. Gauss-Markov-Potts Priors for Inverse Problems in Imaging Systems 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. djafari@lss.supelec.fr http://djafari.free.fr http://www.lss.supelec.fr
Journ´ee Probl`emes Inverses et Optimisation de Forme, 13/12/2007 Laboratoire de math´ematiques Jean Leray, Nantes, france
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Content
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Computed Tomography (CT) as an Invers Problem example
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Classical methods : analytical and algebraic method
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Probabilistic methods
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Bayesian inference approach
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Gauss-Markov-Potts prior moedels for images
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Bayesian computation
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VB with Gauss-Markov-Potts prior moedels
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Application in Computed Tomography
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Conclusions
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Questions and Discussion
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2D and 3D Computed Tomography 3D
2D Projections
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y 40
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gφ (r1 , r2 ) =
Z
Lr1 ,r2 ,φ
f (x, y , z) dl
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gφ (r ) =
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Z
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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) 3 / 36
X ray Tomography and Radon Transform Z I = g (r , φ) = − ln f (x, y ) dl I 0 Lr ,φ ZZ g (r , φ) = f (x, y ) δ(r − x cos φ − y sin φ) dx dy
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f(x,y)
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D
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f (x, y )-
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phi
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-g (r , φ)
RT
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p(r,phi)
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IRT ? =⇒
270 225 180 135 90 45
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0 r
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Analytical Inversion methods
y 6 r @ @ @ @ @ f (x, y )@ @@ @ @ φ @ @ x HH @ H @ @ @ @ •D S•
g (r , φ) = Radon : g (r , φ) = f (x, y ) =
ZZ
R
L f (x, y )
dl
f (x, y ) δ(r − x cos φ − y sin φ) dx dy D
1 − 2 2π
Z
π 0
Z
+∞ −∞
∂ ∂r g (r , φ)
(r − x cos φ − y sin φ)
dr dφ 5 / 36
Filtered Backprojection method Z π Z +∞ ∂ 1 ∂r g (r , φ) f (x, y ) = − 2 dr dφ 2π 0 −∞ (r − x cos φ − y sin φ) Derivation D :
∂g (r , φ) ∂r Z ∞ 1 g (r , φ) ′ dr g1 (r , φ) = π 0 (r − r ′ ) Z π 1 f (x, y ) = g1 (r ′ = x cos φ + y sin φ, φ) dφ 2π 0 g (r , φ) =
Hilbert TransformH : Backprojection B :
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|
−→
IFT
F1−1
g1 (r ,φ)
−→
Backprojection B
f (x,y )
−→
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Limitations : Limited angle or noisy data
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Original
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64 proj.
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16 proj.
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Limited angle or noisy data
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Accounting for detector size
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Other measurement geometries : fan beam, ...
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8 proj. [0, π/2]
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Limitations : Limited angle or noisy data −60
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f(x,y)
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Original
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Data
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Backprojection Filtered Backprojection
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CT as a linear inverse problem Fan beam X−ray Tomography −1
−0.5
0
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1
Source positions
−1
g (si ) =
Z
−0.5
Detector positions
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f (r) dli −→ Discretization −→ g = Hf + ǫ Li 9 / 36
Classical methods in CT
g (si ) =
Z
f (r) dli −→ Discretization −→ g = Hf + ǫ
Li
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H is a huge dimensional matrix of line integrals
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Hf is the forward or projection operation
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H t g is the backward or backprojection operation
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(H t H)−1 H t g is the filtered backprojection minimizing kg − Hf k2 Iterative methods : fb(k+1) = fb(k) + α(k) H t g − H fb(k) is the Least squares iterative reconstruction method Regularization : J(f ) = kg − Hf k2 + λkDf k2 .
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Inversion : Deterministic methods Data matching ◮
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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
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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
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X
|gi − hi (f )|p ,
1
