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Distributed Wavefront Reconstruction with SABRE for
real-time Large Scale Adaptive Optics Control
Elisabeth Brunnera , Cornelius de Visserb , João Silvaa , Michel Verhaegena
a Delft
University of Technology, Delft Center of Systems and Control, Delft, The Netherlands;
University of Technology, Faculty of Aerospace Engineering, Delft, The Netherlands;
b Delft
ABSTRACT
We present advances on Spline based ABerration REconstruction (SABRE) from (Shack-)Hartmann (SH) wavefront measurements for large-scale adaptive optics systems. SABRE locally models the wavefront with simplex
B-spline basis functions on triangular partitions which are defined on the SH subaperture array. This approach
allows high accuracy through the possible use of nonlinear basis functions and great adaptability to any wavefront sensor and pupil geometry. The main contribution of this paper is a distributed wavefront reconstruction
method, D-SABRE, which is a 2 stage procedure based on decomposing the sensor domain into sub-domains each
supporting a local SABRE model. D-SABRE greatly decreases the computational complexity of the method and
removes the need for centralized reconstruction while obtaining a reconstruction accuracy for simulated E-ELT
turbulences within 1% of the global method’s accuracy. Further, a generalization of the methodology is proposed
making direct use of SH intensity measurements which leads to an improved accuracy of the reconstruction
compared to centroid algorithms using spatial gradients.
Keywords: Adaptive Optics, Distributed Wavefront Reconstruction, Extremely Large Telescopes, Splines,
Wavefront Sensing.
1. INTRODUCTION
For the new generation of ground based telescopes, the dimensions of Adaptive Optics (AO) systems are increasing, up to 39m for the European Extremely Large Telescope (E-ELT), such that the computational complexity of
the controller prohibits an (unstructured) centralized implementation. We present the advances on our recently
introduced novel method for distributed wavefront reconstruction from (Shack-)Hartmann (SH) measurements
of the wavefront with application to large-scale AO systems, like the eXtreme Adaptive Optics (XAO) system
of the upcoming E-ELT.1, 2
The new method, indicated as D-SABRE (Distributed Spline based ABerration REconstruction),3 is an extension of the recently presented SABRE method4 for wavefront reconstruction which uses B-spline basis functions
defined on triangular partitions of the wavefront sensor (WFS) domain. D-SABRE performs a decomposition of
the entire WFS domain into a set of sub-domains each supporting a local multivariate spline model. Each local
spline model depends only on local WFS measurements, and information is only shared between directly neighboring models. The disadvantage of most state of the art wavefront reconstrction (WFR) methods5–8 is that,
even though certain computational operations can be parallelized, they are based on non-distributed principles.
Linear computational complexity orders have been reached but it is important to note that these numbers are
WFS-array wide, or global, numbers. With D-SABRE, a truly distributed method is presented which is suitable
to operate on massively parallel hardware architectures. The method is demonstrated on simulated E-ELT turbulence wavefronts and compared with the centralized SABRE algorithm. We show that the D-SABRE obtains a
reconstruction accuracy that is within 1% of the reconstruction accuracy of the global (non-distributed) SABRE
method.
Further, we discuss a generalization of the SABRE methodology making direct use of the intensity measurements in the detector of the SH sensor without the need to first compute the spatial gradients as with the
Further author information: (Send correspondence to E.B.)
E.B.: E-mail: [email protected]
C.d.V.: E-mail: [email protected]
standard centroid algorithms.9 This generalization can be applied in the case of small aberrations and preserves
the parallel linear least squares solution with distributed communication to guarantee global smoothness. In a
simulation study it is shown that this generalization leads to an improved accuracy of the SABRE wavefront
reconstruction for small aberrations, as provided in a closed loop setting, if compared to SABRE making use of
spatial gradient calculations via centroid type of algorithms.
This paper is organized as follows. The SABRE method is summarized in Section 2, where next to the original
WFR for SH slope measurements, the new option for reconstruction from the intensity distributions of the SH
sensor is introduced. The novel distributed solution is presented in Section 3. Both sections are concluded with
numerical results, comparing the two extensions, reconstruction from intensities on the one hand and distributed
reconstruction on the other, with the original centralized SABRE for SH slope measurements.
2. WAVEFRONT RECONSTRUCTION WITH B-SPLINES
In this section, a short overview to B-splines and their application for wavefront modelling in SABRE is given,
based on the proposal of De Visser and Verhaegen who presented SABRE as WFR method from spatial gradient
measurements.4 Next to the SABRE model for SH slope measurements, a linear relationship between the
intensity patterns in the detector plane of the SH sensor and the incoming wavefront represented by a multivariate
spline model is established leading to an extension of the methodology which makes direct use of the intensity
measurements. For both types of measurements, the SABRE method computes an estimate of the wavefront as
a constrained least-squares (LS) solution.
2.1 Preliminaries on B-splines
B-splines are defined locally on simplices, which correspond in the 2-dimensional case to triangles t, each defined
by three non-degenerate vertices (v0 , v1 , v2 ) ∈ R2×3 , and evaluated for (x, y) ∈ R2 with the Barycentric coordinate system. For a triangle t with vertices (v0 , v1 , v2 ), the Barycentric coordinates (b0 , b1 , b2 ) of a point (x, y)
in the Cartesian plane are given by
b1
−1 x
=V
, b0 = 1 − b1 − b2 ,
(1)
b2
y
with transformation matrix V = [v1 − v0 , v2 − v0 ]. On the triangle t, the Bernstein polynomials
(
d!
bκ0 bκ1 bκ2 , (x, y) ∈ t
d
Bκ (b(x, y)) = κ0 !κ1 !κ2 ! 0 1 2
0
, (x, y) ∈
/t
(2)
of degree d, with |κ| = κ0 + κ1 + κ2 = d and κ0 , κ1 , κ2 ≥ 0 give a local basis. A linear combination of the
Bernstein polynomials yields then the B-form polynomial
(P
t d
|κ|=d cκ Bκ (b(x, y)) , (x, y) ∈ t
p(b(x, y)) =
(3)
0
, (x, y) ∈
/t
of degree d on the simplex t. In order to obtain a global B-spline, a triangulation T of J adjacent simplices ti is
constructed on the considered domain as
T =
J
[
ti ,
ti ∩ tj = {∅, t̃} ,
∀ ti , tj ∈ T ,
(4)
i=1
where t̃ is either a vertex or an edge. With (3), a local B-form polynomial is defined on each triangle ti ,
ˆ
ˆ
i = 1, ..., J, by a local coefficient vector cti ∈ Rd×1 and a vector Bdti (b(x, y)) ∈ R1×d containing the local B-spline
(2+d)!
basis functions, where dˆ = 2d! is the number of basis functions per triangle.10 A global B-spline polynomial
on triangulation T is then obtained by
p(b(x, y)) = Bd (x, y)c ,
(5)
ˆ
with global coefficient vector c := [ct1 , ..., ctJ ] ∈ RJ d×1 and B-form vector Bd (x, y) := [Bdt1 (b(x, y), ..., BdtJ (b(x, y)]
ˆ
∈ R1×J d . The local B-form polynomials are joined to a smooth spline polynomial by enforcing continuity for the
first r derivatives of the spline at the edges of neighbouring simplices through equality constraints
Ac = 0 .
(6)
The smoothness matrix A contains equations establishing relationships between the basis coefficients of neighbouring simplices.11 The principle is visualised in Fig. 1, where on the left four spline polynomials pi , i = 1, ..., 4
are constructed on single triangles, which are then joined to an adjacent triangulation. On the domain covered
by this triangulation, a smooth spline polynomial p is obtained after the equality constraints on the coefficients
of the local polynomials pi are enforced.
Figure 1. Concept of B-splines: Local spline polynomials defined on triangles (left) are joint to a global, smooth polynomial
defined over a triangulation (right).
2.2 THE SABRE MODEL FOR (SHACK-) HARTMANN SENSORS
The SABRE method defines a triangulation on the SH subaperture array and the unknown wavefront φ(x, y) is
approximated at any point (x, y) in a the WFS domain with a B-spline polynomial
φ(x, y) ≈ Bd (x, y)c,
(7)
with Bd (x, y) the global vector of B-spline basis functions and with c the global vector of B-coefficients. In
order to retrieve the wavefront estimate of form (7) from SH measurements, a model of the relation between the
measurements and wavefront in terms of the coefficient vector c is needed.
2.2.1 Linear model for slope measurments
A SH sensor consists of an array of N apertures which sample the incoming wavefront by approximating the
local spatial slopes of the wavefront in each subaperture using the center of mass of the intensity measurements
collected by the corresponding area in the detector. This enables the formulation of the wavefront reconstruction
problem as a linear (least-squares) problem.
To obtain the SABRE model for SH slope measurements a tringulation is defined on the (reference) centers of
the subapertures in the SH arry as depicted in the left scheme of Fig. 2. The example of a Type II triangulation
is shown for which four triangles are defined in each rectangular unit marked by the subaperture centers. Note
that no requirements are posed on symmetry, alignment or shape of the SH array. For a vector s ∈ R2N ×1 of
local wavefront slopes, De Visser and Verhaegen4 formulated the SABRE sensor model as follows:
s = dBd−1 Pd,d−1
c + n,
u
(8)
with n ∈ R2N ×1 a residual noise vector. The spline regression matrix built with Bd−1 , the global basis function
matrix of degree d − 1 which is constructed according to Equ. (2), and with Pd,d−1
the full-triangulation de
u
Casteljau matrix which is constructed in a block diagonal fashion using per-triangle de Casteljau matrices.4
Type II Triangulation for SH slopes
Type II Triangulation for SH intensities
Figure 2. Type II triangulations represented by fine black lines for slope (left) and intensity (right) measurements of a
3 × 3 SH array (subapertures and measurements highlighted in grey). The support locations of the triangulation are
depicted as circels and given by the (reference) centers of SH array in case of slope and by subaperture corners in case of
intensity measurements. The corners of all defined triangles give the set of vertices.
2.2.2 Intensity model
The slope measurements provided by the SH sensor are obtained using the center of mass of the pixel pattern
computed with the centroid algorithm, hence not all the information present in the intensity patterns is used.
Silva et al.12, 13 recently proposed a novel method based on a decentralized linearization of the relationship
between the local wavefront aberrations in each subaperture and the corresponding intensity pattern in the
detector. This approach preserves the linearity of the WFR problem, and it makes direct use of the intensity
measurements.
Similar to the previous section, a triangulation is defined on vertices distributed in the SH pupil plane,
whereas now the support points are located at the corners of the subapertures as sketched on the right in Fig.
2. Provided that the aberrations are small, which is given in a closed loop AO system, one can assume that the
part of the wavefront seen by one subaperture has a minimal effect on the intensity pattern in the part of the
detector which corresponds to another subaperture. The literature14 provides a nonlinear model In (m) of the
intensity triggered in each detector pixel m by the local wavefront φn for the subapertures n = 1, ..., N . On each
subaperture n, a local SABRE model is defined as
φn (x, y) ≈ Bdn (x, y)cn
ˆ
ˆ
for local B-form matrix Bdn (x, y) ∈ R1×Jn d and local coefficient vector cn ∈ RJn d×1 , where Jn is the number
of triangles per subaperture. In order to obtain a linear relation between the coefficients cn and the intensity
distribution of subaperture n, a first order Taylor expansion around cn = 0 approximates the nonlinear model
in each pixel m = 1, ..., M :
∂In (m) T
In (m) ≈ In (m)|cn =0 +
cn .
(9)
∂cn cn =0
Hence, we obtain a linear model of the intensity distribution in ∈ RM ×1 of subaperture n in terms of the local
coefficient vector cn with
in = j n + J n c n + e n .
(10)
where the noise vector en ∈ RM ×1 accounts for both the approximation error of the linearization and the readT
out noise. The zero order term in (10) is defined as j n = [In (1)|cn =0 , ..., In (M )|cn =0 ] ∈ RM ×1 and the Jacobian
iT
h
ˆ
, ..., ∂In (M ) ∈ RM ×Jn d .
is the full matrix Jn = ∂In (1) ∂cn
cn =0
∂cn
cn =0
This procedure is repeated independently for each subaperture leading to a global SABRE model of the
wavefront as in (7) and to the global SABRE sensor model for SH intensity measurements
i = j + Jc + e ,
(11)
for which i, j and e ∈ RN M ×1 are simply a concatenation of their subaperture local counterparts and the global
ˆ
Jacobian is defined as J = diag[J1 , ..., Jn ] ∈ RN M ×J d . The blockdiagonal structure of J stems from the fact
that the imaging process is assumed to be not correlated between the subapertures.
2.3 Centralized least-squares solution
With the linear sensor models for SH slope measurements in (8) and for SH intensity measurements in (11)
introduced in the previous section, we can now define the global SABRE wavefront reconstruction problem as
minimize ky − Dck22
subject to Ac = 0 ,
(12)
ˆ
where c ∈ RJ d×1 is the coefficient vector of the global SABRE wavefront model (7). For SH slope measurements,
ˆ
we set regression matrix D := dBd−1 Pd,d−1
∈ RJ d×2N and measurement vector y := s ∈ R2N ×1 ; for intensity
u
ˆ
measurements it is D := J ∈ RJ d×M N and y := i − j ∈ RM N ×1 . The global constraint matrix A enforces
smoothness on the obtained estimate. The constraint optimization problem in Equ. (12) leads to the following
least-squares estimator for the B-coefficients of the simplex B-spline:
−1 T
ĉ = NA (DT
DA s
A DA )
with DA := DNA ,
(13)
where NA is the projection matrix onto the null space of constraint matrix A. The SABRE reconstrucion matrix
−1
Q := NA (DT
DA can be precomputed for a given geometry. The estimated wavefront values can now be
A DA )
computed from SH measurements y as
φ̂(x, y) = Bd (x, y)Qy
(14)
at locations (x, y) for a preevaluated spline B-form matrix Bd (x, y). Note that the B-form matrix and hence the
wavefront estimate can be evaluated at any location in the triangulation (see Fig. 2).
2.4 Numerical results
In a numerical experiment, the two versions of the spline based method for SH slope measurements (SABRE-S)
and SH intensity measurements (SABRE-I) are compared in terms of reconstruction accuracy.
Residual RMS of global SABRE for slopes vs. intensities
2
10
Slopes, d=1, r=0
Intensities, d=1, r=0
1
10
RMS residual [rad]
0
10
−1
10
−2
10
−3
10
−4
10
−6
10
−4
10
−2
10
RMS aberration [rad]
0
10
2
10
Figure 3. Comparison of the average residual RMS as a function of aberration strength between SABRE-I using intensities
and SABRE-S using slope measurements.
A Fourier based Hartmann sensor was used to simulate measurements from a grid of 10 by 10 subapertures
with 25 pixels per subaperture side. The sensor was considered to suffer predominantly the effects of read-out
noise on the detector, which was modelled as white and Gaussian with zero-mean and of standard deviation
σnoise . The noise is additive and affects the normalized intensity distribution values, which are then used for
reconstruction with SABRE. To apply the original version of SABRE for slope measurements the local gradients
are computed with the centroid algorithm from the same intensity distributions.
With this simulation, we show the superior performance of SABRE-I in comparison with SABRE-S for slopes
in the small aberration range. Wavefronts perturbated by aberration of increasing strength, represented in
terms of their RMS values from 10−6 rad to 102 rad, were generated according to the Kolmogorov atmospheric
turbulence model. For each aberration, we computed 100 realisations of Hartmann intensity measurements which
are corrupted by noise of standard deviation σnoise = 4 × 10−4 . A SABRE model using B-Splines of degree d = 1
and continuity order r = 0 was defined on a Type II triangulation (i.e. four triangles per rectangular unit as
depicted in Fig. 2).
In Figure 3, the averaged RMS residual of the reconstructed wavefronts is plotted over the RMS values of
the uncorrected wavefront, comparing the results of SABRE-S with SABRE-I. It shows that SABRE-I yields an
RMS error approximately 1 order of magnitude lower than SABRE-S for aberrations with an RMS value in the
range of 10−3 rad to 1 rad. For aberrations of RMS larger than 10 rad, the diffraction pattern corresponding
to a subaperture will affect the intensity pattern originating from other subapertures. In that case, the locality
assumption presented in Section 2.2.2 is not verified and the SABRE-I performs poorer than reconstruction from
slope measurements. Due to the influence of read out noise on the intensity measurements, the RMS error values
for both versions of SABRE reach a lower threshold when the aberrations imposed are smaller than 0.01 rad.
3. DISTRIBUTED WAVEFRONT RECONSTRUCTION WITH SABRE
In this section, we discuss an extension of SABRE which is suitabe for distributed WFR and introduce a 2 stage
algorithm that forms the spline based distributed wavefront reconstruction method (D-SABRE).3 In the last part
of this section, the results from numerical experiments with the D-SABRE are presented. These experiments are
aimed at validating the D-SABRE method by comparing its reconstruction accuracy to the centralized SABRE.
3.1 Two stage method
The D-SABRE method is based on the decomposition of the global WFR problem from (12) into a set of local
sub-problems. First, we decompose the global triangulation T into a set of G sub-triangulations as follows:
T =
G
[
Ti ,
(15)
i=1
where every sub-triangulation Ti in turn consists, as depicted in Fig. 4, of a core part Ωi and an overlap part Ξi
for i = 1, ..., G such that
Ti = Ωi ∪ Ξi and Ωi ∩ Ξi = ∅ .
(16)
The purpose of Ξi is to overlap neighboring sub-triangulation core parts Ωj in order to increase numerical continuity between neighboring partitions obtained with the local WFR that is performed in parallel and presented
in the next section. The term Overlap Level (OL) defines the size of Ξi . The OL is a scalar, which determines
how many layers of triangles from the neighboring core partitions are included in Ξi as illustrated in Figure 4.3
3.1.1 Local SABRE reconstruction
In the first stage of the D-SABRE method, we approximate the wavefront locally on each sub-triangulation Ti
as follows:
φi (x, y) ≈ Bdi ci , 1 ≤ i ≤ G ,
(17)
where φi (x, y) is the local wavefront phase, Bdi (x, y) the local matrix of B-form regressors and ci the set of local
B-coefficients.
To determine ci for all G local models the following local SABRE WFR problems have to be solved, which
can be done in a fully parallel manner:
minimize ky i − Di ci k22
subject to Ai ci = 0,
1 ≤ i ≤ G,
(18)
Figure 4. An OL-0 partitioning without overlap (left) and an OL-2 partitioning with 2 levels of overlap between partitions
(right) using the same initial triangulation containing 200 triangles.
where the local measurement vector y i and system matrix Di are given by the respective sensor models defined
according to Section 2.2.2. Note, that the local constraint matrices Ai are created for each subproblem i and
cannot be obtained by decomposing the global constraint matrix A into G blocks because it is not block diagonal.
Therefore Ai does not contain any smoothness conditions linking a partition i to any other partition.
The B-coefficients ci of the local spline functions can now be estimated from the local set of WFS measurements y i using the local least squares estimator3
−1 T
ĉi = NA,i (DT
DA,i y i
A,i DA,i )
with DA,i := Di NA,i ,
(19)
−1 T
where the local SABRE reconstruction matrix Qi := NA,i (DT
DA,i and NA,i is the null space projector
A,i DA,i )
of the local constraint matrix Ai . The local estimate (19) does not depend on information from any other
partition, and as a result, each of the G local reconstruction problems can be solved in parallel.
3.1.2 Distributed piston equalisation and smoothing procedure
Applying the distributed local WFR from Section 3.1.1 results in a discontinuous global wavefront, like it is
shown in the upper right plot in Fig. 5. In order to equalize the phase offsets between neighboring D-SABRE
partitions, a process called Distributed Piston Mode Equalization (D-PME) was introduced by De Visser.3 The
D-PME offsets an entire D-SABRE partition i with a single constant ki such that the maximum phase offset
between it and a neighboring partition is minimized. This procedure consists of a distributed iterative algorithm
which takes advantage of a key property of B-splines. The B-coefficients of the local basis in each triangle
correspond to a location in this geometrical unit. The values of the spline function at all vertices are equal to
the value of the coefficient located at those. Hence there is no need to evaluate the local spline polynomials after
the first stage to obtain the values of wavefront estimates φ̂i (x, y) at the edges of the partitions. The respective
B-coefficients computed in (19) can directly be used to compute the piston offsets between the wavefront patches.
The algorithm for the iterative D-PME forms Stage 2 of D-SABRE and is performed in each partition i by
the following algorithm. Let cΩi,m ∈ RJi,m be the vector of Ji,m B-coefficients located at vertices in the core
part of partition i which are shared with the core part of neighboring partition m ∈ Mi , where Mi is the index
set of all neighbor partitions of partition i. As the core parts of neighboring partitions do not overlap, these
vertices are located at the edge of both core parts. Now be cΩm,i a vector containing the respective B-coefficients
computed in partition m at the vertices shared with partition i. After fixing the coefficient vector of one partition
as reference, the distributed D-PME algorithm given by
di,m = cΩi,m (l) − cΩm,i (l) ,
ki (l) =
max {d¯i,m } ,
m∈Mi ,m>i
ci (l + 1) = ci (l) + ki (l) ,
∀m ∈ Mi , m > i
(20)
(21)
(22)
Figure 5. Top: Original wavefront (left); The D-SABRE model after completion of the distributed local reconstruction
stage(right). Bottom: D-SABRE model after completion of D-PME stage, with remaining discontinuities exaggerated
(left); D-SABRE model after completion of the distributed post-smoothing stage (right).
with d¯i,m the average of the piston offsets di,m at the shared vertices, is applied in each partition i and converges
after l = L iterations. In the lower left plot of Fig. 5, the D-PME method has been applied to the discontinuous
reconstructed wavefront obtained with Stage 1.
The D-PME minimizes the unknown phase offsets between local D-SABRE partitions, but in general does
not create continuity at the edges of the partitions. In order to obtain a globally smooth wavefront, a Distributed
Dual Ascent Smoothing (D-DAS) procedure which is based on a dual decomposition approach15 was proposed
for Stage 2 of the D-SABRE.3 The D-DAS not only eliminates the piston offsets, but also enforces smoothness
between local D-SABRE partitions and is defined as follows:
−1
ci (l + 1) = ĉi (l) − (DT
(Hi )T y i (l) ,
A,i DA,i )
(23)
y i (l + 1) = y i (l) + α(l)Hi,M ci,M (l + 1)) ,
(24)
where ĉi is defined as the local LS estimate computed with (19) in Stage 1 and DA,i is the local system matrix
of partition i projected onto the nullspace of local constraint matrix Ai . Two submatrices Hi and Hi,M of the
global constraint matrix A are constructed for each partition i. The submatrix Hi contains blocks that only
influence B-coefficients inside the partition i. The second submatrix Hi,M contains all blocks of A that influence
B-coefficients inside the partition i as well as B-coefficients in neighboring partitions m through the action of
the continuity conditions. We define y i as the dual vector of constraints in A affecting the local B-coefficient
vector ci . Vector ci,M contains all B-coefficients in partition i as well as all B-coefficients in directly neighboring
partitions that are subject to continuity conditions linked to partition i.
Even though more exchange of information is needed for D-DAS than for D-PME in the second post smoothing
stage, the procedure does not require global communication and is fully distributed. The results presented in
the following section were obtained with an updated version of the D-DAS procedure which includes the D-PME
principle into the dual ascent smoothing to speed up the convergence of the latter by merging the operations
from (20) to (24) (publication in preparation16 ). From this point, the D-DAS algorithm is considered as an
optional extension to the D-PME procedure, which is called if a globally smooth wavefront estimate is asked for.
3.2 Numerical results
The D-SABRE algorithm is validated with a numerical experiment in which its reconstruction accuracy is
compared to that of the global SABRE method.4 For this experiment, a Fourier optics based Shack Hartmann
lenslet array is used to obtain wavefront slopes from a set of 100 simulated wavefronts. The SH lenslet array
consists of 2500 lenslets, laid out in a 50 × 50 grid. For each of the 100 wavefront realizations, 10 different signal
to noise ratio (SNR) values are used. The local SABRE models were defined on a Type I triangulation (i.e. two
triangles per rectangular unit instead of four as in Fig. 2) using B-Splines of degree d = 1 and continuity order
r = 0.
Residual RMS of DSABRE vs. global SABRE
1
10
Strehl ratio of DSABRE vs. global SABRE
1
0.9
0.8
0
0.7
RMS [rad]
Strehl ratio [−]
10
−1
10
0.6
0.5
0.4
0.3
global SABRE
DSABRE − No D−DAS (OL−0)
DSABRE − No D−DAS (OL−1)
DSABRE − No D−DAS (OL−4)
DSABRE + D−DAS (OL−1)
DSABRE + D−DAS (OL−4)
−2
10
0
10
20
30
40
Signal to Noise Ratio [dB]
global SABRE
DSABRE − No D−DAS (OL−0)
DSABRE − No D−DAS (OL−1)
DSABRE − No D−DAS (OL−4)
DSABRE + D−DAS (OL−1)
DSABRE + D−DAS (OL−4)
0.2
50
60
0.1
0
10
20
30
40
Signal to Noise Ratio [dB]
50
60
Figure 6. Comparison of average residual RMS (left) and average Strehl ratio (right) as a function of signal to noise ratio
for different D-SABRE variants.
In Figure 6, the average Strehl ratio and the average residual RMS for the 100 reconstructions obtained with
D-SABRE are compared to that of the global SABRE as a function of the SNR. We consider a decomposition
of the global triangulation into 50 partitions. D-SABRE was applied for different overlap levels (OL) and for 30
iterations of either D-PME for piston offset minimization only or for the same number of iterations of D-PME
plus additional smoothing with D-DAS. From these figures it can be seen that the D-SABRE is for all versions
less resilient to noise than the SABRE, but significant performance differences only start to occur for SNR
values below 10dB. Furthermore, it shows that the overlap level has a significant influence on reconstruction
accuracy and noise resilience. By including the D-DAS step in Stage 2 the reconstruction performance is further
improved, but the experiment shows that the improvement obtained by increasing the OL from 1 to 4 can
exceed the benefit of including the D-DAS step. This is also of advantage in a computational sense, since the
communication between the processors which is necessary for the D-DAS step could, for certain parallel hardware
as GPUs, be a bigger limitation to the speed up compared to the additional computational load in each processor
for a local WFR with bigger values for OL.
In Figure 7, the experiment was repeated for D-SABRE with different numbers of partitions in the decomposition of the triangulation and the results compared to the global SABRE. An OL of 1 triangle was used
and 30 iterations of D-PME with D-DAS were performed in the second stage. It can be seen that D-SABRE
shows decreasing resilience to noise for increasing number of partitions, but in all three cases of 25, 100 and
625 partitions significant loss in performance only shows for SNR values below 15 dB. At a signal to noise ratio
of 15 dB, D-SABRE performs with a Strehl ratio of ≈ 0.9 for 625 partitions, which corresponds to WFR on
subtriangulations of only 4 triangles compared to 2500 triangles for the global SABRE, and a drop of only 7
percent points in Strehl compared to 25 partition.
Residual RMS of DSABRE vs. global SABRE
1
10
Strehl Ratio of DSABRE vs. global SABRE
1
0.9
0.8
0
0.7
RMS [rad]
Strehl ratio [−]
10
−1
10
0.6
0.5
0.4
0.3
global SABRE
DSABRE (25 partitions)
DSABRE (100 partitions)
DSABRE (625 partitions)
Uncorrected Wavefront
−2
10
0
10
20
30
40
Signal to Noise Ratio [dB]
global SABRE
DSABRE (25 partitions)
DSABRE (100 partitions)
DSABRE (625 partitions)
0.2
50
60
0.1
0
10
20
30
40
Signal to Noise Ratio [dB]
50
60
Figure 7. Comparison of average residual RMS (left) and average Strehl ratio (right) as a function of signal to noise ratio
of D-SABRE for different number of partitions.
4. CONCLUSION
In this paper, two extensions of the wavefront reconstruction method SABRE (Spline Based ABerration REconstruction) from SH measurements are presented, which aim at reducing the computational complexity of the
method on the one hand and at increasing the accuracy of the reconstruction on the other hand.
Firstly, we discussed a novel method indicated as D-SABRE (Distributed-SABRE) which is based on a
decomposition of the entire SH sensor domain into a set of G sub-domains each supporting a local multivariate
spline model. D-SABRE consists of two stages which are both distributed operations. The first stage is a fully
decentralized linear least squares calculation of the local SABRE models using local wavefront measurements
only. In stage two, iterative distributed optimization requiring communication with direct neighboring models
treats the piston offset between the local reconstructions and the smoothness of the full wavefront profile over
the total SH aperture. The decentralized and distributed solutions can run on multi-core hardware, such as
standard GPU processors. When G partitions have been defined with one processor for each submodel, then
D-SABRE can obtain a speedup factor of O(G2 ) over global centralized, single processor methods for large scale
wavefront reconstruction problems. The method was demonstrated on simulated E-ELT turbulence wavefronts
and a comparison of the results with the centralized SABRE algorithm is done. We showed that the D-SABRE
obtains a reconstruction accuracy that is within 1% of the reconstruction accuracy of a global (non-distributed)
method. Different versions of the D-SABRE method were tested for reconstruction accuracy and the resilience
to noise was analysed for increasing number of partitions in the sensor domain which corresponds to the number
G of processors on which the computational load can be distributed.
Secondly, a generalization of the SABRE method to intensity measurements of the SH sensor was presented
which eliminates the need of spatial gradient computation. A distributed linearization of the model describing
the imaging process in each subaperture allows the estimation of the wavefront from the intensity distribution in
the detector with a linear least squares solution which is valid for small aberrations. This second extension was
demonstrated in simulation for a Hartmann sensor and compared to SABRE for slope measurements. Making
directly use of the intensity distribution without loosing information in the centroid computation to approximate
the spatial gradients, we could show a clear improvement of the reconstruction accuracy for the mentionned
aberration range in comparison with SABRE for slope measurements. Due to the subaperture local nature of
the linearization procedure, this generalization of SABRE to intensity measurements can also be integrated with
the distributed D-SABRE method.
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