overview on pyramid wavefront sensor: forward models ...€¦ · overview on pyramid wavefront...

Introduction Linear models and reconstructors Non-linear models and reconstructors Summary

Overview on pyramid wavefront sensor:forward models, reconstruction algorithms,

practical issues

Iuliia Shatokhina,Victoria Hutterer, Andreas Obereder,Stefan Raffetseder, Ronny Ramlau

Industrial Mathematics Institute, JKU, Linz

WaveFront Sensing in the VLT/ELT era II,Padova, October 2-4, 2017

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Outline

1 Introduction

2 Linear models and reconstructorsRoof WFS: linearized and simplified modelAlgorithms in closed-loop simulations: quality, speed and spidersExtension of algorithms to other linear models

3 Non-linear models and reconstructorsRoof WFS: nonlinear transmission mask modelAlgorithms in closed-loop simulations: quality and speedExtenstion of algorithms to other non-linear models

4 Summary

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Outline

1 Introduction



4 Summary

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Pyramid and Roof WFS

Credit: C. Verinaud

Sx (x , y) =[I1(x , y) + I2(x , y)]− [I3(x , y) + I4(x , y)]

I0

Sy (x , y) =[I1(x , y) + I4(x , y)]− [I2(x , y) + I3(x , y)]

I0I0 – average intensity per subaperture.

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Inverse problem & Remarks

Task: to reconstruct the unknown wavefront φ from non- / modulated pyramidWFS data Sx ,Sy

Sx = Pxφ, Sy = Pyφ

Forward operators Px ,Py are nonlinear singular integral operators

Details omitted (e.g., Sx only)

Any modulation meant; αλ will denote the modulation parameter

αλ =2παλ

=2πrD, d ∈ R+

Omit aperture sometimes (for clarity)

Finite sampling

From simple approximate to complicated models

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Outline

1 Introduction



4 Summary

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Roof WFS: linearized and simplified model

Roof WFS: linearized and simplified operator Rn,l,cs

Sn,l,cx = Rn,l,cs φ

(Rn,l,cs φ)(x , y) =1π

X(y)∫−X(y)

φ(x ′, y)kn,l,c(x ′ − x)

x − x ′dx ′

=[φ ∗mn,l,cx

](x , y)

mn,l,cx (x , y) :=kn,l,c(x)δ(y)

πxkn(x) = 1

kl (x) = sinc(αλ(x))

kc(x) = J0(αλ(x))

J0 – the zero-order Bessel function of the first kind.

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Inversion of Rn,l,cs in spatial domain

Inversion of Finite Hilbert transform: (modulation 0 only, i.e., Rns )

Finite Hilbert Transform Reconstructor (FHTR) [1]Singular Value Type Reconstructor (SVTR) [2]→ SVTR for Rl,c

s ?

Iterative algorithms: adjoint operatorConjugate Gradient for the Normal Equation (CGNE) [1,3,4]Steepest Descent (SD) [3,4]Pyramid Kaczmarz Iteration (PKI) [3,4]

[1] I. Shatokhina, “Fast wavefront reconstruction algorithms for extreme adaptive optics,” Ph.D. thesis (JohannesKepler University Linz, 2014).

[2] V. Hutterer, R. Ramlau, Wavefront Reconstruction from Non-modulated Pyramid Wavefront Sensor Data usinga Singular Value Type Expansion, Inverse Problems, submitted.

[3] V. Hutterer, R. Ramlau, Iu. Shatokhina, Real-time AO with pyramid wavefront sensors: Theoretical analysis ofpyramid forward model, in preparation.

[4] V. Hutterer, R. Ramlau, Iu. Shatokhina, Real-time AO with pyramid wavefront sensors: Accurate wavefrontreconstruction with iterative methods, in preparation.

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FHTR & SVTR

InvertRn

s = Txφ = [φ ∗mnx ]

snx = Rn

s

φ = T−1x sx

FHTR:

φ(x) = −1π

1∫−1

√1− x2

1− x ′2sx (x ′)x − x ′

dx ′

SVTR:

singular value type system (σk , fk , gk )k≥0,fk , gk – weighted Chebyshev polynomials,

σk = 1 ∀k

φ (x , y) = −2∞∑

k=0

1σk〈sx (·, y) , gk 〉ω fk (x).

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Iterative algorithms: CGNE & SD & PKI

Well-known (in applied mathematics) iterative methods

Application of adjoint operators((Rn,c,ls

)∗Ψ)

(x , y) = −1π

p.v .∫

Ωy

Ψ(x ′, y) · kn,c,l(x ′ − x)

x ′ − xdx ′,

Due to discretization largely precomputed −→ fast!

Algorithm: Landweber-Kaczmarz iterationchoose Φ0, set attenuation coefficients β1, β2

for i = 1, . . .K doΦi,0 = Φi−1

Φi,1 = Φi,0 + β1R∗x(sx − Rxφi,0

)Φi,2 = Φi,1 + β2R∗y

(sy − Ryφi,1

)Φi = Φi,2

endfor

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Inversion of Rn,l,cs in Fourier domain

Fourier domain representation of Rn,l,cs φ:

(Fx Sn,l,cx )(u) =[(Fxφ)(u) · gn,l,c(u)

]gn,l,c(u) =

(Fx mn,l,cx

)(u)

Fourier domain based algorithms:Preprocessed Cumulative Reconstructor with domain Decomposition (P-CuReD) [1,2]Convolution with the Linearized Inverse Filter (CLIF) [2,3]Pyramid Fourier Transform Reconstructor (PFTR) [2,3]

[1] Iu. Shatokhina, A. Obereder, R. Rosensteiner, R. Ramlau. Preprocessed cumulative reconstructor with domaindecomposition: a fast wavefront reconstruction method for pyramid wavefront sensor, Applied Optics 52(12),2640-2652 (2013).

[2] I. Shatokhina, R. Ramlau. Convolution and Fourier transform based reconstructors for pyramid wavefrontsensor, Applied Optics 56(22), 6381-6390 (2017).

[3] I. Shatokhina, “Fast wavefront reconstruction algorithms for extreme adaptive optics,” Ph.D. thesis (JohannesKepler University Linz, 2014).

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P-CuReD & CLIF & PFTR

(Fx Sx )(u) = (Fxφ)(u) · g(u)

P-CuReD:

(Fx S)SHpyr (u) = (Fxφ)(u) · gSH

pyr (u)

(Fx SSH )(u) = (Fx Spyr )(u) · gSH/pyr (u)

gSH/pyr (u) :=gSH (u)

gpyr (u)

SSH (x) = (Spyr ∗ F−1x gSH/pyr︸︷︷︸pSH/pyr

)(x)

φ(x, y) = CuReD(SSH )

PFTR:

(Fxφ)(u) = (Fx Sx )(u) · g−1(u)

φ(x, y) =(F−1

x

[(Fx Sx )(u) · g−1(u)

])(x, y)

CLIF:

φ(x, y) =[Sx ∗

(F−1

x g−1)]

(x, y)

p(x, y) =(F−1

x g−1)

(x) δ(y)

φ(x, y) = [Sx ∗ p] (x, y)

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Algorithms in closed-loop simulations: quality, speed and spiders

Considered AO systems

XAO (EPICS on ELT)

aim: direct imaging of exoplanets

D = 42 m telescope

pyramid WFS with 200x200subapertures,with circular modulation

DM update 3000 times per second!

time for reconstruction: 0.3 ms

SCAO (METIS on ELT)

D = 37 m telescope

pyramid WFS with 74x74subapertures,with circular modulation

DM update 500-1000 times persecond!

time for reconstruction: 1-2 ms

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Comparison of quality: linear algorithms

Algorithm Quality in end-to-end simulations (OCTOPUS)METIS mod 0 METIS mod 4 XAO mod 0 XAO mod 4

Photon flux 10000 ph/pix/it 10000 ph/pix/it 50 ph/pix/it 50 ph/pix/itFrame rate 1kHz 1kHz 3kHz 3kHzMatrix-Vector Multiplication (MVM) ≈ 0.62 [1] 0.80 [2] 0.96MMSE (YAO) 0.89 [3]Preprocessed CuReD (P-CuReD) 0.89 0.91 0.96Conv. with Linearized Inverse Filter (CLIF) 0.88 0.94Pyramid FTR (PFTR) 0.88 0.94Finite Hilbert Transform Rec. (FHTR) 0.85Singular Value Type Reconstructor (SVTR) 0.77 0.88Steepest Descent (SD) ≥ 0.79 0.90Pyramid Kaczmarz Iteration (PKI) 0.81 0.92

[1] M. Le Louarn et. al., Latest AO simulation results for the E-ELT, poster AO4ELT5.

[2] Results provided by ESO.

[3] MMSE reconstructor in YAO, results provided by Stefan Hippler.

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Comparison of complexities: linear methods

Algorithm Modulation Complexity Remarksno small large

Matrix-Vector Multiplication (MVM) + + + O(n2) baseline;Fourier Transform Reconstructor (FTR) – – + O(n log n) geometrical modelPreprocessed CuReD (P-CuReD) + + + O(n) Fourier domain basedConv. with Linearized Inverse Filter (CLIF) + + + O(n3/2) (iteartive)Pyramid FTR (PFTR) + + + O(n log n)

Finite Hilbert Transform Rec. (FHTR) + – – O(n3/2) inversion of finiteSingular Value Type Reconstructor (SVTR) + – – O(n3/2) Hilbert transform

Conjugate Gradient for Normal Eq. (CGNE) + + + O(n3/2) iterative algorithms,Steepest Descent (SD) + + + O(n3/2) adjoint operatorsPyramid Kaczmarz Iteration (PKI) + + + O(n3/2)

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Comparison of computational load: linear methods

Algorithm Number of operationsin XAO setting

MVM 4nan 3.4120e+09 100%P-CuReD (4c − 2)n + 20n 1.3248e+06 0.0388%CLIF 4n

√n + n 1.9579e+07 0.5738%

SD(K = 4) K · (12n√

n + 12n + 4) 4 · 5.8996e + 07 4 · 1.73% = 6.92%

PKI(K = 5) K · (8n√

n + 2n) 5 · 3.9158e + 07 5 · 1.15% = 5.75%

na = 29618, n = 28800, c = 7

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Reconstruction in presence of spiders

Residual segmented piston −→ low LE

There are possibilities to control differential pistonfour methodssome provide acceptable qualityextremely fast! Add 6x2na FLOPs

Poke matrix inversion −→ see talk by A. Obereder tomorrow @ 12.40”Keep it simple – Poke Matrix Inversion for a (stable) piston segmentreconstruction”

Split approaches −→ see talk by V. Hutterer tomorrow @ 12.00”Direct piston reconstruction approaches to control segmented ELT-mirrors”

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Important questions

Spiders – theoretical understanding / explanationsign (not possible from linearized roof sensor models)full pyramid model, take interference terms into account?reconstruction of pistons from intensities I1,2,3,4?

identify segments between which piston jumps occurcriteria to identify if the sign of piston between neighbouring segments is the same, or theoppositecriteria for piston sign – work in progress

What is the best possible reconstruction quality?

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Extension of algorithms to other linear models

Other linear models

Model Roof-WFS Pyramid WFS

Linear simplified Rn,l,cs Pn,l,c

s

Linear Rn,l,cl Pn,l,c

l

(Rn,l,cs φ)(x, y) =

1

π

X(y)∫−X(y)

φ(x′, y)kn,l,c(x′ − x)

x − x′dx′

(Rn,l,cl φ)(x, y) = (Rn,l,c

s φ)(x, y)− φ(x, y)(Rn,l,cs 1)(x, y)

(Pn,l,cs φ)(x, y) = (Rn,l,c

s φ)(x, y)−1

π3

X(y)∫−X(y)

Y (x)∫−Y (x)

Y (x)∫−Y (x)

φ(x′, y′)pn,c(x′ − x, y′ − y′′)

(x − x′)(y − y′)(y − y′′)dy′′dy′dx′

pc (x, y) :=1

T

T/2∫−T/2

cos[αλ x sin(2πt/T )] cos[αλ y cos(2πt/T )]dt

(Pn,l,cl φ) = (Rn,l,c

l φ)−1

π3

X(y)∫−X(y)

Y (x)∫−Y (x)

Y (x)∫−Y (x)

[φ(x′, y′)−φ(x, y′′)]pn,c(x′ − x, y′ − y′′)

(x − x′)(y − y′)(y − y′′)dy′′dy′dx′

i-FHTR, i-SVTR; i-PFTR, i-CLIF; CGNE, SD, PKI

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Other linear models – Fourier domain representation

Fourier domain representation of

Rn,l,cs φ: P-CuReD, PFTR, CLIF

(Fx Sn,l,cx )(u) =

[(Fxφ)(u) · gn,l,c(u)

]gn,l,c(u) =

(Fx mn,l,c

x

)(u)

Rn,l,cl φ: i-PFTR, i-CLIF

(Fx Sn,l,cx )(u) =

[(Fxφ)(u) · gn,l,c(u)

]− (Fxφ)(u) ∗

[(FxXΩy×Ωx )(u) · gn,l,c(u)

]Pn,l,c

s

Non-modulated case: i-PFTR, i-CLIF

(Fxy Snx ) = (Fxyφ) · (Fxy mn

x )

−[

(Fxyφ) · (Fxy mnxy )]∗[

(FxyXΩy×Ωx ) · (Fxy mny )]

Modulated case: ?

Pn,l,cl

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Linear simplified Rn,l,cs Pn,l,c

s

Linear Rn,l,cl Pn,l,c

l

Forward operator Algorithm RemarksRn

s → Rnl → Pn

s → Pnl FHTR→ iFHTR Hilbert transform;

iSVTR singular functions

Rn,l,cs P-CuReD Fourier domain based

Rn,l,cs → Rn,l,c

l → Pns → Pn

l CLIF→ i-CLIF

Rn,l,cs → Rn,l,c

l → Pns → Pn

l PFTR→ i-PFTRCGNE iterative algorithms,

Rn,l,cs → Rn,l,c

l → Pn,l,cs → Pn,l,c

l SD adjoint operatorsPKI

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Outline

1 Introduction



4 Summary

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Roof WFS: nonlinear transmission mask model

Roof WFS: nonlinear transmission mask model

Sn,l,cx = Rn,l,c

t φ

(Rn,l,ct φ)(x, y) =

1πXΩy×Ωx (x, y)

X(y)∫−X(y)

sin[φ(x′, y)− φ(x, y)

]kn,l,c(x′ − x)

x − x′dx′

= XΩy×Ωx (x, y) cos(φ(x, y)) ·[XΩy×Ωx (·, y) sin(φ(·, y)) ∗

kn,l,c(·)δ(y)

π·

]− XΩy×Ωx (x, y) sin(φ(x, y)) ·

[XΩy×Ωx (·, y) cos(φ(·, y)) ∗

kn,l,c(·)δ(y)

π·

]

Inversion: Nonlinear Landweber method, nonlinear CG, nonlinear SD, ...

φk+1 = φk +

((Rn,l,ct

)′)∗ (sx − Rn,l,ct φk

), k ∈ N

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Algorithms in closed-loop simulations: quality and speed

Comparison of quality

Algorithm Quality in end-to-end simulations (OCTOPUS)METIS mod 0 METIS mod 4 XAO mod 0 XAO mod 4

Photon flux 10000 ph/pix/it 10000 ph/pix/it 50 ph/pix/it 50 ph/pix/itFrame rate 1kHz 1kHz 3kHz 3kHzMatrix-Vector Multiplication (MVM) ≈ 0.62 [1] 0.80 [2] (1000ph) 0.96

(1000ph) 0.89 [3] 0.96Preprocessed CuReD (P-CuReD) 0.89 0.91 0.96Conv. with Linearized Inverse Filter (CLIF) 0.88 0.94Pyramid FTR (PFTR) 0.88 0.94Finite Hilbert Transform Rec. (FHTR) 0.85Singular Value Type Reconstructor (SVTR) 0.77 0.88Conjugate Gradient for Normal Eq. (CGNE)Steepest Descent (SD) ≥ 0.79 0.90Pyramid Kaczmarz Iteration (PKI) 0.81 0.92Nonlinear Landweber (NL) ≥ 0.83

[1] M. Le Louarn et. al., Latest AO simulation results for the E-ELT, poster AO4ELT5.

[2] Results provided by ESO.

[3] MMSE reconstructor in YAO, results provided by Stefan Hippler.

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Algorithms in closed-loop simulations: quality and speed

Comparison of computational load

Algorithm Modulation Complexity Remarksno small large

Matrix-Vector Multiplication (MVM) + + + O(n2) baseline;Fourier Transform Reconstructor (FTR) – – + O(n log n) geometrical modelPreprocessed CuReD (P-CuReD) + + + O(n) Fourier domain based(i)-Conv. with Linearized Inverse Filter (CLIF) + + + O(n3/2) (iteartive)(i)-Pyramid FTR (PFTR) + + + O(n log n)

Hilbert Transform Reconstructor (HTR) + – – O(n log n) (iterative) inversion of finite(i)-Finite Hilbert Transform Rec. (FHTR) + – – O(n3/2) Hilbert transform;(i)-Singular Value Type Reconstructor (SVTR) + – – O(n3/2) singular functions

Steepest Descent (SD) + + + O(n3/2) adjoint operatorsPyramid Kaczmarz Iteration (PKI) + + + O(n3/2)

Nonlinear Landweber (NL) + + + O(n3/2) Frechet derivativeits adjoint

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Extenstion of algorithms to other non-linear models

Non-linear models


Nonlinear transmission mask Rn,l,ct Pn,l,ct

Nonlinear phase mask without interference Rn,l,cp Pn,l,cp

Nonlinear phase mask withi interference Rn,l,ci Pn,l,ci

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Pyramid WFS: nonlinear transmission mask model

(Pn,l,ct φ)(x, y) = (Rn,l,c

t φ)(x, y)

−1π3

X(y)∫−X(y)

Y (x)∫−Y (x)

Y (x)∫−Y (x)

sin[φ(x′, y ′)−φ(x, y ′′)]pn,c(x′ − x, y ′ − y ′′)(x − x′)(y − y ′)(y − y ′′)

dy ′′dy ′dx′

Inversion: nonlinear Landweber, nonlinear CG, nonlinear SD, ...

ψdet (x , y) =1

2π

(ψaper ∗ F−1OTF t

pyr)

(x , y)

OTF tpyr (ξ, η) =

1∑m=0

1∑n=0

T mn(ξ, η)

T mn(ξ, η) = H2d[(−1)m · ξ, (−1)n · η

]I(x , y) ≈

1∑n=0

1∑m=0

ψn,m(x , y) · ψn,m(x , y)

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Roof & Pyramid WFS: nonlinear phase mask model w/ointerference

Nonlinear, phase mask, no interference:

ψdet (x , y) =1

2π

(ψaper ∗ F−1OTF p

pyr)

(x , y)

OTF ppyr (ξ, η) = exp(−i · Π(ξ, η)) · OTF t

pyr (ξ, η)

I(x , y) ≈1∑

n=0

1∑m=0


Nonlinear, phase mask, with interference:

ψdet (x , y) =1

2π

(ψaper ∗ F−1OTF p

pyr)

(x , y)

OTF ppyr (ξ, η) = exp(−i · Π(ξ, η)) · OTF t

pyr (ξ, η)

I(x , y) =1∑

n=0

1∑m=0


+ 21∑

n=0

1∑m=0

1∑n′=0,n′ 6=n

1∑m′=0,m 6=m

Re[ψn,m(x , y) · ψn′,m′ (x , y)]

Inversion: Nonlinear Landweber method

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Nonlinear transmission mask Rn,l,ct Pn,l,ct

Nonlinear phase mask without interference Rn,l,cp Pn,l,cp

Nonlinear phase mask withi interference Rn,l,ci Pn,l,ci

Forward operator Algorithm Remarks

Rn,l,ct → Rn,l,cp → Rn,l,ci NL, nCG, ... Frechet derivative,Pn,l,ct → Pn,l,cp → Pn,l,ci NL, nCG, ... adjoint

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Outline

1 Introduction



4 Summary

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Summary

Roof −→ pyramid

Linearized models −→ non-linear models

A wide spectrum of algorithms developed and studied: linear and non-linear

Quality and speed better than MVM !

Can handle spiders with a linear method !

Open questions: best reconstruction quality model, ncpa, deeper understanding ofspiders

Go on-sky...

Urban Bitenc et al., On-sky tests of the CuReD and HWR fast wavefront reconstruction algorithms with CANARY.

Monthly Notices of the Royal Astronomical Society 448(2), 1199-1205 (2015).

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Thanks

Thank you for attention!

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overview on pyramid wavefront sensor: forward models ...€¦ · overview on pyramid wavefront...

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