Bayesian Approach for Inverse Problems in Imaging - Ali Mohammad

z(r) q(r). ' &. $. %. A family of Hierarchical Gauss-. Markov-Potts prior models p(f(r)|z(r) = k) = N(mk, vk), k = 1, ··· , K p(z) ∝ exp{ −. ∑ r∈R ∑k αkδ(z(r) − k). − ∑.
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Bayesian Approach for Inverse Problems in Imaging Ali Mohammad-Djafari ` Groupe Problemes Inverses ` Laboratoire des Signaux et Systemes (UMR 8506 CNRS - SUPELEC - Univ Paris Sud 11) ´ Supelec, Plateau de Moulon, 91192 Gif-sur-Yvette, FRANCE. E-mail: [email protected] '

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Absract Inverse problems arise in almost all the imaging systems: denoising, segmentation, deconvolution and restoration, joint restoration, segmentation and contour detection. We proposed and developped methods based on the Bayesian inference for all these problems. In particular, we use a family of Gauss-Markov-Potts prior models within the Bayesian framework which gives us the possibility to perform jointly denoising or restoration, segmentation and contours detection in an optimal way.

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Inverse Problems in Imaging Bayesian Estimation Approach Bayesian Computation Systems ◮ Forward model ◮ Direct computation and use of ◮

Denoising:

M:

? =⇒



Likelihood: Observation model M + Hypothesis on the noise ǫ −→

? =⇒

p(f |g; M) =

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◮ 200

g(r) z(r) ◮ Deconvolution and restoration:

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q(r)

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A family of Hierarchical GaussMarkov-Potts prior models

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Prior models −→ p(f |θ 2; M)



Hyperparameters θ = (θ1, θ 2) −→ p(θ|M)



Bayes:





r∈R

γ

X

δ(z(r) − z(s))

s∈V(r)

p(fk |z(r) = k ) = N (mk 1, Σk ) p(f |z) =

K Y

k =1

p(fk ) =

K Y

Examples of applications

b f (r)

g(r)

p(g|f , θ; M)p(f |θ; M)p(θ|M)

b (r) q

zb(r)

Joint Compted Tomography Reconstruction-Segmentation

p(g|M)

Joint MAP: b = arg max {p(f , θ|g; M)} (fb, θ)

Original

Evidence of the ZZ model: p(g|M) =

p(θ|M) df dθ

Unsupervised:

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b f (r)

References

p(g|f , vǫ) = N (Hf , vǫ)

k =1

 θ = vǫ, (αk , mk , vk ), k = 1, ·, K % &

p(θ)

Conjugate priors

Filtered BP

LS

QR

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p(f , z, θ|g) ∝ p(g|f , vǫ) p(f |z, m, v) p(z|γ, α) p(θ)

Backproj.

Proposed method:

p(g|f , θ; M) p(f |θ; M)

Bayesian estimation with Gauss-Markov-Potts prior

N (mk 1, Σk ).

θ = {(αk , mk , vk ), k = 1, · · · , K , γ}

Joint Restoration-Segmentation

=⇒

Marginalization: R  p(f |g; M) = R p(f , θ|g; M) df p(θ|g; M) = p(f , θ|g; M) dθ ◮ Posterior means: ( R fb = f p(f , θ|g; M) df dθ R θb = θ p(f , θ|g; M) df dθ

p(f , z) = p(f |z) p(z) Model Parameters: &

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(f ,θ)

p(f (r)|z(r) = k ) = N (mk , vk ), k = 1, · · · , K αk δ(z(r) − k )

Choice of approximation criterion : KL(q : p)

g = Hf + ǫ





X

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Forward & errors model: −→ p(g|f , θ1; M)



r∈R k





q(r)

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q(f , z, θ) = q1(f |z) q2(z) q3(θ)

Mode (Maximum A Posteriori) ◮ Mean (Posterior Mean) ◮ Marginal modes ◮ ...

p(f , θ|g; M) =

p(z) ∝ exp

by



g(r) f (r) ◮ Joint Restoration, segmentation and contour estimation:

XX

p(f , z, θ|g; M)

p(g|M)

Full Bayesian approach

z(r)

Main idea in Variational Bayesian methods: Approximate

Estimators:

M:







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Possible approximations : Gauss-Laplace (Gaussian approximation) ◮ Exploration (Sampling) using MCMC methods ◮ Separable approximation (Variational techniques)

p(g|f ; M) p(f |M)

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is too complex



p(f |M)

A priori information ◮ Bayes : ◮

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f (r)



p(g|f ; M) = pǫ(g − Hf )

g(r) f (r) ◮ Segmentation and contour detection:

g(r)

p(f , z, θ|g; M)

g = Hf + ǫ

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zb(r)

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b (r) q

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A. Mohammad-Djafari, Gauss-Markov-Potts Priors for Images in Computer Tomography Resulting to Joint Optimal Reconstruction and segmentation, International Journal of Tomography & Statistics, Vol. 11:W09, pp. 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). % &

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