Phase retrieval of large-scale time-varying aberrations using a non-linear Kalman filtering framework

Pieter Piscaer; Oleg Soloviev; Michel Verhaegen

doi:10.1364/JOSAA.405712

Journal of the Optical Society of America A
Vol. 38,
Issue 1,
pp. 25-35
(2021)
•https://doi.org/10.1364/JOSAA.405712

Phase retrieval of large-scale time-varying aberrations using a non-linear Kalman filtering framework

Pieter Piscaer, Oleg Soloviev, and Michel Verhaegen

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Pieter Piscaer, Oleg Soloviev, and Michel Verhaegen, "Phase retrieval of large-scale time-varying aberrations using a non-linear Kalman filtering framework," J. Opt. Soc. Am. A 38, 25-35 (2021)

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Abstract

This paper presents a computationally efficient framework in which a single focal-plane image is used to obtain a high-resolution reconstruction of dynamic aberrations. Assuming small-phase aberrations, a non-linear Kalman filter implementation is developed whose computational complexity scales close to linearly with the number of pixels of the focal-plane camera. The performance of the method is tested in a simulation of an adaptive optics system, where the small-phase assumption is enforced by considering a closed-loop system that uses a low-resolution wavefront sensor to control a deformable mirror. The results confirm the computational efficiency of the algorithm and show a large robustness against noise and model uncertainties.

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$c_{X}$	Average number of non-zero elements per row/column of any sparse banded matrix $X$
$m$	Number of actuators inside the aperture
$\bar{n}, n$	Size of grid in pupil plane ( $\bar{n} \times \bar{n}$ ) and number of pixels inside aperture, respectively
$\bar{p}, p$	Size of grid in the image plane ( $\bar{p} \times \bar{p}$ ) and $p = {\bar{p}}^{2}$
$\bar{q}$	Size of lenslet grid of the WFS ( $\bar{q} \times \bar{q}$ )
$ϕ (k)$	Aberrated residual wavefront
$a$	Pupil plane amplitude
$u (k)$	DM control signal
$v (k), R (k)$	Measurement noise and its covariance matrix
$y (k)$	Point spread function (PSF)
$A$ , $Q$	Turbulence dynamics model [see Eq. (10)]
$J_{ϕ}, c_{ϕ}$	Taylor expansion of $y (k)$ [see Eq. (20)]
$F$ , $F_{v e c}$	Zero-padded 2D DFT and $F_{v e c} {\cdot} := v e c (F {\cdot})$
$J_{ϕ}, M, B$	Efficient operators [see Eqs. (21), (35), (36)]
$N (μ, P)$	Gaussian random process (mean $μ$ , covariance $P$ )
$S_{X}$	Set of all matrices that satisfy a chosen sparsity pattern for matrix $X$

Step in Algorithm	Computational Complexity
Compute $ϕ (k \| k)$ by solving Eq. (34) via CG method	$O (n (c_{p^{- 1}} + \log n) {L L}_{C G})$
Explicitly computing $p_{c}$ rows of the Jacobian	$O (n p_{c})$
Sparse low-rank approximation of $P^{- 1} (k \| k)$ in Eq. (37)	$O (n p_{c} c_{P^{- 1}})$
Compute $\hat{ϕ} (k + 1 \| k)$ via Eq. (31)	$O (n c_{A} + m c_{B})$
Computing sparse inverse approximation via solving Eq. (39)	$O (n (c_{Γ}^{3} + c_{M_{2}} c_{Γ}^{2}))$
Sparse approximation of $P^{- 1} (k + 1 \| k)$ in Eq. (40)	$O (n c_{M_{1}}^{2} c_{P^{- 1}})$

Parameter	Standard Value
Fried parameter $r_{0}$ [m]	0.2
Wind speed $v$ , layer 1 [m/s]	12
Wind speed $v$ , layer 2 [m/s]	16
Source magnitude $β_{m}$	8
read-out noise $σ_{r}$ [photons]	2
Telescope diameter $D$ [m]	1
WFS lenslet grid size $\bar{q}$	6
DM grid size $\bar{q} + 1$	7
Wavefront grid size $\bar{n}$	30
Sample frequency $f_{s}$ [Hz]	500
Outer scale $L_{0}$ [m]	15

$c_{X}$	Average number of non-zero elements per row/column of any sparse banded matrix $X$
$m$	Number of actuators inside the aperture
$\bar{n}, n$	Size of grid in pupil plane ( $\bar{n} \times \bar{n}$ ) and number of pixels inside aperture, respectively
$\bar{p}, p$	Size of grid in the image plane ( $\bar{p} \times \bar{p}$ ) and $p = {\bar{p}}^{2}$
$\bar{q}$	Size of lenslet grid of the WFS ( $\bar{q} \times \bar{q}$ )
$ϕ (k)$	Aberrated residual wavefront
$a$	Pupil plane amplitude
$u (k)$	DM control signal
$v (k), R (k)$	Measurement noise and its covariance matrix
$y (k)$	Point spread function (PSF)
$A$ , $Q$	Turbulence dynamics model [see Eq. (10)]
$J_{ϕ}, c_{ϕ}$	Taylor expansion of $y (k)$ [see Eq. (20)]
$F$ , $F_{v e c}$	Zero-padded 2D DFT and $F_{v e c} {\cdot} := v e c (F {\cdot})$
$J_{ϕ}, M, B$	Efficient operators [see Eqs. (21), (35), (36)]
$N (μ, P)$	Gaussian random process (mean $μ$ , covariance $P$ )
$S_{X}$	Set of all matrices that satisfy a chosen sparsity pattern for matrix $X$

Step in Algorithm	Computational Complexity
Compute $ϕ (k \| k)$ by solving Eq. (34) via CG method	$O (n (c_{p^{- 1}} + \log n) {L L}_{C G})$
Explicitly computing $p_{c}$ rows of the Jacobian	$O (n p_{c})$
Sparse low-rank approximation of $P^{- 1} (k \| k)$ in Eq. (37)	$O (n p_{c} c_{P^{- 1}})$
Compute $\hat{ϕ} (k + 1 \| k)$ via Eq. (31)	$O (n c_{A} + m c_{B})$
Computing sparse inverse approximation via solving Eq. (39)	$O (n (c_{Γ}^{3} + c_{M_{2}} c_{Γ}^{2}))$
Sparse approximation of $P^{- 1} (k + 1 \| k)$ in Eq. (40)	$O (n c_{M_{1}}^{2} c_{P^{- 1}})$

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Equations (42)

Journal of the Optical Society of America A