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1
Outline
  • AO Imaging
  • Constrained Blind Deconvolution
  • Algorithm
  • Application
  • - Quantitative measurements
  • Future Directions

2
References
  • 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.

3
Adaptive Optics Imaging
Adaptive Optics systems do NOT produce perfect
images (poor compensation)
Seeing disc
Without AO
  • With AO

4
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)

5
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.

6
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) of 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

7
The Imaging Equation
  • Shift invariant imaging equation
  • (Image Domain)
  • (Fourier Domain)

g(r) f(r) h(r) n(r)
G(f) F(f) H(f) N(f)
g(r) Measurement h(r) Point Spread Function
(PSF) f(r) Target n(r) Contamination - Noise
8
Deconvolution
  • 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

9
Blind Deconvolution
Solve for both object PSF
  • g(r) f(r) h(r) n(r)

contamination
Measurement
unknown object irradiance
unknown or poorly known PSF
Single measurement Under determined - 1
measurement, 2 unknowns Never really blind
10
Blind Deconvolution Physical Constraints
  • How to minimize the search space for a solution?
  • 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) ?(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, N1 unknowns.

11
Multiple Frame Constraints
Multiple Observations of a common object
  • Reduces the ratio of unknown to measurements from
    21 to n1n
  • The greater the diversity of h(r),the easier the
    separation of the PSF and object.

12
An MFBD Algorithm
  • 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.

13
An MFBD Algorithm
  • Object non-negativity
  • Reparameterize the object as the square of
    another variable HARD
  • or penalize the object against negativity.
  • SOFT
  • PSF Constraints (when pupil is not known)
  • - Non-negativity
  • Reparameterize - or penalize
  • - Band-limit

14
PSF Constraints
  • Use as much prior knowledge of the PSF as
    possible.
  • Transfer function is band-limited
  • PSF is positive and real

MTF
fc D/?
15
An MFBD Algorithm
  • PSF Constraints (Using the Pupil)
  • - Parameterize the PSF as the power spectrum of
    the complex wavefront at the pupil, i.e.
  • where

PSF
Pupil
16
PSF Constraints
  • 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.

17
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)

18
Object Constraints
  • 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.

19
Object Constraints
  • GGMRF
  • example

20
Object Prior Information
  • Planetary/hard-edged objects (avoids ringing)
  • Use of the finite-difference gradients ?f(r) to
    generate an extra error term which preserves hard
    edges in f(r).
  • ? ? are adjustable parameters.

21
An MFBD Algorithm
  • Myopic Deconvolution (using known PSF
    information)
  • - 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.

22
An MFBD Algorithm
  • Further Constraints
  • Truncated Iterations (Tikenhov)
  • Support Constraints
  • In many cases, a limited field is available and
    it is important to compute the error metric only
    over a specific region M of the observation
    space, i.e.

23
idac iterative deconvolution algorithm in c
  • SNR Regularization (Fourier Domain)
  • Minimize in the Fourier domain rather than the
    image domain, i.e.
  • where

24
An MFBD Algorithm
  • Forward Modeling of Imaging Process
  • Compute Error Metric based on Measurement
  • where data is not pre-processed

Noise terms
Measurement
Signal
Background (sky dark)
Gain (flatfield)
25
idac an MFBD Algorithm
  • idac is a generic physically constrained
    blind-deconvolution algorithm written in C and is
    platform independent on UNIX systems.
  • Maximum-likelihood with Gaussian statistics
    error metric minimization using a conjugate
    gradient algorithm.
  • It can handle single or multiple observations of
    the same source.
  • It allows masking of the observation (convolution
    image) permitting the saturated regions to make
    no contribution to the final results for both the
    target and the PSF.
  • It has the option to fit a the strength of a bias
    term in the image (skydark) asik
  • The algorithm can be run as with either a fixed
    PSF or a fixed object or both unknown.
  • idac was written by Keith Hege Matt Chesalka
    (as part of a collaborative effort with Stuart
    Jefferies and Julian Christou) and is made
    available via Steward Observatory and the CfAO.

26
idac iterative deconvolution algorithm in c
  • Conjugate Gradient Error Metric Minimization
  • Convolution Error
  • Band-limit Error
  • Non-negativity
  • PSF Constraint (for multiple images)

27
idac Software Page
http//cfao.ucolick.org/software/idac/ http//bach
.as.arizona.edu/hege/docs/docs/IDAC27/idac_packag
e.tar.gz
28
Application of idac
  • Investigation of relative photometry and
    astrometry in deconvolved image.
  • - Gemini/Hokupaa Galactic Center data
  • - PSF reconstruction
  • Application to various astronomical AO images.
  • - Resolved Galactic Center sources (bow-shocks)
  • - Solar imaging
  • - Solar system object (Io) comparison with
    Mistral
  • Artificial satellite imaging
  • Non-astronomical AO imaging.

29
Application of idac
  • How well does the deconvolved image retain the
    photometry and astrometry of the data?
  • - It has been suggested that it is better to
    measure the photometry especially from the raw
    data.
  • - Investigated using dense crowded field data
    from Gemini/Hokupaa commisioning data.
  • - Comparison of Astrometry and Photometry from
    these data to that measured directly via
    StarFinder.
  • - Comparison of both techniques to simulated
    data.

30
Hokupaa Galactic Center Imaging
Crowded Stellar Field with partial
compensation Difficult to do photometry and
astrometry because of overlapping PSFs - Field
Confusion Need to identify the sources for
standard data-reduction programs.
See Poster
31
Observed GC Field
  • Gemini /Hokupaa 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 of
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 stars position and
intensity.
32
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)

33
Gemini Imaging of the Galactic Center -
Deconvolution
4 frame average for each of the sub-fields.
idac reductions. FWHM 0.07"
  • Note residual PSF halo

34
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).

35
Gemini Imaging of the Galactic Center
  • Data Reduction Outline
  • Blind Deconvolution to obtain target PSF
  • Estimate PSF from isolated star and h(r)
  • Fixed deconvolution using estimated PSF
  • Blind Deconvolution to relax PSF estimates

36
Gemini Imaging of the Galactic Center Object
Recovery
  • Average observation
  • initial idac
    result

  • fixed PSF result

37
Gemini Imaging of the Galactic Center Image
Sharpening
  • FWHM
  • Compensated 0.20 arcsec
  • Initial - 0.07 arcsec
  • Final - 0.05 arcsec
  • Diffraction-limit
  • a 0.06 arcsec

38
Observed GC Field Reconstructions
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.
39
Observed GC Field Reconstructions
  • The fainter the point source, the broader it
    is.
  • Magnitude measurement depends upon measuring
    area and not peak.

40
Observed GC Field - Photometry
Common Stars
  • 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.

41
Observed GC Field - Astrometry
Common Stars
  • 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.

42
Observed GC Field PSF Reconstructions
Blind Deconvolution
StarFinder
  • 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.

43
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.
44
Observed GC Field PSF Reconstructions
  • 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.

45
Extended Sources near the Galactic Center
  • Point sources show strong uncompensated halo
    contribution.
  • Bow shock structure is clearly seen in the
    deconvolutions.

Data from Angelle Tanner, UCLA
46
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
47
ADONIS AO Imaging of Io
  • 3.8 ?m
  • Two distinct hemispheres
  • 11 frames/hemisphere
  • Co-added initial object
  • PSF reference as initial PSF
  • Surface structure visible showing volcanoes.
  • (Marchis et. al., Icarus, 148, 384-396, 2000.)

48
Keck Imaging of Io
  • Why is deconvolution important? This is why

(Data obtained by D. LeMignant F. Marchis et
al.)
49
Keck Imaging of Io
Why is deconvolution important? This is why
(Data obtained by D. LeMignant F. Marchis et
al.)
50
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

51
Artificial Satellite Imaging
256 frames per apparition
52
Summary
  • Blind/Myopic Deconvolution is well suited to AO
    imaging where the PSFs are not well known.
  • Incorporate as many physical constraints about
    the imaging process as possible.
  • Building a specific algorithm to match the
    application is advantageous.
  • This algorithm (idac) suffers from the same
    problem as others in that the PSF get wider as
    the dynamic range increases ( a problem of
    half-wave rectification of the noise with hard
    positivity constraint?)
  • Aperture photometry yields good relative
    photometry (lt0.1m for ?m lt 4 and ? 0.2m for 4.0 lt
    ?m lt 7.0.
  • A general algorithm has limitations.Can one build
    a modular algorithm to incorporate as much prior
    information as possible for the data?
  • Deconvolution algorithms are not necessarily
    user-friendly. How can we do this?
  • Assumption of isoplanatism is assumed, how to
    incorporate anisoplanatism for wide field imaging?

53
References
  • 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.
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