High Angular Resolution School

35 Pages·2010·0.37 MB·French

High Angular Resolution School
THE FRENCH AEROSPACE LAB
Introduction to inverse problems.
Application to deconvolution
t
ONERA / Dep Optique Théorique & Appliquée, HAR team
0
1
0
2
r
e
b
m
http://www.onera.fr/dota
e
v
o
n
–
r [email protected]
e
i
n
g
u
M
.
L
1
High Angular Resolution School
Context
o
Aim of this talk: introduction to methods for solving “inverse problems”,
including image deconvolution;
o
Common framework for problems and methods in a wide variety of fields:
+
X-ray tomography;
+
ultrasound echography (medical diagnosis, Non-Destructive Testing);
+
computer vision (stereo, etc.);
+
geophysics;
0
1
0
2 +
r wavefront sensing (Hartmann-Shack, curvature, phase diversity);
e
b
m
+
e image restoration, e.g., for imaging through turbulence (w/ or without AO);
v
o
n
+
– image reconstruction in interferometry,
r
e
i
n +
g etc.
u
M
.
L
THE FRENCH AEROSPACE LAB
2
High Angular Resolution School
Inverse problem: a contrario definition
o
Wavefront sensing by Shack-Hartmann
Unknowns: wavefront (phase) Measurements: local slopes
Lenslet array
Sensor
0 o
1 image restoration
0
2
r
e object → atmosphere+telescope+camera → image
b
m
e
v
o o
n image reconstruction in long baseline optical interferometry
–
r
e
ni object → atmosphere+interferometer → visibilities
g
u
M
L. Physics → model the data, i.e., compute the direct problem.
THE FRENCH AEROSPACE LAB
3
High Angular Resolution School
Considered “direct model”, i.e. model of the data
o
linear model:
y = Hx + n, H linear operator (matrix in finite dimension),
y
, vector of measurements (image pixels, S-H slopes,...),
x
, vector of unknowns (object, wavefront, etc.),
n
measurement noise.
o
convolution case:
0
1
20 + H linear and shift-invariant ⇔ Hx = h ⋆ x
r
e
b +
m h : point spread function (PSF) : x = δ ⇒ h ⋆ x = h.
e
v
no + h˜ = FT(h) : transfer function.
–
r + ˜
ie Direct model: y = h ⋆ x + n where y˜ = h.x˜ + n˜
n
g
u
M
.
L
THE FRENCH AEROSPACE LAB
4
High Angular Resolution School
“Classical” resolution of an inverse problem
o
One searches for xˆ ; y = Hxˆ
existence
o
Invertible (linear) problem ∀y:
( uniqueness
o
If dim(x) > dim(y), not enough measurements for a unique solution.
o
0 If dim(x) < dim(y), usually no solution: noise makes it impossible to fit all
1
0
2
r data.
e
b
m
e o
v existence and uniqueness ∀y ⇔ H square and invertible
o
n
–
er o
i Not often the case + accomodating more data is desirable!
n
g
u
M
.
L
THE FRENCH AEROSPACE LAB
5
High Angular Resolution School
Least Squares method
One searches for xˆ that minimizes the Least Squares criterion:
2 2
ky − Hxˆk = inf ky − Hxk (1)
x
Legendre (1805) and Gauss (between 1795 and 1809)
→
(estimation of the Earth’s ellipticity from arc measurements definition of the meter)
o
existence guaranteed (in finite dimension):
(1) ⇔ Py = Hxˆ, where P projector on Im(H).
o
uniqueness guaranteed:
⊥
if N(H) 6= {0} one projects xˆ onto N(H)
→ LS solution of minimal norm xˆ , or Generalized Inverse.
min
0 norm
1
20 o analytical solution: xˆ = (HtH)−1Hty (generally)
r
e
b
m
e
v x y
o T
N(H)
n
–
r H Im H
nie x^min x^
g norm Py
u
M H
N(H)
.
L
THE FRENCH AEROSPACE LAB
6
High Angular Resolution School
Hartmann-Shack Wave Front Sensor
incident wavefront Sub−images
Microlens array
(cid:0)(cid:1)
Sensor
(cid:0)(cid:1)
(cid:0)(cid:1)
(cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1)
(cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
0 αf
1
0
2 α f
r
e
b
m
o
e
Principle: measurement of local slopes of wavefront;
v
o
n
–
o
r
e Works with point sources and extended objects, w/ large spectral bandwidth;
i
n
g
u
M o
Estimation of wavefront from local slopes.
.
L
THE FRENCH AEROSPACE LAB
7
High Angular Resolution School
Hartmann-Shack: direct problem and LS solution
o
Data: K sub-apertures → 2 K slopes y = y
u/v,i
1 ∂φ(u, v)
(cid:8) (cid:9)
y = du dv + noise
u,i
S ∂u
i S
Z i
o
Linear direct model: y = Hx + n with:
← −→
j
j max
max
x = {x }, φ(u, v) = x Z (u, v), H = 2Kx . 
j j j

0
1 
0 j=2
2 X  . 
er  
b
m  
e o 
ov Problem: estimate the wavefront x from the measureyd slopes y
n
–
r
nie o LS solution: xˆ = (HtH)−1Hty
g
u
M
.
L
THE FRENCH AEROSPACE LAB
8
High Angular Resolution School
Hartmann-Shack simulation
o
20 × 20 sub-apertures,
central obstruction d/D = 0.33 ;
o
K = 276 illuminated sub-apertures
⇒ 2.K = 552 slope measurements;
o
n
zero-mean white Gaussian (and
0
1
0
2 approximately stationary), such that
r
e
b
m SNR = 1 ;
e slopes
v
o
n
–
r
e
i
n
g
u
M
.
L
THE FRENCH AEROSPACE LAB
9
High Angular Resolution School
LS wavefront reconstructions
True phase LS j = 3 LS j = 21
max max
0
1
0
2
r
e
b
m
e
v
o LS j = 55 LS j = 105 LS j = 210
n max max max
–
r
e
i Reconstruction quality depends on j
n max
g
u
M optimal j depends on SNR .
max slopes
.
L
THE FRENCH AEROSPACE LAB
10