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**Outline**• AO Imaging • Constrained Blind Deconvolution • Algorithm • Application - Quantitative measurements • Future Directions Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**Core**Artifacts • Halo Binary Star components Adaptive Optics Imaging Adaptive Optics systems do NOT produce perfect images (poor compensation) With AO Seeing disc Without AO Mathematical Challenges in Astronomical Imaging**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) Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**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 Mathematical Challenges in Astronomical Imaging**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 Mathematical Challenges in Astronomical Imaging**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 Mathematical Challenges in Astronomical Imaging**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” Mathematical Challenges in Astronomical Imaging**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, N+1 unknowns. Mathematical Challenges in Astronomical Imaging**Multiple Frame Constraints**Multiple Observations of a common object • 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. Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**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 Mathematical Challenges in Astronomical Imaging**MTF**Normalized Spatial Frequency 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/ Mathematical Challenges in Astronomical Imaging**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 Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**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) Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**truth raw**over under Object Constraints GGMRF example Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**idac – iterative deconvolution algorithm in c**• SNR Regularization (Fourier Domain) Minimize in the Fourier domain rather than the image domain, i.e. where Mathematical Challenges in Astronomical Imaging**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) Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**idac – iterative deconvolution algorithm in c**• Conjugate Gradient Error Metric Minimization • Convolution Error • Band-limit Error • Non-negativity • PSF Constraint (for multiple images) Mathematical Challenges in Astronomical Imaging**idac Software Page**http://cfao.ucolick.org/software/idac/ http://bach.as.arizona.edu/~hege/docs/docs/IDAC27/idac_package.tar.gz Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical 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. Mathematical Challenges in Astronomical Imaging**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 Mathematical Challenges in Astronomical Imaging**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 StarFinderis 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. Mathematical Challenges in Astronomical Imaging**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) Mathematical Challenges in Astronomical Imaging**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" Mathematical Challenges in Astronomical Imaging**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). Mathematical Challenges in Astronomical Imaging**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 Mathematical Challenges in Astronomical Imaging**Gemini Imaging of the Galactic Center**Object Recovery Average observation initial idac result fixed PSF result Mathematical Challenges in Astronomical Imaging**Gemini Imaging of the Galactic Center**Image Sharpening FWHM Compensated – 0.20 arcsec Initial - 0.07 arcsec Final - 0.05 arcsec Diffraction-limit α = 0.06 arcsec Mathematical Challenges in Astronomical Imaging**The BD reconstruction solves for the common object from all**four observed frames. 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. Mathematical Challenges in Astronomical Imaging**Observed GC Field Reconstructions**• The fainter the point source, the broader it is. • Magnitude measurement depends upon measuring area and not peak. Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**Observed GC Field – PSF Reconstructions**Blind Deconvolution 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. StarFinder Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**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. Mathematical Challenges in Astronomical Imaging**IRS 10**IRS 5 IRS 10 IRS 1W IRS 21 IRS 1W IRS 21 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] Mathematical Challenges in Astronomical Imaging**AO**Deconvolved 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] Mathematical Challenges in Astronomical Imaging**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.) Mathematical Challenges in Astronomical Imaging**Keck Imaging of Io**Why is deconvolution important? This is why … (Data obtained by D. LeMignant & F. Marchis et al.) Mathematical Challenges in Astronomical Imaging**Keck Imaging of Io**Why is deconvolution important? This is why … (Data obtained by D. LeMignant & F. Marchis et al.) Mathematical Challenges in Astronomical Imaging**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 Mathematical Challenges in Astronomical Imaging