overview on pyramid wavefront sensor: forward models ...€¦ · overview on pyramid wavefront...
TRANSCRIPT
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
1 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
2 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
3 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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.
4 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
5 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
6 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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.
7 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Roof WFS: linearized and simplified model
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.
8 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Roof WFS: linearized and simplified model
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).
9 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Roof WFS: linearized and simplified model
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
10 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Roof WFS: linearized and simplified model
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).
11 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Roof WFS: linearized and simplified model
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)
12 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
13 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Algorithms in closed-loop simulations: quality, speed and spiders
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.
14 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Algorithms in closed-loop simulations: quality, speed and spiders
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)
15 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Algorithms in closed-loop simulations: quality, speed and spiders
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
16 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Algorithms in closed-loop simulations: quality, speed and spiders
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”
17 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Algorithms in closed-loop simulations: quality, speed and spiders
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?
18 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
19 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Extension of algorithms to other linear models
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
20 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Extension of algorithms to other linear models
Extension of algorithms to 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
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
21 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
22 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
23 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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.
24 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
25 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Extenstion of algorithms to other non-linear models
Non-linear models
Model Roof-WFS Pyramid WFS
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
26 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Extenstion of algorithms to other non-linear models
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)
27 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Extenstion of algorithms to other non-linear models
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
ψn,m(x , y) · ψn,m(x , y)
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
ψn,m(x , y) · ψn,m(x , y)
+ 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
28 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Extenstion of algorithms to other non-linear models
Extenstion of algorithms to other non-linear models
Model Roof-WFS Pyramid WFS
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
29 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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
30 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
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).
31 / 32
Introduction Linear models and reconstructors Non-linear models and reconstructors Summary
Thanks
Thank you for attention!
32 / 32