outline

IPAM 2004 January 28 Mathematical Challenges in Astronomical Imaging 1

Outline

• AO Imaging

• Constrained Blind Deconvolution

• Algorithm

• Application

- Quantitative measurements

• Future Directions


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


Adaptive Optics Imaging

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

Without AO With AO

Seeing disc Halo Artifacts

Binary Starcomponents

Core


Adaptive Optics Imaging

• Quality of compensation depends upon:– Wavefront sensor – Signal strength & signal stability– Speckle noise - d / r0

– Duty cycle - t / t0

– Sensing & observing - λ – Wavefront reconstructor & geometry– Object extent– Anisoplanatism (off-axis)


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.


Why Deconvolution and PSF Calibration?

• Better looking image

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

• PSF calibrationRemoves 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 resolutionIn specific cases depending upon algorithms and

SNR


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)


• 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


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

Blind Deconvolution

Measurement

unknown object irradiance unknown or poorly

known PSF

contamination

Solve for both object & PSF

Single measurement:Under – determined - 1 measurement, 2 unknownsNever really “blind”


• 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, N+1 unknowns.

Blind Deconvolution – Physical Constraints

• How to minimize the search space for a solution?


Multiple Observations of a common object

)()()(

)()()(

)()()(

22

11

rhrfrg

rhrfrg

rhrfrg

nn

• 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 of the PSF and object.

Multiple Frame Constraints


• 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.

222

~~~ ik

ikik

ikiikik

ikik rhfgggE

An MFBD Algorithm

2 ik

ikik rrE


• 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

0

~

2

Obj

~

ifuifE

2~iif

2,,

~kikih

An MFBD Algorithm

0

~

2

,PSF

,

~

kihukihE

cuuk

ukHE,

2

,bl

~


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

MT

F fc = D/

MTF


• 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

ikikik aah ~~~

vk

vvik N

ivjWa 2

exp~

Pupil

PSF


• 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

mvkmvk Zq

1

vkvk

2

2exp

v


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)

vv mf


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


GGMRF

example

truth raw

over under

Object Constraints


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.

r

rfrfE

1lnFD


• 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.

An MFBD Algorithm

2SAASAA

SAA

~ ik

ikik hhE

i H

iiHH

EPSD

~ 2

PSD


• Further Constraints

– Truncated Iterations (Tikenhov)

– Support ConstraintsIn 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.

ik ik

nikkiikiikik

ikiik knnhfghfgE 222

,

2

conv ~~

ˆ ~~

An MFBD Algorithm

ikik

ikik MggE2~


• SNR Regularization (Fourier Domain)

Minimize in the Fourier domain rather than the image domain, i.e.

where

idac – iterative deconvolution algorithm in c

uk uukuk GGE

2

convˆ

2

22

u

uu

uG

NG


• Forward Modeling of Imaging Process:

• Compute Error Metric based on Measurement

where data is not pre-processed

An MFBD Algorithm

ikikikikik nsGgg ˆ

Measurement

Signal

Gain (flatfield)Background (sky + dark)

Noise terms

2~ˆˆ

ikikik ggE


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 (sky+dark) – 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.


• Conjugate Gradient Error Metric Minimization

– Convolution Error

– Band-limit Error

– Non-negativity

– PSF Constraint (for multiple images)

idac – iterative deconvolution algorithm in c

SAAblconv EEEE

ik ikikiikik ashfgmE

2

convˆˆ

2

blˆ

k uu uk

cHE

2 saasaaSAA

ˆ i ii hhE

22 ˆ and ˆikikii bhaf


idac Software Page

http://cfao.ucolick.org/software/idac/http://bach.as.arizona.edu/~hege/docs/docs/IDAC27/idac_package.tar.gz


Application of idac

• Investigation of relative photometry and astrometry in deconvolved image.- Gemini/Hokupa’a 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.


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/Hokupa’a commisioning data.

- Comparison of Astrometry and Photometry from these data to that measured directly via StarFinder.

- Comparison of both techniques to simulated data.


Hokupa’a 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


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 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 star’s position and intensity.


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)


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"


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).

hfg ˆˆˆ PSFPSF

PSFg

PSFf

h


• 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


Average observation

initial idac result

fixed PSF result

Gemini Imaging of the Galactic CenterObject Recovery


FWHM

Compensated – 0.20 arcsec

Initial - 0.07 arcsec

Final - 0.05 arcsec

Diffraction-limit

α = 0.06 arcsec

Gemini Imaging of the Galactic CenterImage Sharpening


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.


Observed GC Field Reconstructions

• The fainter the point source, the broader it is.

• Magnitude measurement depends upon measuring area and not peak.


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


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


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


Simulated GC Field ComparisonsComparison 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.


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


IRS 5IRS 10

IRS 1W IRS 21IRS 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]


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


= 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.)

ADONIS AO Imaging of Io


Why is deconvolution important? This is why …

Keck Imaging of Io

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


Keck Imaging of Io Why is deconvolution important? This is why …

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


Io in Eclipse

Two Different BD Algorithms

Keck observations to identify hot-spots.

K-Band

19 with IDAC17 with MISTRAL

L-Band

23 with IDAC12 with MISTRAL


Artificial Satellite Imaging

256 frames per apparition


• 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 (<0.1m for m < 4 and 0.2m for 4.0 < m < 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?

Summary


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

outline

Documents

adaptive optics psfs

interferometric images

stellar field deconvolution

iterative blind deconvolution

myopic deconvolution

discmathematical challenges

function psf

spread function