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:
? =⇒
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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)
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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
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◮
X
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Forward & errors model: −→ p(g|f , θ1; M)
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r∈R k
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◮
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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250 50
Possible approximations : Gauss-Laplace (Gaussian approximation) ◮ Exploration (Sampling) using MCMC methods ◮ Separable approximation (Variational techniques)
p(g|f ; M) p(f |M)
100
is too complex
◮
p(f |M)
A priori information ◮ Bayes : ◮
50
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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