Fast iterative tomographic wavefront estimation with recursive Toeplitz reconstructor structure for large-scale systems

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Fast iterative tomographic wavefront estimation with

recursive Toeplitz reconstructor structure for large-scale

systems

Yoshito H. Ono, Carlos Correia, Rodolphe Conan, Leonardo Blanco, Benoit

Neichel, Thierry Fusco

To cite this version:

Yoshito H. Ono, Carlos Correia, Rodolphe Conan, Leonardo Blanco, Benoit Neichel, et al.. Fast

iterative tomographic wavefront estimation with recursive Toeplitz reconstructor structure for

large-scale systems. Journal of the Optical Society of America. A Optics, Image Science, and Vision,

Optical Society of America, 2018, 35 (8), pp.1330-1345. �10.1364/JOSAA.35.001330�. �hal-01842351�

Fast Iterative Tomographic Wave-front Estimation with

Recursive Toeplitz Reconstructor Structure for Large

Scale Systems

Y

OSHITO

H. O

NO1,2

, C

ARLOS

C

ORREIA2

, R

ODOLPHE

C

ONAN3

, L

EONARDO

B

LANCO4

, B

ENOIT

N

EICHEL2

,

AND

T

HIERRY

F

USCO2,4

1

_{Subaru Telescope, National Astronomical Observatory of Japan, 650 North A’ohoku Place, Hilo, HI 96720, U.S.A.}

2

_{Aix Marseille Univ, CNRS, LAM, Laboratoire d’Astrophysique de Marseille, Marseille, France}

3

_{GMTO Corporation, 465 N. Halstead Street, Suite 250. Pasadena, CA 91107}

4

_{ONERA, the French Aerospace Laboratory, F-92322 Chatillon, France}

*

_{Corresponding author: [email protected]}

Compiled June 22, 2018

Tomographic wave-front reconstruction is the main computational bottleneck to realize real-time

correc-tion for turbulence-induced wave-front aberracorrec-tions in future laser-assisted tomographic adaptive-optics

(AO) systems for ground-based Giant Segmented Mirror Telescopes (GSMT), because of its

unprece-dented number of degrees of freedom, N, i.e. the number of measurements from wave-front sensors

(WFS). In this paper, we provide an efficient implementation of the minimum-mean-square error (MMSE)

tomographic wave-front reconstruction mainly useful for some classes of AO systems not requiring a

multi-conjugation, such as laser-tomographic AO (LTAO), multi-objcet AO (MOAO) and ground-layer

AO (GLAO) systems, but also applicable to multi-conjugate AO (MCAO) systems. This work expands

that by R. Conan [ProcSPIE, 9148, 91480R (2014)] to the multi-wave-front, tomographic case using natural

and laser guide stars. The new implementation exploits the Toeplitz structure of covariance matrices used

in a MMSE reconstructor, which leads to an overall O

(

N log N

)

real-time complexity compared to O

(

N

2

)

of the original implementation using straight vector-matrix multiplication. We show that the

Toeplitz-based algorithm leads to 60 nm rms wave-front error improvement for the European Extremely Large

Telescope Laser-Tomography AO system over a well-known sparse-based tomographic reconstruction,

but the number of iterations required for suitable performance is still beyond what a real-time system

can accommodate to keep up with the time-varying turbulence.

© 2018 Optical Society of America

OCIS codes: (010.1080) Active or adaptive optics; (100.3190) Inverse problems

http://dx.doi.org/10.1364/ao.XX.XXXXXX

1. INTRODUCTION

The tomographic wave-front reconstruction (WFR) in

adaptive-optics (AO) systems using multiple guide-stars (GS) and

wave-front sensors (WFS) represents a computational challenge for

real-time atmospheric turbulence correction at a few hundreds

to thousands Hertz frame-rates. This is especially so for AO

systems in future giant segmented mirror telescopes (GSMT)

with primary diameters in the 20 m–40 m range, because of its

unprecedented number of degrees of freedom, N, i.e. the

num-ber of measurements from WFSs. A flurry of methods has been

developed, providing reduced complexity algorithms that could

accelerate simulation of large systems and later be mapped onto

real-time computers [

1

–

6

].

The tomographic WFR problem can be divided into two steps:

i) the estimation of the wave-front in the pupil plane along GS

directions and ii) the three-dimensional estimation in the

turbu-lence volume from its projections. Efficient methods to solve for

the first step are for instance the Fourier-domain reconstructor

[

1

,

2

] and more recently [

3

] that promise to decrease the

com-putational complexity from its original O

(

N

2

)

to respectively

O

(

N log

(

N

))

and O

(

N

)

. Work reviewed in Ramlau et al [

4

] falls

under this category.

A somewhat different path has been followed by others in

that an explicit minimum-mean-square error (MMSE) residual

cost-functional is solved for leading to formulations that are

amenable to sparse representations under some reasonable

ap-proximations (and therefore to efficient computational

recon-struction methods). The most representative examples are

re-viewed in Ellerbroek et al [

5

], with some more additions along

the same lines from [

6

] using a fractal approximation of the

regularizing stratified phase covariance term.

More recently, for some classes of AO systems not requiring

multi-conjugation, it has been noted that the MMSE

reconstruc-tion can be further simplified if we skip the explicit estimareconstruc-tion

of the 3D wave-front profiles to estimate instead the pupil-plane

wave-front in the directions of interest only [

7

–

10

] (hereinafter,

referred to as spatio-angular WFR), which is suitable for

multi-object AO (MOAO), laser-tomography AO (LTAO) and

option-ally ground-layer AO (GLAO) systems. The spatio-angular WFR

doesn’t require any approximation and thus can provide more

accurate estimation than the sparse reconstructor. However, for

large-scale systems, this explicit formulation requires

instan-tiating huge covariance matrices with the overall complexity

remaining O(N

2

) which is still a cause of computational

bottle-neck.

An efficient implementation of the spatio-angular WFR is

pro-posed by R. Conan [

10

] for a natural GS (NGS) based classical

single-conjugate AO system (SCAO). This implementation

re-duces the computational complexity to O

(

N log

(

N

))

from its

original O

(

N

2

)

by exploiting the Toeplitz structure of the

covari-ance matrices. In this paper we generalize this method to the

tomographic system using NGSs and laser GSs (LGS). Especially

for tomographic systems with multiple LGSs, we develop a way

to deal with the spatial sampling change at high altitudes due to

the cone effect of LGSs and the removal of low-order modes not

measurable by LGSs, whilst keeping the Toeplitz structure of the

covariance matrices. In passing, although this is not the main

motivation of this paper, in doing so we can show that MCAO

systems are also covered by the implementation presented here.

This paper is organized as follows. In §

2

we review the

re-construction formulations and the fast implementations they are

amenable to when considering large systems for GSMT. We give

a thorough account of the development of a spatio-angular WFR

algorithm exploiting the Toeplitz block structure of the

recon-structor matrices. In §

3

we compare performance on 8 m class

telescopes by analytic and Monte-Carlo, physical-optics

simu-lations and extend to the European Extremely Large Telescope

in §

4

. In §

5

, the real time readiness of the Toeplitz algorithm

using parallel computing with graphical processing unit (GPU)

is discussed. Final discussion and remarks are laid out in §

6

.

2. TOMOGRAPHIC WAVE-FRONT RECONSTRUCTION

We assume a situation shown in Fig.

1

, where the atmospheric

turbulence is expressed as N

layer

thin turbulent layers at

mul-tiple altitudes h

k

(

k

=

1,

· · ·

, N

layer

)

, and the phase aberration

on the atmospheric layer grids (circle symbols in Fig.

1

) is noted

as ϕ. This term is referred to as the layered phase. The

tomo-graphic computation is made based on N

gs

GSs in directions

of α

j

= (α

x

_j

, α

y

_j

) (

j

=

1,

· · ·

, N

gs)

using measurements from N

gs

WFSs with DM corrections applied over N

target

target directions

Layered

Phase

'

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Directional Phase

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N

layer

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・・・

・・・

Fig. 1.

Schematic figure of the assumed situation with N

layer

turbulence layers, N

gs

GSs and N

target

targets. The layered

phase grids of N

layer

turbulence layers are represented by the

circles. The cross symbols show the directional phase grid in a

GS direction α

j

. The target direction is denoted by β

i

.

β

i

(

i

=

1,

· · ·

, N

target

)

. The directional phase φ

α

j

is integrated

on the pupil-plane by ray-tracing the layered contributions ϕ

and interpolating at the intercepts. In other words, the phase

values of the propagating wave-fronts (cross symbols in Fig.

1

)

are interpolated from ϕ using a bi-linear interpolation matrix

P

_α

_j

_,k

at h

k

[

11

], and the final wave-front phase at pupil-plane is

given by an integration over altitudes, i.e.

φ

α

j

=

P

α

j

ϕ

=

N

layer

∑

k=1

P

_α

_j

_,k

ϕ

,

(1)

WFS measurements are modeled by a linear WFS operator G

and a noise vector η as

s

α

i

,η

=

GP

α

i

ϕ

+

η

,

(2)

where s

α

i

=

GP

α

i

ϕ

is denoted as the noiseless measurement

component. The concatenation of measurements from all N

gs

WFSs is denoted by s

α,η

.

In the remainder of the paper, variables with a hat symbol

represent an estimated quantity,

Σ

xy

represents auto- and

cross-covariance matrices of the vectors indicated in subscript, i.e.

Σ

xy

= h

xy

i

where

h·i

stands for ensemble average over time.

A. Minimum Mean Square Error reconstruction

We will address MMSE reconstructors which minimize the

aperture-plane residual wave-front error in a single direction

which is given by the Euclidean norm L

2

over the telescope

pupil

Ω of the difference between the input and the correction

phases

σ

2

_β

_i

=











φ

β

i

−

φ

b

β

i













2

L

2

(

Ω)

.

(3)

where the estimated directional phase

φ

b

_β

_i

=

R

_β

_i

s

α

with the

reconstructor matrix R

β

i

minimizing

R

_β

_i

=

arg min

R

_βi

D

σ

2

_β

_i

E

.

(4)

for the direction of optimization β

i

by solving for ∂σ

2

/∂R

β

i

=

0

.

i.e. SR

≥ (

1

−

σ

2

/2

)

2

, minimizing Eq.(

3

) is equal to maximize

image quality in terms of the Strehl ratio.

The MMSE solution was developed in [

12

,

13

] and others

re-maining general and applicable to multi-conjugate AO (MCAO)

systems, which requires an explicit estimation of the layered

phase ϕ for the multiple DM conjugation:

R

=

Σ

ϕsα

Σ

s

α

s

α

+

Σ

ηη

−1

(5)

=

Σ

ϕϕ

P

T

α

G

T



GP

α

Σ

ϕϕ

P

T

α

G

T

+

Σ

ηη



−1

(6)

= (

P

_α

T

G

T

Σ

−1

_ηη

GP

α

+

Σ

ϕϕ

−1

)

−1

_P

T

α

G

T

_Σ

−1

ηη

.

(7)

The reconstructor for the directional phase in Eq.(

4

) is given

by R

_β

_i

=

P

_β

_i

R. Although the WFR is followed by the fitting

process to determine commands sent to DM(s) [

11

], in this paper

we focus only on the WFR process.

For LGSs-based systems the removal of low-order modes

(tip/tilt/focus) needs be taken into account in the reconstructor

because these modes are not measurable by LGSs. We will

address this point later in §

2

.

B. Sparse reconstructor formulations

It has early been recognized that the formulation of Eq.(

7

) is

amenable to a sparse representation in an attempt to reduce

the computational burden with iterative implementations [

13

]

since the number of operations in a matrix-vector multiplication

(MVM) with a sparse matrix is proportional to the number of

non-zero elements in the matrix. The directional phase is

esti-mated iteratively using the formulation of Eq.(

7

) divided into

the following three steps:

ζ

=

P

_α

T

G

T

Σ

−1

_ηη

s

α

i

,η

(8)

and a layered phase estimation

ϕ

b

with an iterative method

(

P

_α

T

G

T

Σ

−1

_ηη

GP

α

+

Σ

ϕϕ

−1

)

ϕ

b

=

ζ

(9)

thus avoiding inverting explicitly

(

P

_α

T

G

T

Σ

_ηη

−1

GP

α

+

Σ

ϕϕ

−1

)

which

would cause both off-line issues to compute and store the matrix

and on-line increased burden since it is a full matrix. Finally,

the phase is ray-traced to the aperture

φ

b

_β

_i

=

P

_β

_i

ϕ

b

, using a low

complexity step, since only 4 elements per phase sample per

layer have non-zero values [

11

].

Assuming the use of Shack-Hartmann WFS (SH-WFS) with a

subaperture size of d, measurements are the spatial derivatives

of the wave-front averaged on each subaperture. The linear

SH-WFS operator G admits a discrete approximation from uniform

3

×

3 stencils [

14

] given by

stencil

(

G

)x

=

1

2d











−

1/4

0

1/4

−

1/2

0

1/2

−

1/4

0

1/4











(10)

stencil

(

G

)y

=

stencil

(

G

)

T

x

(11)

and therefore has only 6 non-zero elements per subaperture. The

noise covariance matrix is generally considered as a diagonal

matrix by assuming a zero-mean additive Gaussian noise. In the

LGSs case, although x- and y-measurements are correlated due

to the spot elongation, only 3 central diagonals have non-zero

values [

15

]. The inverse covariance matrix of the layered phase

Σ

−1

ϕϕ

is a dense matrix but it can be approximated as a sparse

matrix with a discrete Laplacian operator L as

Σ

_ϕϕ

−1

≈

L

T

L [

13

].

C. Spatio-angular tomographic formulation

We now enter the core matter of this paper. We start from the

formulation in Eq.(

5

) to estimate the directional phase. The

di-rectional phase φ

β

i

is directly connected with the measurements

s

α,η

through the covariance matrix

Σ

φ

_βi

s

α

=

P

β

i

Σ

ϕsα

, and the

explicit estimation of the layered phase is circumvented, thus

al-lowing us to make the size of the reconstruction matrix compact.

The required covariance matrices are derived theoretically

along with the measurement model. The slope-slope and

phase-slope covariance matrices are computed through numerical

integration [

16

] or the fast Fourier transform (FFT) [

7

].

Ap-proximated measurement models are also proposed for the fast

computation of the covariance matrices [

8

]. In this paper, we

investigate three measurement models to derive the theoretical

covariance matrices: the accurate FFT model, Hudgin-like model

and Fried model. The details of the models are summarized in

Appendix. The Hudgin-like model and Fried model define a

measurement with two and four discrete phase points,

respec-tively. Although models defining a measurement with more

than 4 points would give better approximations, these models

result in more computations and loose their advantage for a

fast computation. Therefore, we only focus on the Hudgin-like

model and Fried model as fast approximated gradient models

in this paper.

The iterative implementation of Eq.(

5

) is given by the

follow-ing two steps:

Σ

s

α

s

α

+

Σ

ηη

ζ

=

s

α,η

(12)

and

b

φ

_β

_i

=

Σ

φ

_βi

s

α

ζ

,

(13)

where ζ in Eq.(

12

) is computed iteratively. As it stands it

re-quires a MVM per iteration which is prohibitive. Therefore, a

fast algorithm for the MVM with these covariance matrices is

necessary. In the rest of §2, we investigate how it can be suitably

mapped into an efficient runtime algorithm.

D. Toeplitz Structure in Covariance Matrices

An efficient implementation for solving Eq.(

12

) has been

pro-posed by R. Conan [

10

] for large-scale NGS-based SCAO systems

by exploiting the Toeplitz structure of the covariance matrices.

Here, we show how these structure is kept in a tomographic

setting.

In order to avoid an overly complicated notation, s

α,η

, s

α

and

s

α

i

are simplified into s

η

, s and s

i

, respectively. In addition, we

only focus on one target direction β, which is a case of LTAO

systems. For multiple target cases such as MOAO systems,

WFR for each target direction is performed in parallel. The

number of subapertures and phase points in one SH-WFS are

n

×

n and

(

n

+

1

) × (

n

+

1

)

, and the geometric arrangement

of the measurement and the phase points shown in Fig.

2

as

n

=

4. First, a square aperture without vignetted subapertures

is assumed. The impact of more complicate apertures, such as

circular and annular, on the Toeplitz covariance matrix will be

discussed in 2.F.

Slope-Slope Covariance Matrix

As shown in Fig.

3

, the slope-slope covariance matrix

Σ

ss

is

de-composed into N

gs

×

N

gs

blocks, and the

(

i, j

)

-th block can also

be decomposed into 2

×

2 components:

Σ

s

i

s

j

=







Θ

x

i

,x

j

Θ

x

i

,y

j

Θ

y

i

,x

j

Θ

y

i

,y

j





 .

(14)