Adaptive optics and atmospheric tomography: An inverse problem in telescope imaging

Transcription

1 Adaptive optics and atmospheric tomography: An inverse problem in telescope imaging Jonatan Lehtonen Bayesian Inversion guest lecture, / 19

2 Atmospheric turbulence Atmospheric turbulence = random temperature fluctuations random index of refraction light travels along random paths 2 / 19

3 Present and future telescopes Very Large Telescope (VLT): 8 m built/operated by European Southern Observatory (ESO) located on the Cerro Paranal mountain in Chile, 2635 metres above sea level 3 / 19

4 Present and future telescopes ESO European Extremely Large Telescope (E-ELT): 100 m, 42 m, 39 m Planned first light: 2018, 2019, 2022, 2024? Cerro Armazones in Chile, 3046 metres above sea level 4 / 19

5 Present and future telescopes ESO Mirror diameter increases effect of turbulence dominates diffraction 4 / 19

6 Possible future telescopes? Randall Munroe 5 / 19

7 Adaptive optics comes into play UCLA Galactic Center Group - W.M. Keck Observatory Laser Team Adaptive optics = real-time deblurring by hardware 6 / 19

8 Main tools: deformable mirror and wavefront sensor A. Tokovinin 7 / 19

9 Main tools: deformable mirror and wavefront sensor ESO Randall Munroe 8 / 19

10 Video time! Yay, a video! 9 / 19

11 Atmospheric tomography ESO 10 / 19

12 Atmospheric tomography for MOAO/MCAO use several guidestars goal: quality in the field of view Aφ = s atmospheric turbulence noisy tomography layers measurements operator φ = (φ j) 3 j=1 Atmospheric tomography: Fitting step: WFS measurements layers layers DM commands 11 / 19

13 Atmospheric tomography for MOAO/MCAO use several guidestars goal: quality in the field of view Aφ = s atmospheric turbulence noisy tomography layers measurements operator φ = (φ j) 3 j=1 Atmospheric tomography: Fitting step: WFS measurements layers layers DM commands 11 / 19

14 Atmospheric tomography for MOAO/MCAO use several guidestars goal: quality in the field of view A is given by projecting the layers to the telescope pupil in the direction of each guide star and summing their contributions Aφ = s atmospheric turbulence noisy tomography layers measurements operator φ = (φ j) 3 j=1 Atmospheric tomography: Fitting step: WFS measurements layers layers DM commands 11 / 19

15 Bayesian solution to the open loop problem The tomography step is extremely ill-posed regularization needed Firstly, the likelihood is given by ( π like (s φ) exp 1 ) C 1 2 E (Aφ s) 2. 2 L 2 The prior (more on its definition later) is given by ( π prior (φ) exp 1 C 1 2 φ 2 φ 2 and the posterior is then of course L 2 ), π post (φ s) π like (s φ)π prior (φ) ( exp 1 C 1 2 E (Aφ s) ) C 1 2 L 2 φ 2 φ 2 L 2 12 / 19

16 Bayesian solution to the open loop problem Due to the extreme time constraints in atmospheric tomography, typically only the Bayesian maximum a posteriori estimate is of interest. Maximizing the posterior distribution is equivalent to the following minimization problem: C φ atm = argmin 1 2 φ E (Aφ s) 2 C L 2 φ φ 2 where C E and C φ describe the covariances of the noise and the prior distribution, respectively. In the fitting step the deformable mirror commands a DM are optimized by a DM = argmin a E (P γ a)(r) (Q γ φ atm )(r) 2 drdγ F OV Above, P γ and Q γ are projections towards direction γ. Ω L 2 13 / 19

17 MAP estimate analytic solution MAP estimate inverse noise covariance inverse turbulence covariance (A C 1 E A + C 1 φ )φ = A C 1 E s atmospheric tomography turbulence layers noisy measurements 14 / 19

18 Atmospheric turbulence model The noise covariance C E is just a diagonal matrix obtained from estimating the wavefront sensor measurement noise. So, the last missing piece is the prior covariance C φ. Phase aberration φ from light propagation through a layer of turbulence: stationary Gaussian stochastic process with zero mean, and covariance given by C φ ( x) = φ(x + x, y + y)φ(x, y). The power spectral density is the Fourier transform of this covariance, and is given by the classical von Kármán model: Ĉ φ (κ) = c 0 ( κ 2 + 1/L 2 0) 11/6. Discretizing the layer of turbulence as a grid of points and assuming unit turbulence strength c 0, we can compute the covariance between each pair of points to obtain a covariance matrix C 15 / 19

19 Example of turbulent layer 16 / 19

20 Atmospheric turbulence model Turbulence layers are assumed to be uncorrelated and zero-mean Gaussian random fields with block diagonal covariance matrix γ 1 C 0 C φ =..., 0 γ n C where the vector γ = (γ 1,..., γ n ) describes the turbulence strength profile (so-called C 2 n-profile). The vector γ is usually estimated by external measurements. However, all current techniques involve difficulties (large uncertainties) that push towards estimation of γ within AO, which is what we ll quickly look at next. 17 / 19

21 Atmospheric profiling The atmospheric profiles at Paranal So-called SLODAR techniques are based on computing correlations within AO data. Our work aims to integrate these methods to the RTC. Formally the problem becomes dependent on the profile C (φ atm, γ atm ) = argmin 1 2 (φ,γ) E (Aφ s) 2 + n 1 C φj L 2 γ j leading to hierarchical Bayesian modelling j=1 L 2 18 / 19

22 Atmospheric profiling Formally the problem becomes dependent on the profile C (φ atm, γ atm ) = argmin 1 2 (φ,γ) E (Aφ s) 2 + L 2 leading to hierarchical Bayesian modelling n j=1 1 γ j C 1 2 φj 2 The natural thing to do would then be to add a hyperprior on the turbulence strengths γ j, to obtain e.g. C (φ atm, γ atm ) = argmin 1 2 (φ,γ) E (Aφ s) 2 n 1 C φj + γ L 2 γ L j L 2 1 j=1 L 2 where the term γ L 1 promotes sparsity in the turbulence profile γ 19 / 19