Introduction to MOMFBD
a short overview
Mats Löfdahl
Institute for Solar Physics
Stockholm University
1st CASSDA-SOLARNET Workshop
Freiburg 18-20 February, 2014
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
1 / 13
Multi-Object Multi-Frame Blind Deconvolution
1
What is MOMFBD?
MOMFBD – Deconvolution
MOMFBD – Blind Deconvolution
MOMFBD – Multiple Frames
MOMFBD – Phase Diversity
MOMFBD – Multiple Objects
2
History
3
Problems and solutions
Alignment
Truncated wavefront expansion
Model mismatch
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
2 / 13
Multi-Object Multi-Frame Blind Deconvolution
1
What is MOMFBD?
MOMFBD – Deconvolution
MOMFBD – Blind Deconvolution
MOMFBD – Multiple Frames
MOMFBD – Phase Diversity
MOMFBD – Multiple Objects
2
History
3
Problems and solutions
Alignment
Truncated wavefront expansion
Model mismatch
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
2 / 13
Multi-Object Multi-Frame Blind Deconvolution
1
What is MOMFBD?
MOMFBD – Deconvolution
MOMFBD – Blind Deconvolution
MOMFBD – Multiple Frames
MOMFBD – Phase Diversity
MOMFBD – Multiple Objects
2
History
3
Problems and solutions
Alignment
Truncated wavefront expansion
Model mismatch
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
2 / 13
What is MOMFBD?
MOMFBD – Deconvolution
Blurring and unblurring
Convolution
d =f ∗s
Mats Löfdahl (Institute for Solar Physics)
=
Introduction to MOMFBD
∗
Freiburg 2014-02-18–20
3 / 13
What is MOMFBD?
MOMFBD – Deconvolution
Blurring and unblurring
Convolution
d =f ∗s
=
∗
=
∗−1
Deconvolution
f = d ∗−1 s
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
3 / 13
What is MOMFBD?
MOMFBD – Blind Deconvolution
BD: Joint estimation of object and aberrations
Image:
Image = mix of two unknown quantities
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
How do they separate?
Freiburg 2014-02-18–20
4 / 13
What is MOMFBD?
MOMFBD – Blind Deconvolution
BD: Joint estimation of object and aberrations
Object:
PSF:
Perfect optics and fuzzy object?
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Or...
Freiburg 2014-02-18–20
4 / 13
What is MOMFBD?
MOMFBD – Blind Deconvolution
BD: Joint estimation of object and aberrations
Object:
PSF:
Stellar object and weird PSF?
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Or...
Freiburg 2014-02-18–20
4 / 13
What is MOMFBD?
MOMFBD – Blind Deconvolution
BD: Joint estimation of object and aberrations
Object:
PSF:
Constrain the solution and find ... A star and a proper PSF!
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
4 / 13
What is MOMFBD?
MOMFBD – Multiple Frames
Data collection model: Multiple samples of seeing
Extended target
Turbulence
☛
❘
Optics
❄
Filter
✗
Collected images:
Detector
...
t = 1,
2,
Mats Löfdahl (Institute for Solar Physics)
3,
...
T – Varying seeing, one object
Introduction to MOMFBD
Freiburg 2014-02-18–20
5 / 13
What is MOMFBD?
MOMFBD – Multiple Frames
Maths
Image formation model
dj = f ∗ sj + nj
sj = | F
−1
{Pj
Image
}|2
PSF
Pj = A · exp(iφj )
P
ˆ
φj ≈ M
m αmj ψm = φj
Pupil function
Pupil phase
Image
=
Mats Löfdahl (Institute for Solar Physics)
∗
+
Introduction to MOMFBD
Freiburg 2014-02-18–20
6 / 13
What is MOMFBD?
MOMFBD – Multiple Frames
Maths
Image formation model
dj = f ∗ sj + nj
sj = | F
−1
{Pj
}|2
Pj = A · exp(iφj )
P
ˆ
φj ≈ M
m αmj ψm = φj
Image
PSF
Pupil function
Pupil phase
PSF and pupil function
−1
= F
· exp i ·
Mats Löfdahl (Institute for Solar Physics)
2
Introduction to MOMFBD
Freiburg 2014-02-18–20
6 / 13
What is MOMFBD?
MOMFBD – Multiple Frames
Maths
Image formation model
dj = f ∗ sj + nj
sj = | F
−1
{Pj
Image
}|2
PSF
Pj = A · exp(iφj )
P
ˆ
φj ≈ M
m αmj ψm = φj
Pupil function
Pupil phase
Pupil phase
≈ α4 ·
+ α5 ·
Mats Löfdahl (Institute for Solar Physics)
+ α6 ·
Introduction to MOMFBD
+ · · · + αM ·
=
Freiburg 2014-02-18–20
6 / 13
What is MOMFBD?
MOMFBD – Multiple Frames
Maths
Model fitting
Image formation model
dj = f ∗ sj + nj
sj = | F
−1
{Pj
Image
}|2
PSF
Pj = A · exp(iφj )
P
ˆ
φj ≈ M
m αmj ψm = φj
Pupil function
Pupil phase
Minimize difference
between data and model
data:
P P
minα
|dj − ˆf ∗ sˆj |2
j
pixels
Pupil phase
≈ α4 ·
+ α5 ·
+ α6 ·
+ · · · + αM ·
=
Non-linear optimization methods, linear equality constraints
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
6 / 13
What is MOMFBD?
MOMFBD – Multiple Frames
Maths
Model fitting
Image formation model
dj = f ∗ sj + nj
sj = | F
−1
{Pj
Image
}|2
PSF
Pj = A · exp(iφj )
P
ˆ
φj ≈ M
m αmj ψm = φj
Pupil function
Pupil phase
Minimize difference
between data and model
data:
P P
minα
|dj − ˆf ∗ sˆj |2
j
pixels
Pupil phase
≈ α4 ·
+ α5 ·
+ α6 ·
+ · · · + αM ·
=
Karhunen–Loève functions statistically uncorrelated for atmosphere
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
6 / 13
What is MOMFBD?
MOMFBD – Phase Diversity
Data collection model: Multiple samples of seeing
Extended target
Turbulence
☛
❘
Optics
❄
Filter
✗
Collected images:
Detector
...
t = 1,
2,
Mats Löfdahl (Institute for Solar Physics)
3,
...
T – Varying seeing, one object
Introduction to MOMFBD
Freiburg 2014-02-18–20
7 / 13
What is MOMFBD?
MOMFBD – Phase Diversity
Data collection model: Multiple samples of seeing with PD
Extended target
Turbulence
☛
❘
Shutter
Optics
❄
Beam splitter
❯ Filter
☛
✗
Collected images:
d = 0:
...
d = 1:
...
t = 1,
2,
Mats Löfdahl (Institute for Solar Physics)
3,
...
✐
Detector
Two cameras w/ identical seeing,
intentional focus difference
T – Varying seeing, one object
Introduction to MOMFBD
Freiburg 2014-02-18–20
7 / 13
MOMFBD – Multiple Objects
What is MOMFBD?
Data collection model: Multiple samples of seeing with MO
Extended target
Shutter
Turbulence
☛
❘
Optics
❄
Beam splitter
❯
❯
Filter i = 1
Filter i = 2
Collected images:
i = 1:
...
i = 2:
...
t = 1,
2,
Mats Löfdahl (Institute for Solar Physics)
3,
...
✗
❨
Detector
Two cameras w/ identical seeing
T – Varying seeing, two objects
Introduction to MOMFBD
Freiburg 2014-02-18–20
8 / 13
History
MOMFBD time line
MFBD and MOMFBD
Phase Diversity
R.A. Gonsalves. Optical Engineering,
21(5):829–832, 1982.
T.J. Schulz. JOSA A, 10:1064–1073,
1993.
R.G. Paxman, T.J. Schulz, and J.R.
Fienup. JOSA A, 9(7):1072–1085,
1992a.
M.G. Löfdahl. Proc. SPIE, 4792:146,
2002.
R.G. Paxman, T.J. Schulz, and J.R.
Fienup. OSA Technical Digest Series,
11:5, 1992b.
M.G. Löfdahl and G.B. Scharmer. A&A
Suppl. Ser., 107:243, 1994.
R.G. Paxman, J.H. Seldin, M.G.
Löfdahl, G.B. Scharmer, and C.U.
Keller. ApJ, 466:1087, 1996.
M. van Noort, L. Rouppe van der Voort,
and M.G. Löfdahl. Solar Physics,
228(1–2):191, 2005.
M.J. van Noort and L.H.M. Rouppe van
der Voort. ApJ, 489:429, 2008.
R. Schnerr, J. de la Cruz Rodríguez,
and M. van Noort. A&A, 534:A45, 2011.
V.M.J. Henriques. A&A, 548:A114,
2012.
Phase Diversity
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
9 / 13
History
MOMFBD time line
MFBD and MOMFBD
Phase Diversity
R.A. Gonsalves. Optical Engineering,
21(5):829–832, 1982.
T.J. Schulz. JOSA A, 10:1064–1073,
1993.
R.G. Paxman, T.J. Schulz, and J.R.
Fienup. JOSA A, 9(7):1072–1085,
1992a.
M.G. Löfdahl. Proc. SPIE, 4792:146,
2002.
R.G. Paxman, T.J. Schulz, and J.R.
Fienup. OSA Technical Digest Series,
11:5, 1992b.
M.G. Löfdahl and G.B. Scharmer. A&A
Suppl. Ser., 107:243, 1994.
R.G. Paxman, J.H. Seldin, M.G.
Löfdahl, G.B. Scharmer, and C.U.
Keller. ApJ, 466:1087, 1996.
M. van Noort, L. Rouppe van der Voort,
and M.G. Löfdahl. Solar Physics,
228(1–2):191, 2005.
M.J. van Noort and L.H.M. Rouppe van
der Voort. ApJ, 489:429, 2008.
R. Schnerr, J. de la Cruz Rodríguez,
and M. van Noort. A&A, 534:A45, 2011.
V.M.J. Henriques. A&A, 548:A114,
2012.
Phase Diversity, Solar Phase Diversity
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
9 / 13
History
MOMFBD time line
MFBD and MOMFBD
Phase Diversity
R.A. Gonsalves. Optical Engineering,
21(5):829–832, 1982.
T.J. Schulz. JOSA A, 10:1064–1073,
1993.
R.G. Paxman, T.J. Schulz, and J.R.
Fienup. JOSA A, 9(7):1072–1085,
1992a.
M.G. Löfdahl. Proc. SPIE, 4792:146,
2002.
R.G. Paxman, T.J. Schulz, and J.R.
Fienup. OSA Technical Digest Series,
11:5, 1992b.
M.G. Löfdahl and G.B. Scharmer. A&A
Suppl. Ser., 107:243, 1994.
R.G. Paxman, J.H. Seldin, M.G.
Löfdahl, G.B. Scharmer, and C.U.
Keller. ApJ, 466:1087, 1996.
M. van Noort, L. Rouppe van der Voort,
and M.G. Löfdahl. Solar Physics,
228(1–2):191, 2005.
M.J. van Noort and L.H.M. Rouppe van
der Voort. ApJ, 489:429, 2008.
R. Schnerr, J. de la Cruz Rodríguez,
and M. van Noort. A&A, 534:A45, 2011.
V.M.J. Henriques. A&A, 548:A114,
2012.
Phase Diversity, Solar Phase Diversity, MFBD & MOMFBD
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
9 / 13
History
MOMFBD time line
MFBD and MOMFBD
Phase Diversity
R.A. Gonsalves. Optical Engineering,
21(5):829–832, 1982.
T.J. Schulz. JOSA A, 10:1064–1073,
1993.
R.G. Paxman, T.J. Schulz, and J.R.
Fienup. JOSA A, 9(7):1072–1085,
1992a.
M.G. Löfdahl. Proc. SPIE, 4792:146,
2002.
R.G. Paxman, T.J. Schulz, and J.R.
Fienup. OSA Technical Digest Series,
11:5, 1992b.
M.G. Löfdahl and G.B. Scharmer. A&A
Suppl. Ser., 107:243, 1994.
R.G. Paxman, J.H. Seldin, M.G.
Löfdahl, G.B. Scharmer, and C.U.
Keller. ApJ, 466:1087, 1996.
M. van Noort, L. Rouppe van der Voort,
and M.G. Löfdahl. Solar Physics,
228(1–2):191, 2005.
M.J. van Noort and L.H.M. Rouppe van
der Voort. ApJ, 489:429, 2008.
R. Schnerr, J. de la Cruz Rodríguez,
and M. van Noort. A&A, 534:A45, 2011.
V.M.J. Henriques. A&A, 548:A114,
2012.
Phase Diversity, Solar Phase Diversity, MFBD & MOMFBD,
MOMFBD strategies
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
9 / 13
Problems and solutions
Alignment
Pinhole calibrated camera alignment
arcseconds
10
5
a
0
b
arcseconds
10
5
c
0
0
d
5
arcseconds
0
5
arcseconds
a. LCP; b. RCP; c. MOMFBD
magnetogram; d. Traditional
magnetogram; note artifacts
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
10 / 13
Problems and solutions
Single WB object
Alignment
D
iv
er
sit
y
k
van Noort et al. (2005)
Fe 630.2 Wide
Object i
Fe 630.2 LCP
Fe 630.2 RCP
1
Realization t
9
11
19
All WB frames ⇒ single restored
image
NB frames co-restored with some
WB frames
Different NB states aligned by the
common WB image
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
11 / 13
Problems and solutions
Single WB object
Alignment
Extra WB objects
Henriques (2012)
D
iv
er
sit
y
k
van Noort et al. (2005)
Fe 630.2 Wide
Object i
Fe 630.2 LCP
Fe 630.2 RCP
1
Realization t
9
11
19
All WB frames ⇒ single restored
image
NB frames co-restored with some
WB frames
Different NB states aligned by the
common WB image
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Allow for residual
misalignment
Freiburg 2014-02-18–20
11 / 13
Encir
0.4
0.4
0.4
Problems and solutions
0.2
5 cm
Truncated wavefront expansion
0.2
7 cm
0.2
5 cm
A finite number of wavefront modes
0.0
0
1
2
3
0.0
5
10
15
r0 / 1 cm
r / 1"
Fig. 2: Encircled PSF energy for different r0 as
indicated in the figure. Red: PSFs based on S ;
Blue: PSFs based on Sˆ ; Black: diffraction limited PSF; Black dashed: 90% level.
20
25
Fig. 3: Strehl ratios as a function of r0 . Solid
line: Eq. (11); Plus (+) symbols: PSFs based
on S ; Cross (×) symbols: PSFs based on Sˆ .
0.0
0.0
0.2
0.4
0.6
0.8
1.0
u
Fig. 4: Power spectra (angular averages) of S
(red) and Sˆ (blue) for r0 as indicated.
Perfect correction of 36 KL modes restores resolution, not contrast
13.29%
12.78%
11.83%
9.77%
7.23%
4.63%
14.51%
14.51%
14.50%
14.48%
14.41%
14.29%
14.52%
= ∞ Far left: Original
25 cmimage. Top: Low-pass
20 cm images degraded
15 cmby high-order aberrations,
10 cm
Fig. 5:r0Images.
i.e., by S7, cm
corresponding to5r0cm
=25, 20, 15,
10, 7 and 5 cm, resp. (same layout as Fig. 1). Bottom: Degraded images compensated by use of the method described, i.e., by Sˆ . All images are
scaled between min and max of the original image and low-pass filtered to 90% of the SST diffraction limit. The numbers above and below the
image tiles are the RMS contrasts in percent of the mean intensity (100 × RMS/mean).
Solution
The synthetic images were correction
processed with the based
MOMFBD on
parameters,
without and withstatistics.
added noise). In this
section
Post-restoration
atmospheric
Not
in we
program in various ways (MFBD or JPDS, different numbers of summarize the results of these simulations.
!!
pipeline
yet.
Scharmer & Löfdahl (2010)
realizations,
different
subfield sizes (256 pixels = 15 , 128 pixels
!!
!!
= 7. 6, 80 pixels = 4. 7), different number of estimated wavefront
Mats Löfdahl (Institute for Solar Physics)
Figure 6 shows RMS intensity errors of restored images using different techniques (MFBD, JPDS), number of aberration
parameters (M=35, 50 and 100) without and with Sˆ compensa-
Introduction to MOMFBD
Freiburg 2014-02-18–20
12 / 13
Problems and solutions
Model mismatch
Model mismatch hurts the fit
Pupils
Non-common paths
Telescope
FL
Field stop/pinholes
CT
DC
DM
Blue
path
TM
RL
Prefilter
Shutter
E
LR
Cs
SL
PB
E
HR
Other problems
AO WFS
Optical parameters
Not used
PI
al F
Du
Finite exposures
WB
R
NB
T
NB
ISP
CR
Flats: time-dependent,
polarized, IR
Symptoms
Under-estimated wavefronts ⇒ under-corrected images
Small-scale image artifacts amplified by deconvolution
Mats Löfdahl (Institute for Solar Physics)
Introduction to MOMFBD
Freiburg 2014-02-18–20
13 / 13
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