Blind/Myopic Deconvolution Julian Christou UCSC CfAO PSF Reconstruction Workshop May 2004
Adaptive Optics Imaging • Quality of compensation depends upon: – Wavefront sensor – Signal strength & signal stability – Speckle noise - d / r 0 – Duty cycle - t / t 0 – Sensing & observing - _ – Wavefront reconstructor & geometry – Object extent – Anisoplanatism (off-axis) Victoria 10th May 2004 PSF Reconstruction Meeting 2
Adaptive Optics Imaging Adaptive Optics systems do NOT produce perfect images (poor compensation) Seeing disc Core Artifacts Halo Binary Star components Without AO With AO Victoria 10th May 2004 PSF Reconstruction Meeting 3
Adaptive Optics Imaging Simulated AO imaging of a Galaxy with different Strehl ratios 11% 45% 86% Strehl Ratio Illustrates the need of knowing the PSF Deconvolution removes the effect of the imperfect PSF and replaces it with a perfect PSF Victoria 10th May 2004 PSF Reconstruction Meeting 4
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 Victoria 10th May 2004 PSF Reconstruction Meeting 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 r 0 & t 0 ). - 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 Victoria 10th May 2004 PSF Reconstruction Meeting 6
The Imaging Equation Shift invariant imaging equation (Image Domain) g(r) = f(r) * h(r) + n(r) (Fourier Domain) G(f) = F(f) • H(f) + N(f) g(r) – Measurement h(r) – Point Spread Function (PSF) f(r) – Target n(r) – Contamination - Noise Victoria 10th May 2004 PSF Reconstruction Meeting 7
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 Victoria 10th May 2004 PSF Reconstruction Meeting 8
Blind Deconvolution Solve for both object & PSF g(r) = f(r) * h(r) + n(r) contamination Measurement unknown unknown or poorly object irradiance known PSF Single measurement: Under – determined - 1 measurement, 2 unknowns Never really “blind” Victoria 10th May 2004 PSF Reconstruction Meeting 9
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) = 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. Victoria 10th May 2004 PSF Reconstruction Meeting 10
Multiple Frame Constraints Multiple Observations of a common object g ( r ) f ( r ) h ( r ) = ∗ 1 1 g ( r ) f ( r ) h ( r ) = ∗ 2 2 M M M g ( r ) f ( r ) h ( r ) = ∗ n n • 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. Victoria 10th May 2004 PSF Reconstruction Meeting 11
An MFBD Algorithm • Uses a Conjugate Gradient Error Metric Minimization scheme - Least squares fit. • Error Metric – minimizing the residuals (convolution error): ~ ~ 2 ~ ( ) 2 2 E g g g f h r ∑ ∑ ∑ = − = − ∗ = ik ik ik i ik ik ik ik ik • Alternative error metric – minimizing the residual autocorrelation: Autocorrelation of residuals 2 E r r ∑ = ⊗ ik ik Reduces correlation in the residuals ik (minimizes “print through”) So not sum over the 0 location. Victoria 10th May 2004 PSF Reconstruction Meeting 12
An MFBD Algorithm • Object non-negativity ~ 2 f = α Reparameterize the object as the square of another variable HARD i i or penalize the object against negativity. ~ 2 E f ∑ = Obj i SOFT ~ u f 0 ∈ < i • PSF Constraints (when pupil is not known) - Non-negativity ~ 2 ~ E h ∑ 2 h = Reparameterize - or penalize – = β PSF i , k i , k i , k ~ u h 0 ∈ < i , k ~ 2 - Band-limit E H ∑ = bl k , u k , u u > c Victoria 10th May 2004 PSF Reconstruction Meeting 13
PSF Constraints Use as much prior knowledge of the PSF as possible . • Transfer function is band-limited MTF f c = D / λ MTF Normalized Spatial Frequency • PSF is positive and real Victoria 10th May 2004 PSF Reconstruction Meeting 14
An MFBD Algorithm • PSF Constraints (Using the Pupil) - Parameterize the PSF as the power spectrum of the complex wavefront at the pupil, i.e. ~ 2 π iv ~ ~ ~ ∗ h a a a W exp j where = ∑ = − ϕ ik ik ik ik v vk N v PSF Pupil Victoria 10th May 2004 PSF Reconstruction Meeting 15
PSF Constraints • PSF Constraints (Using the Pupil) - Modally - express the phases as either a set of Zernike modes of order M M q Z ∑ ϕ = vk m vk m 1 = 2 v ( ) exp - or zonally as where which η = − ϕ = φ ∗ η vk vk 2 σ 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. Victoria 10th May 2004 PSF Reconstruction Meeting 16
PSF Constraints • Myopic Deconvolution (using known PSF information) - For MFBD penalize PSFs for departure from a “typical” PSF or model (good for multi-frame measurements) ~ 2 SAA SAA E h h ∑ = − SAA ik ik ik - Penalize PSF on power spectral density (PSD) ~ ~ 2 H H − i i E ∑ = PSD PSD i H where the PSD is based upon the atmospheric conditions and AO correction. Victoria 10th May 2004 PSF Reconstruction Meeting 17
PSF Constraints • Myopic Deconvolution ~ ~ 2 H H − i using the reconstructed PSF i E ∑ = PSD ( ) PSD f i H residual phase structure function Perfect optics transfer function ~ 1 − ( ) H exp D T f - where = f ( ) φ λ 2 Speckle transfer function ~ 2 τ ( ) 2 ( ) PSD f s STF f H - and = σ = − H turb T i r Integration time - where 0 . 36 0 τ = s v Victoria 10th May 2004 PSF Reconstruction Meeting 18
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. ( ) f m = ∗ γ v v • Parametric information about the object structure can be used (Model Fitting): - Multiple point source - Planetary type-object (elliptical uniform disk) Victoria 10th May 2004 PSF Reconstruction Meeting 19
Object Constraints Local Gradient across the object defines the object texture (Generalized Gauss-Markov Random Field Model), i.e. | f i – f j | p where p is the shape parameter. Victoria 10th May 2004 PSF Reconstruction Meeting 20
Object Constraints GGMRF example truth raw over under Victoria 10th May 2004 PSF Reconstruction Meeting 21
Object Prior Information • Planetary/hard-edged objects (avoids ringing) Use of the finite-difference gradients Δ f ( r ) to generate an extra error term which preserves hard edges in f (r). α & β are adjustable parameters. ( ) ( ) f r f r Δ Δ E ln 1 ∑ = α − + FD β β r Victoria 10th May 2004 PSF Reconstruction Meeting 22
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