for XAO with pyramid WFS. Iuliia Shatokhina1, Andreas Obereder2,. Matthias Rosensteiner2, Ronny Ramlau1. 1 â Industrial Mathematics Institute, JKU, Linz.
Problem
Preprocessed CuReD
Quality and speed performance
P-CuReD – a fast wavefront reconstruction algorithm for XAO with pyramid WFS Iuliia Shatokhina1 , Andreas Obereder2 , Matthias Rosensteiner2 , Ronny Ramlau1 1 – Industrial Mathematics Institute, JKU, Linz 2 – MathConsult GmbH
AO4ELT3, Florence, Italy 26-31 May 2013 This research was partially supported by the Austrian Science Fund (FWF): W1214-N15, project DK8.
Problem
Preprocessed CuReD
Outline
1
Problem
2
Preprocessed CuReD
3
Quality and speed performance
Quality and speed performance
Problem
Preprocessed CuReD
Quality and speed performance
Problem
XAO (EPICS on E-ELT) D = 42 m telescope pyramid WFS with 200 × 200 subapertures, without / with linear / with circular modulation frame rate 3 kHz, time for reconstruction: 0.3 ms Task To determine the unknown wavefront φ from pyramid WFS data Sx , Sy Sx = Qx φ,
Sy = Qy φ
with Qx , Qy – nonlinear singular integral operators.
(1)
Problem
Preprocessed CuReD
Quality and speed performance
Linearized forward models
To linearize Qx , Qy , we use assumptions: roof approximation, infinite telescope size, small wavefront perturbations (closed loop) Sxn (x, ·) = φ(x, ·) ∗
1 , πx
sinc(αλ x) , πx J0 (αλ x) Sxc (x, ·) = φ(x, ·) ∗ , πx
Sxl (x, ·) = φ(x, ·) ∗
{n, l, c} – no / linear / circular modulation case, αλ = 2πα/λ,
α = bλ/D – modulation radius,
b – positive integer, J0 – zero-order Bessel function of the first kind.
(2) (3) (4)
Problem
Preprocessed CuReD
Quality and speed performance
P-CuReD
P-CuReD = Data Preprocessing + CuReD Two-step method
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Data preprocessing: Transform the modulated P-WFS data to SH-like data according to the analytical relation in the Fourier domain.
2
CuReD: To the modified data apply the CuReD – very efficient reconstructor for SH-WFS data. Details on CuReD: Matthias Rosensteiner, Poster 13208, Fast wavefront reconstruction with the CuReD algorithm
Problem
Preprocessed CuReD
Quality and speed performance
Step 1: Data preprocessing Representation of the measurements in the Fourier domain (FSpyr )(u) = (Fφ)(u) · gpyr (u) · sinc(du)
(5)
(FSsh )(u) = (Fφ)(u) · gsh (u) · sinc(du)
(6)
u – spatial frequency, d – subaperture size. Fourier domain relation between the two sensors (FSsh )(u) = (FSpyr )(u) · gsh/pyr (u),
gsh/pyr (u) :=
gsh (u) . gpyr (u)
(7)
Fourier convolution theorem → relation between the two sensors in the space domain 1 Ssh (x) = √ Spyr (x) ∗ F −1 gsh/pyr (x) . 2π | {z } psh/pyr (x)
(8)
Problem
Preprocessed CuReD
Quality and speed performance
Fourier domain filters
SH gsh (u) = 2iπdu,
(9)
Pyramid n gpyr (u) = i sgn(u), ∀u ∈ [−ucut , ucut ]. ( i sgn(u), |u| > umod , l gpyr (u) = iu/umod , |u| ≤ umod , ( i sgn(u), |u| > umod , c gpyr (u) = 2i arcsin(u/u ), |u| ≤ umod . mod π
(10) (11)
(12)
where the parameter umod > 0 is defined as umod = α/λ = b/D, and b is a positive integer.
Problem
Preprocessed CuReD
Quality and speed performance
Step 1: SH-Pyr transmission filters n gsh/pyr (u) = 2πdu sgn(u), ∀u ( 2πdu sign(u), l gsh/pyr (u) = 2πdumod , ( 2πdu sgn(u), c gsh/pyr (u) = π 2 du , arcsin(u/u )
∈ [−ucut , ucut ] |u| > umod , |u| umod , |u| ≤ umod ,
mod
SH/pyramid transmission filters in the Fourier domain 3.5
no modulation linear modulation, α = 4λ /D circular modulation, α = 4λ /D
3
Fourier SNR
2.5 2 1.5 1 0.5 0 -2.5
-2
-1.5
-1
-0.5
0
0.5
frequency, m-1
1
1.5
2
2.5
(13) (14) (15)
Problem
Preprocessed CuReD
Quality and speed performance
Step 1: Space domain kernels n psh/pyr (x) =
πx 2πdb2 π + sinc2 = sinc d d D2
πxb D
−
Analytical kernel 2
1.5
1
kernel
l psh/pyr (x)
πx πx π π − . sinc sinc2 d d 2d 2d
0.5
0
-0.5
-1
-1.5
-2
-1
0 space variable x, m
1
2
πx π . sinc2 2d 2d
(16)
(17)
Problem
Preprocessed CuReD
Quality and speed performance
Step 1: Convolution Convolve Sx row-wise and Sy column-wise with 1d kernel psh/pyr 1 Ssh (x) = √ Spyr (x) ∗ psh/pyr (x). 2π
Computationally very cheap, highly parallelizable and pipelinable.
(18)
Problem
Preprocessed CuReD
Quality and speed performance
Step 2: CuReD
To the modified data apply the CuReD – method for wavefront reconstruction from SH-WFS data CuReD has a linear complexity, is parallelizable and pipelinable Details on CuReD: Matthias Rosensteiner, Poster 13208, Fast wavefront reconstruction with the CuReD algorithm
Problem
Preprocessed CuReD
Quality and speed performance
LE Strehl
LE Strehl in K band: MVM and P-CuReD vs. the detected NGS photon flux.
ESO median atmosphere
ESO bad/good atmospheres
Problem
Preprocessed CuReD
Quality and speed performance
LE PSF
LE PSF in the K-band for median atmosphere and high-flux case.
Problem
Preprocessed CuReD
Quality and speed performance
Comparison: P-CuReD vs MVM
Complexity: Data preprocessing requires 26N operations. CuReD requires 20N operations. Parallelizable and pipelinable.
Strehl ratio Complexity Number of megaFLOP for ∼ 29600 actuators to be performed within 0.3ms Percentage of MVM flop Parallelizability and pipelinability
P-CuReD 0.965 46n
MVM 0.965 4n2
1.36 0.04% yes
3500 100% yes
Problem
Preprocessed CuReD
Quality and speed performance
Speed of the C-prototype System configuration: Intel(R) Xeon(R) CPU X5650 @ 2.67GHz, 12 Cores (dual hexacore) MVM: 402 ms Reconstruction time 2500
P-CuReD (2) P-CuReD (3) P-CuReD (4) P-CuReD (5)
2000
time (µs)
1500
1000
500
0
0
2
4
6
8
number of processor cores used
10
12
Problem
Preprocessed CuReD
Quality and speed performance
Paper
More quality results in: Iu. Shatokhina, A. Obereder, M. Rosensteiner, R. Ramlau. Preprocessed cumulative reconstructor with domain decomposition: a fast wavefront reconstruction method for pyramid wavefront sensor. Applied Optics, 52(12), 2640–2652 (2013). Derivation of analytical kernel for different modulation scenarios SE Strehl (convergence) LE Strehl for non-modulated case Noise propagation analysis
Problem
Preprocessed CuReD
Quality and speed performance
Summary
P-CuReD – fast reconstruction method for P-WFS with / without modulation. Linear complexity O(N). Strehl ratios are the same as / better than MVM. PSF is suitable for extrasolar planet search. Method suits the XAO requirements on ELTs.
Thanks for your attention!
Problem
Preprocessed CuReD
Quality and speed performance
Summary
P-CuReD – fast reconstruction method for P-WFS with / without modulation. Linear complexity O(N). Strehl ratios are the same as / better than MVM. PSF is suitable for extrasolar planet search. Method suits the XAO requirements on ELTs.
Thanks for your attention!