

Fourier Synthesis in Diffraction tomography ωy

y ψ(r, φ)

^ f (ωx , ω y )

FT 1

2 2 1

f (x, y)

x Incident plane wave Diffracted wave

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-k 0

k0

ωx

Fourier Synthesis in Diffraction tomography

F (ωx , ωy ) =

ZZ

f (x, y ) exp {−jωx x, ωy y } dx dy

v 50

100

150

u

? =⇒

200

250

300 50

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100

150

200

250

300

350

400

Fourier Synthesis in different imaging systems

F (ωx , ωy ) =

ZZ

v

f (x, y ) exp {−jωx x, ωy y } dx dy

v

v

u

X ray Tomography

u

Diffraction

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u

Eddy current

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u

Invers Problems: other examples and applications ◮

X ray, Gamma ray Computed Tomography (CT)

Microwave and ultrasound tomography

Positron emission tomography (PET)

Magnetic resonance imaging (MRI)

Photoacoustic imaging

Geophysical imaging

Non Destructive Evaluation (NDE) and Testing (NDT) techniques in industry

Hyperspectral imaging

Earth observation methods (Radar, SAR, IR, ...)

Survey and tracking in security systems

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Computed tomography (CT) A Multislice CT Scanner Fan beam X−ray Tomography −1

−0.5

0

0.5

g (si ) = 1

Source positions

−1

−0.5

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f (r) dli + ǫ(si )

Li

Detector positions

0

Z

Discretization g = Hf + ǫ

1

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Magnetic resonance imaging (MRI) Nuclear magnetic resonance imaging (NMRI), Para-sagittal MRI of the head

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X ray Tomography   Z I = g (r , φ) = − ln f (x, y ) dl I0 Lr ,φ ZZ

150

100

y

f(x,y)

f (x, y ) δ(r − x cos φ − y sin φ) dx dy

g (r , φ) =

50

D

0

x

−50

−100

f (x, y)

−150

−150

phi

−100

−50

0

50

100

g (r , φ)

RT

150

60

p(r,phi)

40 315

IRT ? =⇒

270 225 180 135 90 45

0

−20

−40

−60

0

−60

r

20

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−20

0

20

40

60

Analytical Inversion methods S•

y 6

r



f (x, y ) φ

-

x

•D g (r , φ) Radon:

ZZ

f (x, y ) δ(r − x cos φ − y sin φ) dx dy   Z π Z +∞ ∂ 1 ∂r g (r , φ) f (x, y ) = − 2 dr dφ 2π 0 −∞ (r − x cos φ − y sin φ)

g (r , φ) =

D

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Filtered Backprojection method f (x, y ) =



1 − 2 2π

Z

π

0

Z

∂ ∂r g (r , φ)

+∞ −∞

(r − x cos φ − y sin φ)

dr dφ

∂g (r , φ) ∂r Z ∞ 1 g (r , φ) ′ dr Hilbert TransformH : g1 (r , φ) = π (r − r ′ ) Z π 0 1 g1 (r ′ = x cos φ + y sin φ, φ) dφ Backprojection B : f (x, y ) = 2π 0 Derivation D :

g (r , φ) =

f (x, y ) = B H D g (r , φ) = B F1−1 |Ω| F1 g (r , φ) • Backprojection of filtered projections: g (r ,φ)

−→

FT

F1

−→

Filter

|Ω|

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g1 (r ,φ)

−→

Backprojection B

f (x,y )

−→

Limitations : Limited angle or noisy data

60

60

60

60

40

40

40

40

20

20

20

20

0

0

0

0

−20

−20

−20

−20

−40

−40

−40

−40

−60

−60

−60

−60

−40

−20

0

20

40

60

−60

−40

Original

−20

0

20

40

64 proj.

60

−60

−60 −40

−20

0

20

40

16 proj.

Limited angle or noisy data

Accounting for detector size

Other measurement geometries: fan beam, ...

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60

−60

−40

−20

0

20

40

8 proj. [0, π/2]

60

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

f (r) dli + ǫ(si ) −→ Discretization −→ g = Hf + ǫ

Li

1

g, f and H are huge dimensional

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Algebraic methods: Discretization S•

Hij

y 6

r



f1 fj

f (x, y )

gi

φ

-

fN

x

•D g (r , φ) g (r , φ) =

Z

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

f (x, y ) dl

gi =

L

j=1

g = Hf + ǫ A. Mohammad-Djafari,

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Hij fj + ǫi

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)) bf = 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.

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Deterministic Inversion Algorithms Least Squares Based Methods bf = arg min {J(f)} f

with J(f) = kg − Hfk2

∇J(f) = −2Ht (g − Hf)

Initialize:

f (0)

f (k+1) = f (k) − α∇J(f (k) )   At each iteration: f (k+1) = f (k) + αHt g − Hf (k) we have to do the following operations: ◮ Compute g b = Hf (Forward projection) ◮

Iterate:

Compute

Distribute δf = Ht δg (Backprojection of error)

Update

δg = g − b g (Error or residual)

f (k+1) = f (k) + δf ISIP 2012,

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Gradient based algorithms Operations at each iteration:

  f (k+1) = f (k) + αHt g − Hf (k)

b g = Hf (Forward projection) b (Error or residual) δg = g − g

Compute

Compute

Distribute δf = Ht δg (Backprojection of error)

Update

f (k+1) = f (k) + δf

projections of Initial estimated Forward guess −→ image −→ projection −→ estimated image −→ H b g = Hf (k) f (0) f (k) ↑ update ↑

↓ compare ↓

correction term Backprojection in image space ←− ←− Ht δf = Ht δg

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Measured ← projections g

Fixed step gradient:   f (k+1) = f (k) + αHt g − Hf (k)

  f (k+1) = f (k) + α(k) Ht g − Hf (k)

with α(k) = arg minα {J(f + αδf)} ◮

Conjugate Gradient f (k+1) = f (k) + α(k) d(k) The successive directions d(k) have to be conjugate to each other.

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Algebraic Reconstruction Techniques

Main idea: Use the data as they arrive   f (k+1) = f (k) + α(k) [Ht ]i ∗ gi − [Hf (k) ]i

which can also be written as:   gi − [Hf (k) ]i hti∗ f (k+1) = f (k) + hti∗ hi ∗   P (k) X gi − j Hij fj = f (k) + Hij P 2 i Hij i

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Algebraic Reconstruction Techniques ◮

Use the data as they arrive f (k+1) = f (k) +

  gi − [Hf (k) ]i

hti∗ hti∗ hi ∗   P (k) g − H f X i j ij j Hij = f (k) + P 2 i Hij i

Update each pixel at each time   P (k) gi − j Hij fj (k) (k+1) = fj + fj Hij P 2 i Hij

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Algebraic Reconstruction Techniques (ART)

f (k+1) or fj

  P (k) g − H f X i j ij j = f (k) + Hij P 2 i Hij i

(k+1)

  P (k) gi − j Hij fj (k) = fj + Hij P 2 i Hij

projections of Initial estimated Forward image guess −→ image −→ projection −→ estimated P (k) (0) (k) H bi = g f f j Hij f j

−→

↑ update ↑

Measured ← projections gi

↓ compare ↓

correction term in image space P δg P i δfj = i Hij H j

←−

Backprojection ←− Ht

ij

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Algebraic Reconstruction using KL distance ◮

bf = arg min {J(f)} f fj

(k+1)

with fj =P

J(f) =

(k)

i

Hij

X i

P

i

gi ln P gHi ij fj

Hij P

j

gi j

Hij fj

(k)

Interestingly, this is the OSEM (Ordered subset Expectation-Maximization) algorithm which is based on Maximum Likelihood and proposed first by Shepp & Vardi. estimated Initial image f (k) guess −→ (k) f (k+1) f (0) fj = Pj H i

ij

projections of Forward image −→ projection −→ estimated P (k) H b gi = j Hij f j

↑ update ↑

−→

↓ compare ↓

correction term in image space P δfj = P 1H i Hij δgi j

←−

Backprojection ←− Ht

ij

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Measured ← projections gi

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,

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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) : bf = 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,

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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 − Hfk2 2σǫ Hypothesis on f : f ∼ N (0, σf2 I)   1 2 p(f) ∝ exp − 2 kfk 2σf A posteriori:   1 1 2 2 p(f|g) ∝ exp − 2 kg − Hfk − 2 kfk 2σǫ 2σf MAP : bf = arg maxf {p(f|g)} = arg minf {J(f)} with

J(f) = kg − Hfk2 + λkfk2 ,

Advantage : characterization of the solution b with bf = PH b t g, f|g ∼ N (bf, P)

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λ=

σǫ2 σf2

 b = Ht H + λI −1 P

MAP estimation with other priors: bf = arg min {J(f)} with J(f) = kg − Hfk2 + λΩ(f) f Separable priors:  P 2 ◮ Gaussian: p(fj ) ∝ exp −α|fj |2 −→ Ω(f) = kfk2 = α j |fj | P ◮ Gamma: p(fj ) ∝ f α exp {−βfj } −→ Ω(f) = α j 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 ) −→   i ∈Nj

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Ω(f) = α

Ω(f) = α

P

XX j

i ∈Nj

j

|fj |p ,

φ(fj , fi ),

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

Meaning and tools for model selection

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 ...

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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 − HWzk2 + 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 − HWzk2 + λ

z z = arg minz {J(z)} −→ bf = Wb

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P

j

|z j |β

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) p(g |f ,θ ;M) p(f |θ ;M) p(θ |M) Bayes: −→ p(f, θ|g; M) = p(g |M) b = arg max {p(f, θ|g; M)} Joint MAP: (bf, 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 bf = 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

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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: ◮ ◮ ◮

Gaussian approximation (Laplace) Numerical exploration MCMC Variational Bayes (Separable approximation)

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Which images I am looking for? 50 100 150 200 250 300 350 400 450 50

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200

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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

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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 Mixture of Gaussians k , vk )X  p(z(r)|z(rk′ ), r′ ∈ V(r)) ∝ exp γ δ(z(r) − z(r′ ))  ′  r ∈V(r) A. Mohammad-Djafari,

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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)

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f (r)

z(r) 42/56

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

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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)

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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)

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July23-28, 2012,

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MCMC based algorithm p(f, z, θ|g) ∝ p(g|f, z, θ) p(f|z, θ) p(z) p(θ) 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)

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|bf, b b p(z) Sample z from p(z|bf, θ, z, θ) Needs sampling of a Potts Markov field.

z, (mk , vk )) p(θ) Sample θ from p(θ|bf, b z, g) ∝ p(g|bf, σǫ2 I) p(bf|b Conjugate priors −→ analytical expressions.

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July23-28, 2012,

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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

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Application in CT 20

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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,

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July23-28, 2012,

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Results: 2D case

Original

Backprojection

Gauss-Markov+pos

Filtered BP

GM+Line process

GM+Label process

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LS

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c

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z

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Some results in 3D case (Results obtained with collaboration with CEA)

M. Defrise

Phantom

FeldKamp

Proposed method

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July23-28, 2012,

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Some results in 3D case

FeldKamp

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July23-28, 2012,

Proposed method

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Some results in 3D case Experimental setup

A photograpy of metalique esponge

Reconstruction by proposed method

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July23-28, 2012,

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Application: liquid evaporation in metalic esponge

Time 0

Time 1

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July23-28, 2012,

Time 2

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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)

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July23-28, 2012,

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Some references ◮ ◮ ◮ ◮ ◮ ◮ ◮ ◮ ◮

◮ ◮ ◮

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Thanks, Questions and Discussions Thanks to:

◮ ◮ ◮ ◮

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:

◮ ◮ ◮ ◮ ◮

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:

◮ ◮ ◮ ◮ ◮ ◮ ◮

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)