## Gauss-Markov-Potts Priors for Inverse Problems in Imaging Systems

Gauss-Markov-Potts Priors for Inverse Problems in Imaging ... Computed Tomography (CT) as an Invers Problem example .... Accounting for detector size.
. 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. [email protected] 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

1 / 36

Content

Computed Tomography (CT) as an Invers Problem example

Classical methods : analytical and algebraic method

Probabilistic methods

Bayesian inference approach

Gauss-Markov-Potts prior moedels for images

Bayesian computation

VB with Gauss-Markov-Potts prior moedels

Application in Computed Tomography

Conclusions

Questions and Discussion

2 / 36

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

Lr1 ,r2 ,φ

f (x, y , z) dl

−60

gφ (r ) =

−40

Z

−20

0

20

40

60

80

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

150

100

y

f(x,y)

50

D

0

x

−50

−100

f (x, y )-

−150

−150

−100

−50

phi

0

50

100

-g (r , φ)

RT

150

60

p(r,phi)

40 315

IRT ? =⇒

270 225 180 135 90 45

20

0

−20

−40

−60

0 r

−60

−40

−20

0

20

40

60

4 / 36

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 )

−→

6 / 36

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

Original

40

60

−60

−40

−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, ...

60

−60

−40

−20

0

20

40

60

8 proj. [0, π/2]

7 / 36

Limitations : Limited angle or noisy data −60

−60

−60

−40

−40

−20

−20

0

0

20

20

40

40

−150

−40 −100

f(x,y)

y

−20 −50

0

x

0

50

20

100

40 150

60

60 −60

−40

−20

0

20

40

60

−150

−100

−50

0

50

100

60 −60

150

−40

−20

0

20

40

60

−60

−60

−40

−40

−20

−20

−60

−40

−20

0

20

40

60

−60

−40

−20

0

20

40

60

−150

−100

f(x,y)

y

−50

x

0

50

0

0

20

20

40

40

100

150

60 −150

Original

−100

−50

0

50

Data

100

150

60 −60

−40

−20

0

20

40

60

Backprojection Filtered Backprojection

8 / 36

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 −→ Discretization −→ g = Hf + ǫ Li 9 / 36

Classical methods in CT

g (si ) =

Z

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

Li

H is a huge dimensional matrix of line integrals

Hf is the forward or projection operation

H t g is the backward or backprojection operation

(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 .

10 / 36

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