Copula and Tomography Ali Mohammad-Djafari & Doriano-Boris Pougaza ` 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
StatImage, 19-20/01/2009, Univ. Paris 1
1 / 16
Content
◮
Tomography
◮
Radon Transform
◮
Classical methods of Tomography
◮
Tomography and Copula
◮
Some preliminary results
◮
Conclusions
2 / 16
Tomography : seeing 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 (r ), i = 1, · · · , M find f (x, y) 3 / 16
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) 4 / 16
X ray Tomography and Radon Transform Z I = g(r , φ) = − ln f (x , y ) dl I0 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
5 / 16
Analytical Inversion methods 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
π 0
Z
+∞ −∞
∂ ∂r g(r , φ)
(r − x cos φ − y sin φ)
dr dφ 6 / 16
Filtered Backprojection method f (x, y) =
1 − 2 2π
Z
0
π
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 )
−→
7 / 16
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, ...
8 / 16
Tomography with only Two projections
Forward problem : Given f (x, y) find f1 (x) and f2 (y)
Inverse problem : Given f1 (x) and f2 (y) find f (x, y)
9 / 16
Link between Tomography and Copula Tomography : Given the two horizontal and vertical projections f1 (x) and f2 (y), find f (x, y) Copula : Given the two marginal pdfs f1 (x) and f2 (y), find the joint pdf f (x, y) Ill-posed Inverse problems : Infinite number of solutions There is a need to reduce the space of possible solutions
10 / 16
Link between Tomography and Copula Copula and Tomography Given f1 (x) and f2 (y) find f (x, y) such that Z Z f1 (x) = f (x, y) dy and f2 (y) = f (x, y) dx Tomography : If we find a solution f0 (x, y), we can add any function ω(x, y) such that Z Z ω(x, y) dx = ω(x, y) dy = 0 and we get again another solution. Copula : Any function f (x, y) given by f (x, y) = f1 (x) f2 (y) c (F1 (x), F2 (y)) where c(u, v) is any copula, is a solution to the problem. Z Z c(x, y) dx = c(x, y) dy = 1 11 / 16
Link between Tomography and Copula Tomography : Back Projection (BP) : f (x, y) = f (x) + f (y) Copula : Multiplicative Back Projection (MBP) : f (x, y) = f (x) f (y) Copula Back Projection (MBP) : f (x, y) = f (x) f (y) c (F1 (x), F2 (y))
12 / 16
Preliminary results
f(x,y, f (x) and f (y) 1
2
fh(x,y), fh (x) and fh (y) 1
fh(x,y), fh (x) and fh (y)
2
1
2
fh(x,y), fh1(x) and fh2(y)
Originals f (x, y)
BP bf (x, y)
FBP bf (x, y)
MBP bf (x, y) 13 / 16
Preliminary results
Originals f (x, y)
BP bf (x, y)
FBP bf (x, y)
MBP bf (x, y) 14 / 16
Conclusions
◮
There is a link between Copula and Tomography.
◮
Both problems are ill-posed.
◮
We are sure that the methods in one domain can be useful in other one.
◮
With only two projections we can only reconstruct simple images
◮
There is a need for more data or for more constraintes (maximum entropy, minimum energy, ...)
◮
How to use more than 2 projections with Copula ?
15 / 16
