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Statistical methods for inverses problems in signal and image processing

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Ali Mohammad-Djafari Ph.D. Students & collaborators: M. Ichir, O. F´eron, P. Brault, A. Mohammadpour, Z. Chama H. Snoussi, F. Humblot, S. Moussaoui, N. Bali, H. Akbari Laboratoire des Signaux et Syst`emes ´lec-Ups Cnrs-Supe Sup´elec, Plateau de Moulon 91192 Gif-sur-Yvette, FRANCE. [email protected] &

http://djafari.free.fr 1

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Contents

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• Inverses problems in image processing • Summary of different statistical methods • Multi sensor image processing problems • Basics of Bayesian approach • HMM modeling of images • Examples of applications – Single channel image restoration – Fourier synthesis in optical imaging – Multi channel data fusion and joint segmentation – Video movie segmentation with motion estimation – Blind source (image) separation (BSS) – Hyperspectral image segmentation • Bayesian image processing in wavelet domain • Some results and conclusions &

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Inverse problems H

model

( g, measurement (Data)

f, unknown quantities

z, other unknowns

² ) errors and noise

Explicit relation:

g = H(f , z, ²)

Additive output error model:

g = H(f , z) + ²   g = H (f , z) + ²

Relation between f and z:

Non linear+ additive errors: Linear + additive errors: &

=

0

1

 H2 (f , z) = 0

gi = hi (f ) + ²i hi (f ) =

X j

3

or

g = H(f ) + ²

hij fj −→ g = Hf + ²

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Inverses problems in image processing

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• General non linear inverse problem: g(r) = [Hf (r 0 )](r) + ²(r), • Linear model: g(r) =

Z

r = (x, y) ∈ R,

r 0 = (x0 , y 0 ) ∈ R0

f (r 0 )h(r, r 0 ) dr 0 + ²(r)

R0

• Linear and translation invariante (convolution) model: Z f (r 0 )h(r − r 0 ) dr 0 + ²(r) = h(r) ∗ f (r) + ²(r) g(r) = R0

• Discretized version g = Hf + ² where g = {g(r), r ∈ R}, ² = {²(r), r ∈ R} and f = {f (r 0 ), r 0 ∈ R0 }

&

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Deconvolution ²(t) f (t)

? - + -

h(t)

-

Observation model:

g(t) = h(t) ∗ f (t) + ²(t)

g = Hf + ²

x(t)

yb(t)

0.9

0.9

0.8

0.8

? ⇐=

0.7 0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

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0.1

0.1

0 −0.1 0

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Single channel image restoration

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²(x, y)

? f (x, y) -

h(x, y)

- +

Observation model :

- g(x, y) = f (x, y) ∗ h(x, y) + ²(x, y)

g = Hf + ²

? ⇐=

&

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Fourier synthesis in optical imaging Z £ ¤ t g(ω) = f (r) exp −jω r dr + ²(ω)

• Coherent imaging:

G(g) = g

−→

g = Hf + ²

• Non coherent imaging:

G(g) = |g|

−→

g = H(f ) + ²

g = {g(ω), ω ∈ Ω},

² = {²(ω), ω ∈ Ω}

?

20

40

and

f = {f (r), r ∈ R}

20

40

⇐=

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Deterministic methods

Data matching • Observation model

gi = hi (f ) + ²i ,

i = 1, . . . , M

• Misatch between data and output of the model ∆(g, H(f ))

• Examples: – LS

b = arg min {∆(g, H(f ))} f f 2

∆(g, H(f )) = kg − H(f )k =

X

|gi − hi (f )|

2

i

– Lp – KL

p

∆(g, H(f )) = kg − H(f )k = ∆(g, H(f )) =

X i

X

p

|gi − hi (f )| ,

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