Blind/Myopic Deconvolution Julian Christou UCSC CfAO PSF - - PowerPoint PPT Presentation

Blind/Myopic Deconvolution

Julian Christou UCSC

CfAO PSF Reconstruction Workshop May 2004

Victoria 10th May 2004 PSF Reconstruction Meeting 2

Adaptive Optics Imaging

• Quality of compensation depends upon:

– Wavefront sensor – Signal strength & signal stability – Speckle noise - d / r0 – Duty cycle - t / t0 – Sensing & observing - _ – Wavefront reconstructor & geometry – Object extent – Anisoplanatism (off-axis)

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Adaptive Optics Imaging

Adaptive Optics systems do NOT produce perfect images (poor compensation)

Without AO With AO Seeing disc Halo Artifacts Binary Star components Core

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Adaptive Optics Imaging

Simulated AO imaging of a Galaxy with different Strehl ratios Strehl Ratio

11% 45% 86%

Deconvolution removes the effect of the imperfect PSF and replaces it with a perfect PSF

Illustrates the need of knowing the PSF

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Why Deconvolution and PSF Calibration?

• Better looking image
• Improved identification

Reduces overlap of image structure to more easily identify features in the image (needs high SNR)

• PSF calibration

Removes artifacts in the image due to the point spread function (PSF)

• f the system, i.e. extended halos, lumpy Airy rings etc.
• Improved Quantitative Analysis

e.g. PSF fitting in crowded fields.

• Higher resolution

In specific cases depending upon algorithms and SNR

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Adaptive Optics: PSF Variability

• Science Target and Reference Star typically observed at

different times and under different conditions.

• Differences in Target & Reference compensation due to:
• Temporal variability of atmosphere(changing r0 & t0).
• Object dependency (extent and brightness) affecting centroid

measurements on the wavefront sensor (SNR).

• Full & sub-aperture tilt measurements
• Spatial variability (anisoplanatism)
• In general: Adaptive Optics PSFs are poorly determined.
• Need PSF for the observation

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Shift invariant imaging equation

(Image Domain) (Fourier Domain)

The Imaging Equation

g(r) – Measurement h(r) – Point Spread Function (PSF) f(r) – Target n(r) – Contamination - Noise

g(r) = f(r) * h(r) + n(r) G(f) = F(f) • H(f) + N(f)

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• Invert the shift invariant imaging equation

i.e. solve for f(r) INVERSE PROBLEM given both g(r) and h(r).

• But h(r) is generally poorly determined.
• Need to solve for f(r) and improve the h(r) estimate simultaneously.

Unknown PSF information Some PSF information Blind/Myopic Deconvolution

Deconvolution

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g(r) = f(r) * h(r) + n(r)

Blind Deconvolution

Measurement unknown

• bject irradiance

unknown or poorly known PSF contamination

Solve for both object & PSF

Single measurement: Under – determined - 1 measurement, 2 unknowns Never really “blind”

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• Uses Physical Constraints.

– f(r) & h(r) are positive, real & have finite support. – h(r) is band-limited – symmetry breaking prevents the simple solution of h(r) = δ(r)

• a priori information - further symmetry breaking (a * b = b * a)

– Prior knowledge (Physical Constraints) – PSF knowledge: band-limit, known pupil, statistical derived PSF – Object & PSF parameterization: multiple star systems – Noise statistics – Multiple Frames: (MFBD)

• Same object, different PSFs.
• N measurements, N+1 unknowns.

Blind Deconvolution – Physical Constraints

• How to minimize the search space for a solution?

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Multiple Observations of a common object

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

2 2 1 1

r h r f r g r h r f r g r h r f r g

n n

∗ = ∗ = ∗ = M M M

• Reduces the ratio of unknown to measurements from 2:1

to n+1:n

• The greater the diversity of h(r),the easier the separation
• f the PSF and object.

Multiple Frame Constraints

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• Uses a Conjugate Gradient Error Metric Minimization scheme
• Least squares fit.
• Error Metric – minimizing the residuals (convolution error):
• Alternative error metric – minimizing the residual autocorrelation:

Autocorrelation of residuals Reduces correlation in the residuals (minimizes “print through”) So not sum over the 0 location.

( )

2 2 2

~ ~ ~

∑ ∑ ∑

= ∗ − = − =

ik ik ik ik i ik ik ik ik

r h f g g g E

An MFBD Algorithm

2

∑

⊗ =

ik ik ik

r r E

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• Object non-negativity

Reparameterize the object as the square of another variable HARD

• r penalize the object against negativity.

SOFT

• PSF Constraints (when pupil is not known)
• Non-negativity

Reparameterize -

• r penalize –
• Band-limit

∑

< ∈

=

~ 2 Obj

~

i

f u i

f E

2

~

i i

f α =

2 , ,

~

k i k i

h β =

An MFBD Algorithm

∑

< ∈

=

~ 2 , PSF

,

~

k i

h u k i

h E

∑

>

=

c

u u k u k

H E

, 2 , bl

~

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Use as much prior knowledge of the PSF as possible.

• Transfer function is band-limited
• PSF is positive and real

PSF Constraints

Normalized Spatial Frequency MTF

fc = D/λ

MTF

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• PSF Constraints (Using the Pupil)
• Parameterize the PSF as the power spectrum of the

complex wavefront at the pupil, i.e. where

An MFBD Algorithm

∗

=

ik ik ik

a a h ~ ~ ~       −       =∑

vk v v ik

N iv j W a ϕ π 2 exp ~

Pupil PSF

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• PSF Constraints (Using the Pupil)
• Modally - express the phases as either a set of Zernike modes of order M
• or zonally as where which

enforces spatial correlation of the phases.

• Phases can also be constrained by statistical knowledge of the AO system

performance.

• Wavefront amplitudes can be set to unity or can be solved for as an

unknown especially in the presence of scintillation.

PSF Constraints

∑

=

=

M m vk m vk

Z q

1

ϕ

( )

vk vk

η φ ϕ ∗ =

              − =

2

2 exp σ η v

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• Myopic Deconvolution (using known PSF information)
• For MFBD penalize PSFs for departure from a “typical” PSF or model

(good for multi-frame measurements)

• Penalize PSF on power spectral density (PSD)

where the PSD is based upon the atmospheric conditions and AO correction.

2 SAA SAA SAA

~

∑

− =

ik ik ik

h h E

∑

          − =

i H i i

H H E PSD ~ ~

2 PSD

PSF Constraints

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• Myopic Deconvolution

using the reconstructed PSF

• where
• and
• where

( )

( )

f T D H

f 

    − =

λ φ

2 1 exp ~

( )

∑

          − =

i H i i

f H H E PSD ~ ~

2 PSD

PSF Constraints

v r

s

36 . = τ

residual phase structure function

( ) ( )

      − = =

2 2 turb

~ STF PSD H f T f

i s H

τ σ

Speckle transfer function Perfect optics transfer function Integration time

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Object Constraints

• In an incoherent imaging system, the object is also real and positive.
• The object is not band-limited and can be reconstructed on a pixel-by-pixel basis –

leads to super-resolution (recovery of power beyond spatial frequency cut-off).

• Limit resolution (and pixel-by-pixel variation) by applying a smoothing operator in the

reconstruction.

• Parametric information about the object structure can be used (Model Fitting):
• Multiple point source
• Planetary type-object (elliptical uniform disk)

( )

v v

m f γ ∗ =

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Local Gradient across the object defines the object texture (Generalized Gauss-Markov Random Field Model), i.e. | fi – fj | p where p is the shape parameter.

Object Constraints

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GGMRF example

truth raw

• ver under

Object Constraints

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Object Prior Information

• Planetary/hard-edged objects (avoids ringing)

Use of the finite-difference gradients Δf(r) to generate an extra error term which preserves hard edges in f(r). α & β are adjustable parameters.

( ) ( )

∑

              Δ + − Δ =

r

r f r f E β β α 1 ln

FD

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Adaptive Optics Imaging – WDS 00310+2839

• 12 arcsecond binary observed at Lick
• LGS – between components
• tip-tilt - fainter component

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Adaptive Optics Imaging – WDS 00310+2839

Using single component for deconvolution

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Adaptive Optics Imaging – WDS 00310+2839

Δm = 1.68 ± 0.02 a = 0.875 ± 0.001 arcseconds

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Hokupa’a Galactic Center Imaging

Crowded Stellar Field with partial compensation Difficult to do photometry and astrometry because of

• verlapping PSFs
• Field Confusion

Need to identify the sources for standard data- reduction programs.

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Observed GC Field

Gemini /Hokupa’a infrared (K′ with texp = 30s) observations of a sub-field near the Galactic Center. 4 separate exposures Note the density of stars in the field. FOV = 4.6 arcseconds Reduced with idac & StarFinder

StarFinder is a semi-analytic program in IDL which reconstructs AO PSF and synthetic fields

• f very crowded images based on relative

intensity and superposition of a few bright stars arbitrarily selected. It extracts the PSF numerically from the crowded field and then fits this PSF to solve for the star’s position and intensity.

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Gemini Imaging of the Galactic Center - Deconvolution Initial Estimates:

Object – 4 frames co-added PSF – K' 20 sec reference (FWHM = 0.2")

4.8 arcsecond subfield

256 x 256 pixels

(This is a typical start for this algorithm)

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Gemini Imaging of the Galactic Center - Deconvolution

Note residual PSF halo 4 frame average for each of the sub-fields. idac reductions. FWHM = 0.07"

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Gemini Imaging of the Galactic Center – PSF Recovery

Frame PSF recovered by isolating individual star from f(r) and convolving with recovered PSFs, h(r).

h f g ˆ ˆ ˆ

PSF PSF

∗ =

PSF

ˆ g

PSF

ˆ f h ˆ ∗ =

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• Data Reduction Outline
• 1. Blind Deconvolution to obtain target & PSF
• 2. Estimate PSF from isolated star and h(r)
• 3. Fixed deconvolution using estimated PSF
• 4. Blind Deconvolution to relax PSF estimates

Gemini Imaging of the Galactic Center

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Average observation initial idac result fixed PSF result

Gemini Imaging of the Galactic Center Object Recovery

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FWHM

Compensated – 0.20 arcsec Initial - 0.07 arcsec Final - 0.05 arcsec Diffraction-limit _ = 0.06 arcsec

Gemini Imaging of the Galactic Center Image Sharpening

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Observed GC Field Reconstructions

The BD reconstruction solves for the common object from all four observed frames.

Reconstructed star field distributions from StarFinder as applied to the four separate observations. StarFinder is a

photometric fitting packages which solves for a numerical PSF.

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Observed GC Field Reconstructions

• The fainter the point source, the

broader it is.

• Magnitude measurement depends upon

measuring area and not peak.

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Comparison of Photometry and for the 55 common stars in the 4 frame StarFinder and IDAC

• reductions. There is close agreement between the two up to 3.5 magnitudes. Then there is a

trend for the IDAC magnitudes to be fainter than the StarFinder ones. This can be explained by the choice of the aperture size used for the photometry due to the increasing size of the fainter sources. Even so, the rms difference between them is still ≈ 0.25 magnitudes. A more sophisticated photometric fitting algorithm than imexamine is therefore suggested.

Common Stars

Observed GC Field - Photometry

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Comparison of Astrometry and for the 55 common stars in the 4 frame StarFinder and IDAC reductions. The x and y differences are shown by the appropriate

• symbols. The dispersion of ≈ 10-14 mas is small, less than a pixel, and a factor of

four less than the size of the diffraction spot.

Observed GC Field - Astrometry

Common Stars

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Reconstructed PSFs for the four frames using IDAC (top) and StarFinder (bottom). The PSF cores (green and white) are essentially identical with the StarFinder PSFs generally having larger wings.

Blind Deconvolution StarFinder

Observed GC Field – PSF Reconstructions

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Simulated GC Field Comparisons

Comparison of aperture photometry from blind deconvolution to true magnitudes for the simulated GC field. Comparison of aperture photometry from blind deconvolution to StarFinder analysis for the simulated GC field.

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Reconstructed PSFs for the four frames using IDAC (top) and StarFinder (bottom). The PSF cores (green and white) are essentially identical with the StarFinder PSFs generally having larger wings.

Observed GC Field – PSF Reconstructions

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IRS 5 IRS 10 IRS 1W IRS 21 IRS 1W IRS 21 IRS 10

• Point sources show

strong uncompensated halo contribution.

Extended Sources near the Galactic Center

• Bow shock structure

is clearly seen in the deconvolutions.

[Data from Angelle Tanner, UCLA]

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Adaptive Optics Imaging – NGC6240

Deconvolved using a PSF estimated from the bright core and a separate PSF star.

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Adaptive Optics Solar Imaging

Low-Order AO System

• Lack of PSF information.
• Sunspot and granulation

features show improved contrast, enhancing detail showing magnetic field structure

[Data from Thomas Rimmele, NSO-SP]

AO Deconvolved

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Limitations of Blind Deconvolution

• Both sets of images are significantly improved by

deconvolution.

• Enhanced contrast
• PSFs are not significantly modified from original

estimate.

• Not enough PSF diversity from frame-to-frame. AO stabilizes the

PSF limiting diversity.

• sensitive to original PSF estimate
• further constraints needed (especially for the PSF).
• MISTRAL – Mugnier et al.
• uses AO information

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Why is deconvolution important? This is why …

Keck Imaging of Io

(Data obtained by D. LeMignant & F. Marchis et al.)

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Keck Imaging of Io

Why is deconvolution important? This is why …

(Data obtained by D. LeMignant & F. Marchis et al.)

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Io in Eclipse

Two Different BD Algorithms Keck observations to identify hot-spots. K-Band 19 with IDAC 17 with MISTRAL L-Band 23 with IDAC 12 with MISTRAL

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Artificial Satellite Imaging

256 frames per apparition

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• Blind/Myopic Deconvolution is well suited to AO imaging where the

PSFs are not well known.

• AO PSFs are stable which reduces the effectiveness of MFBD
• Incorporate as many physical constraints about the imaging process as

possible.

• Object Constraint
• PSF Constraint
• Assumption of isoplanatism is assumed, how to incorporate

anisoplanatism for wide field imaging?

Summary

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S.M. Jefferies & J.C. Christou, “Restoration of astronomical images by iterative blind deconvolution”, Astrophys. J., 415, 862-874, 1993.

• E. Thiébaut & J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution”, J. Opt. Soc.

Am., A, 12, 485-492, 1995. J.-M. Conan, L.M. Mugnier, T. Fusco,, V. Micheau & G. Rousset, “Myopic deconvolution of adaptive optics images by use of object and point-spread-function power spectra”, App. Optics, 37, 4614-4622, 1998. B.D. Jeffs & J.C. Christou, “Blind Baysian Restoration of Adaptive Optics images using generalized Gaussian Markov random field models”, Adaptive Optical System Technologies, D. Bonacinni & R.K. Tyson, Ed., Proc. SPIE, 3353, 1998. E.K. Hege, J.C. Christou, S.M. Jefferies & M. Chesalka, “Technique for combining interferometric images”, J. Opt.

• Soc. Am. A, 16, 1745-1750, 1999.
• T. Fusco, J.-P. Véran, J.-M. Conan, & L.M. Mugnier, ”Myopic deconvolution method for adaptive optics images of

stellar fields”, Astron. Astrophys. Suppl. Ser., 134, 193-200, 1999. J.C. Christou, D. Bonaccini, N. Ageorges, & F. Marchis, “Myopic Deconvolution of Adaptive Optics Images”, ESO Messenger, 1999.

• T. Fusco, J.-M. Conan, L.M. Mugnier, V. Micheau, & G. Rousset, “Characterization of adaptive optics point spread

function for anisoplanatic imaging. Application to stellar field deconvolution.”, Astron. Astrophys. Suppl. Ser., 142, 149-156, 2000.

• E. Diolaiti, O. Bendinelli, D. Bonaccini, L. Close, D. Currie, & G. Parmeggiani, “Analysis of isoplanatic high

resolution stellar fields by the StarFinder code”, Astron. Astrophys. Suppl. Ser., 147, 335-346 , 2000. S.M. Jefferies, M. Lloyd-Hart, E.K. Hege & J. Georges, “Sensing wave-front amplitude and phase with phase diversity”, Appl. Optics, 41, 2095-2102, 2002.

References