Outline

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.