Inverse Problems in Imaging systems and Computer Vision: From

h(x,y) : Point Spread Function (PSF) of the imaging ... Inverse problem : gφ(r) or gφ(r1,r2) −→ f(x,y) or f(x,y,z) ..... Needs optimisation of a quadratic criterion.
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. Inverse Problems in Imaging systems and Computer Vision: From Regularization to Bayesian Inference Ali Mohammad-Djafari ` Groupe Problemes Inverses Laboratoire des Signaux et Syst`emes UMR 8506 CNRS - SUPELEC - Univ Paris Sud 11 ´ Supelec, Plateau de Moulon, 91192 Gif-sur-Yvette, FRANCE. [email protected] http://djafari.free.fr http://www.lss.supelec.fr

´ inter-instituts d’Alembert/Farman Journee ` et du vivant : Imagerie de la matiere de l’instrument a` l’algorithme : 18 mars 2010, ENS Cachan 1 / 37

Content ◮

Invers problems : Examples and general formulation



Inversion methods : Analytical and Algebraic methods



Deterministic methods : Least squares and Regularization



Probabilistic methods : Bayesian inference Three main steps in Bayesian inference :



◮ ◮ ◮

Forward models and likelihood Prior models for images Bayesian computation



Applications (Computed Tomography)



Conclusions



Questions and Discussion 2 / 37

Inverse problems in imaging systems ◮

Example 1 : Measuring variation of temperature with a thermometer ◮ ◮



Example 2 : Making an image with a camera, a microscope or a telescope ◮ ◮



f (x , y ) real scene g(x , y ) observed image

Example 3 : Making an image of the interior of a body ◮ ◮



f (t) variation of temperature over time g(t) variation of length of the liquid in thermometer

f (x , y ) a section of a real 3D body f (x , y , z) gφ (r ) a line of observed radiographe gφ (r , z)

Example 4 : Microwave, ultrasound or optical imaging ◮ ◮

f (x , y ) a section of a real 3D body gφ (r ) a line of diffracted wave measured at a given angle

3 / 37

Measuring variation of temperature with a thermometer ◮

f (t) variation of temperature over time



g(t) variation of length of the liquid in thermometer



Forward model : Convolution Z g(t) = f (t ′ ) h(t − t ′ ) dt ′ + ǫ(t) h(t) : impulse response of the measurement system



Inverse problem : Deconvolution Given the forward model H (impulse response h(t))) and a set of data g(ti ), i = 1, · · · , M find f (t)

4 / 37

Making an image with a camera, a microscope or a telescope ◮

f (x, y) real scene



g(x, y) observed image



Forward model : Convolution ZZ g(x, y) = f (x ′ , y ′ ) h(x − x ′ , y − y ′ ) dx ′ dy ′ + ǫ(x, y) h(x, y) : Point Spread Function (PSF) of the imaging system



Inverse problem : Image restoration Given the forward model H (PSF h(x, y))) and a set of data g(xi , yi ), i = 1, · · · , M find f (x, y) 5 / 37

Making an image with an unfocused camera Forward model : 2D Convolution ZZ g(x, y) = f (x ′ , y ′ ) h(x − x ′ , y − y ′ ) dx ′ dy ′ + ǫ(x, y) ǫ(x, y) ?

f (x, y)- h(x, y) - +m-g(x, y)

Inversion : Deconvolution ? ⇐=

6 / 37

Making an image of the interior of a body ◮

f (x, y) a section of a real 3D body f (x, y, z)



gφ (r ) a line of observed radiographe gφ (r , z)



Forward model : Line integrals or Radon Transform Z gφ (r ) = f (x, y) dl + ǫφ (r ) L

ZZ r ,φ = f (x, y) δ(r − x cos φ − y sin φ) dx dy + ǫφ (r )



Inverse problem : Image reconstruction Given the forward model H (Radon Transform) and a set of data gφi (rj ), i = 1, · · · , M, j = 1, · · · N find f (x, y) 7 / 37

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

f (x, y, z) dl Lr1 ,r2 ,φ

−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) 8 / 37

Microwave or ultrasound imaging Mesaurs : scattered wave by the object φd (ri ) Incident

plane Wave Unknown quantity : f (r) = k02 (n2 (r) − 1) Intermediate quantity : φ(r) (total field)

y

Measurement

plane

Object

r'

r x

TM : 2D Case ZZ :

Gm (ri , r ′ )φ(r ′ ) f (r ′ ) dr ′ , ri ∈ S z ZZ r r r r Go (r, r ′ )φ(r ′ ) f (r ′ ) dr ′ , r ∈ D r φ(r) = φ0 (r) + ! ! D L r r , a , φd (ri ) =

D

a

E - E r Born approximation (φ(r ′ ) ≃ φ0 (r ′ ))) : D e (φ, x ) φ0 ZZ r % r ′ ′ ′ ′ % r Gm (ri , r )φ0 (r ) f (r ) dr , ri ∈ S φd (ri ) = r D

r

r

r

S

(y )

9 / 37

General formulation of inverse problems ◮

General non linear inverse problems : g(s) = [Hf (r)](s) + ǫ(s),



r ∈ R,

s∈S

Z Linear models : g(s) = f (r) h(r, s) dr + ǫ(s) If h(r, s) = h(r − s) −→ Convolution.



Discrete dataZ: g(si ) =

h(si , r) f (r) dr + ǫ(si ),

i = 1, · · · , m



Inversion : Given the forward model H and the data g = {g(si ), i = 1, · · · , m)} estimate f (r)



Well-posed and Ill-posed problems (Hadamard) : existance, uniqueness and stability



Need for prior information 10 / 37

Analytical methods in X ray Tomography y 6

S•

r

 @ @ @ @ @ @ @ f (x, y)   @ @  @ φ @ @ x HH @ H @ @ @ @ •D

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 2π 2

Z

0

π

Z

+∞

−∞

∂ ∂r g(r , φ)

(r − x cos φ − y sin φ)

dr dφ 11 / 37

Filtered Backprojection method f (x, y) =



1 2π 2

Z

π 0

Z

∂ ∂r g(r , φ)

+∞ −∞

(r − x cos φ − y sin φ)

dr dφ

∂g(r , φ) ∂r Z 1 ∞ g(r , φ) ′ dr Hilbert TransformH : g1 (r , φ) = π 0 (r − r ′ ) Z π 1 g1 (r ′ = x cos φ + y sin φ, φ) dφ Backprojection B : f (x, y) = 2π 0 Derivation D :

g(r , φ) =

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 )

−→

12 / 37

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 −40

−20

0

20

Original

40

60

−60

−60 −40

−20

0

20

40

60

64 proj.

−60

−60 −40

−20

0

20

16 proj.

40

60

−60

−40

−20

0

20

40

60

8 proj. [0, π/2]



Limited angle or noisy data



Accounting for detector size



Other measurement geometries : fan beam, ...

13 / 37

Algebraic methods g(si ) = ◮

Z

h(si , r) f (r) dr + ǫ(si ),

i = 1, · · · , M

f (r) is assumed to be well approximated by N X f (r) ≃ fj bj (r) j=1

with {bj (r)} a basis or any other set of known functions Z N X g(si ) = gi ≃ fj h(si , r) bj (r) dr, j=1

g = Hf + ǫ with Hij = ◮

Z

i = 1, · · · , M

h(si , r) bj (r) dr

H is huge dimensional

14 / 37

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



LS :

Examples :

∆(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