Image restoration — Convex approaches: penalties and constraints —
An example in astronomy
Jean-Fran¸cois Giovannelli
[email protected] Groupe Signal – Image Laboratoire de l’Int´egration du Mat´eriau au Syst`eme Univ. Bordeaux – CNRS – BINP
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Summary
Direct model and inverse problem Interpolation-extrapolation / deconvolution / Fourier synthesis Indetermination, non inversibility
Prior information and regularized solution Positivity and possible support Point sources onto smooth background and double model
Algorithmic aspects and numerical optimisation Data processing results Simulated Data NRH Data
Conclusions et extensions
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Interferometry: principles of measurement Physical principle [Thompson, Moran, Swenson, 2001] Antenna array
large aperture
Frequency band, e.g., 164 MHz Couple of antennas interference
Antenna positions
Fourier plane
0
Frequency (v )
North-South (km)
Picture site (NRH)
one measure in the Fourier plane
−1
−2
20
40
60
80
100
120
−1
0
1
2
3
West-Est (km)
20
40
60
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100
120
Frequency (u)
Knowledge of the sun, magnetic activity, eruptions, sunspots,. . . Forecast of sun events and their impact,. . . 3 / 28
Interferometer: example of measurements
ES
PS x
Fx
TFx
y = TFx + ε
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Instrument model Truncated and noisy Fourier transform y = TFx + ε x ∈ RN : unknown image y, ε ∈ CM : measurements, errors F : Fourier matrix (N × N) T : troncature matrix (M × N), e.g., T =
h
0 0 0
1 0 0
0 1 0
0 0 0
0 0 1
0 0 0
0 0 0
i
ε x
H
+
y
Difficulties: M N, noise 5 / 28
Different formulations Fourier synthesis (original formulation) y = TFx + ε ◦
Interpolation – extrapolation: change of variable x = F x ◦
y = Tx + ε Deconvolution: transformation of data ◦
y ¯ = F †T ty H = F †T tT F ◦
y ¯ = Hx + e ε A few simple properties F † F = F F † = IN : orthonormality T t : zero-padding matrix, M t
N (T t extends)
T T : (diagonal) projection matrix, N
N (T t T nullifies)
T T t = IM 6 / 28
Interferometry: illustration True map ES
True map PS
Dirty beam
Dirty map ES
Dirty map PS
Dirty map PS + ES
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Data based inversion and ill-posed character
Deficient rank, missing data F T , T , H: 1 singular value order M and 0 order N − M
Infinity of Least-Squares solution QLS (x) = ky − T F xk2
Other solutions: minimum norm solution, TSVD, (quasi) Wiener . . . ◦
QLS (y ¯ )=0 ◦ ¯ + u − F † T t T F u = 0, QLS y
for all map u
Necessity of other information
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Taking constraints into account: positivity and support Notation M: index set of the image pixels S, D ⊂ M: index set of a part of the image pixels
Investigated constraints here Positivity Cp : ∀p ∈ M ,
xp > 0
Support Cs : ∀p ∈ S¯ ,
xp = 0
Extensions (non investigated here) Template ∀p ∈ M ,
tp− 6 xp 6 tp+
Partially known map ∀p ∈ D ,
xp = mp 9 / 28
Point sources + extended source
Double-model [Ciuciu02, Samson03] et [Magain98, Pirzkal00] x = xe + xp Direct model y = T F (xe + xp ) + ε New indeterminations
Appropriate regularisation Re (xe ) =
X
[xe (n + 1) − xe (n)]2
Rp (xp ) =
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Point sources + extended source
Double-model [Ciuciu02, Samson03] et [Magain98, Pirzkal00] x = xe + xp Direct model y = T F (xe + xp ) + ε New indeterminations
Appropriate regularisation X
[xe (n + 1) − xe (n)]2 X Rp (xp ) = |xp (n)| = Re (xe ) =
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Point sources + extended source
Double-model [Ciuciu02, Samson03] et [Magain98, Pirzkal00] x = xe + xp Direct model y = T F (xe + xp ) + ε New indeterminations
Appropriate regularisation X
[xe (n + 1) − xe (n)]2 X X Rp (xp ) = |xp (n)| = xp (n) Re (xe ) =
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Frequential analysis
1
0.5
0 0
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0.1
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0.5
1
0.5
0 0
1
0.5
0 0
0.1
Reduced frequency 13 / 28
Regularized criterion y regularized solution Criterion: penalized, quadratic, strictly convex Q(xe , xp )
2
= ky − T F (xe + xp )k + λe
X
2
[xe (n + 1) − xe (n)] + λp
+ εe
X
xe (n)2 + εp
X
X
xp (n)
xp (n)2
Solution: unique constrained minimizer (x = [xe ; xp ])
cp ) = (c xe , x
arg min Q(xe , xp )
s.t. (C )
1 arg min xt Q x + q t x 2 = ( xp = 0 for p ∈ S¯ s.t. xp ≥ 0 for p ∈ M 14 / 28
Constraints: some illustrations
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Positivity: one variable One variable: α(t − ¯t )2 + γ 250
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50
0 −10
0 −5
0
5
10
−10
t
−5
0
5
10
t
Non-constrained solution: bt = ¯t Constrained solution: bt = max [ 0, ¯t ] Active and inactive constraints 16 / 28
Positivity: two variables (1) Two variables: α1 (t1 − t¯1 )2 + α2 (t2 − t¯2 )2 + β(t2 − t1 )2 + γ
Glop
Pas glop
10
10
5
5
0
0
−5 −5
0
5
10
−5 −5
0
5
10
Sometimes / often difficult to deduce the constrained minimiser from the non-constrained one
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Positivity: two variables (2) Two variables: α1 (t1 − t¯1 )2 + α2 (t2 − t¯2 )2 + β(t2 − t1 )2 + γ 10
10
10
5
5
5
0
0
0
−5 −5
0
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1
10
−5 −5
0
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−5 −5
2a
0
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2b
Constrained solution = Non-constrained solution (1) Constrained solution 6= Non-constrained solution (2) . . . so active constraints
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Positivity: two variables (3) Two variables: α1 (t1 − t¯1 )2 + α2 (t2 − t¯2 )2 + β(t2 − t1 )2 + γ 10
10
5
5
0
0
−5 −5
0
5
2a
10
−5 −5
0
5
10
2b
Constrained solution 6= Non-constrained solution (2) . . . so active constraints Constrained solution 6= Projected non-constrained solution (2a) tb1 ; tb2 6= (max [0, t¯1 ] ; max [0, t¯2 ]) Constrained solution = Projected non-constrained solution (2b) tb1 ; tb2 = (max [0, t¯1 ] ; max [0, t¯2 ]) 19 / 28
Numerical optimisation: state of the art Problem Quadratic optimisation with linear constraints Difficulties N ∼ 1 000 000 Constraints ⊕ non-separable variables
Existing algorithms Existing tools with guaranteed convergence [Bertsekas 95,99; Nocedal 00,08; Boyd 04,11] Gradient projection methods, constrained gradient method Broyden-Fletcher-Goldfarb-Shanno (BFGS) and limited memory Interior points and barrier Pixel-wise descent Augmented Lagrangian, ADMM Constrained but separated + non-separated but non-constrained Partial solutions still through FFT
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Lagrangians und penalisation Equality constraint: xp = 0 −
X
`p xp +
p∈S¯
Inequality constraint: (xp ≥ 0) −
X p∈S
`p (xp − sp ) +
1 X 2 c xp 2 ¯ p∈S
(sp − xp = 0 ; sp ≥ 0) 1 X c (xp − sp )2 2 p∈S
Globally L(x, s, `) =
1 1 t x Q x + q t x − `t (x − s) + c (x − s)t (x − s) 2 2 21 / 28
Iterative algorithm L(x, s, `) =
1 t 1 x Q x + q t x − `t (x − s) + c (x − s)t (x − s) 2 2
Iterate three steps 1
Unconstrained minimization of L w.r.t. x e = −(Q + cIN )−1 (q + [` + cs]) x
2
3
Minimization of L w.r.t. s, s.t. sp ≥ 0, ( max (0, cxp − `p )/c e sp = 0
(≡ fft)
for p ∈ S for p ∈ S¯
Update ` `ep =
( max (0, `p − cxp ) `p − cxp
for p ∈ S for p ∈ S¯ 22 / 28
Details about Q and q q
dirty map:
∂J q= ∂x xe ,xp =0
Q
∂J ∂xe = ∂J ∂xp
◦ y ¯ = −2 ◦ y ¯ + λc 1/2
dirty beam:
∂2J Q= = ∂x2
∂2J ∂x2e 2
∂ J ∂xp ∂xe
∂2J H + λs D t D ∂xe ∂xp = ∂2J H 2 ∂xp
H
H + εs I
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Simulated data results
True extended object x?e
ce Estimated extended object x
20
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60
Dirty map
80
100
80
100
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120
120
40 20
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120
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60
True point object x?p
cp Estimated point object x
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Simulated data results
True extended object x?e
ce Estimated extended object x
120
120
100
100
80
80
Dirty map
60
40
60
40
120
20
20
100
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True point object x?p
80
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40
120
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cp Estimated point object x
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Real data: first results ce Estimated extended object x 20
40
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Dirty map 80
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120 20
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cp Estimated point object x
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120
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Real data: first results ce Estimated extended object x 120
100
80
Dirty map
60
120
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100
20
80
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cp Estimated point object x
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120
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Conclusions Synthesis Direct model and inverse problem Interpolation-extrapolation/deconvolution/Fourier synthesis Double-model and appropriate regularisation: point/background Positivity and support
Optimisation: lagrangians Simulations et real data: interferometry in radio-astronomy Perspectives Quantitative assessment Case of a single map: an extended source / a set of point sources Non quadratic penalty background resolution enhancement Data and/or sources “out grid” Hyperparameter estimation 28 / 28
