Adaptive optics implementation with a Fourier reconstructor.
ABSTRACT Adaptive optics takes its servo feedback error cue from a wavefront sensor. The common Hartmann-Shack spot grid that represents the wavefront slopes is usually analyzed by finding the spot centroids. In a novel application, we used the Fourier decomposition of a spot pattern to find deviations from grid regularity. This decomposition was performed either in the Fourier domain or in the image domain, as a demodulation of the grid of spots. We analyzed the system, built a control loop for it, and tested it thoroughly. This allowed us to close the loop on wavefront errors caused by turbulence in the optical system.
- SourceAvailable from: E. N. Ribak[Show abstract] [Hide abstract]
ABSTRACT: Recently we developed and tested different algorithms for wave front reconstruction from dense Hartmann-Shack patterns. All depend on the recognition of a main frequency in the patterns, whose distortion from wave aberrations can be construed as slight phase changes in the pattern. An alternative description of these aberrations is a slight frequency change in Fourier domain. The slopes can thus be found by demodulation in either the image or the Fourier domain. These slopes can then be integrated in the Fourier domain again for the wave front itself. For smooth slopes both demodulation and integration can be performed in the Fourier domain. In addition, commands for the adaptive optics loop can be taken directly in the Fourier domain, saving on processing time. We modeled and tested these algorithms thoroughly in simulation and in laboratory experiments on two separate adaptive optics systems.Proceedings of SPIE - The International Society for Optical Engineering 07/2006; · 0.20 Impact Factor
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ABSTRACT: The estimation of the centroid offset can have an effect on the accuracy of wavefront measurements conducted by highly sensitive Shack-Hartmann wavefront sensors. In this paper, a novel offset estimation algorithm processed in the Fourier domain is proposed. This method can be used to process the offset estimation in the Fourier domain and is efficient in noise suppression. The principle of the algorithm is described in detail. Comparisons between the technique and two other widely used algorithms, the best-threshold center of gravity algorithm and the correlation algorithm, are performed theoretically using numerical simulation and experimentally using a Shack-Hartmann wavefront sensor. The results show that the proposed offset estimation algorithm is unbiased, as robust as the correlation algorithm, as fast as the best-threshold center of gravity algorithm, and achieves a good balance between precision and speed.Journal of Optics 05/2013; 15(5):5702-. · 2.01 Impact Factor
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ABSTRACT: The process of Zernike mode detection with a Shack-Hartmann wavefront sensor is computationally extensive. A holographic modal wavefront sensor has therefore evolved to process the data optically by use of the concept of equal and opposite phase bias. Recently, a multiplexed computer-generated hologram (CGH) technique was developed in which the output is in the form of bright dots that specify the presence and strength of a specific Zernike mode. We propose a wavefront sensor using the concept of phase biasing in the latter technique such that the output is a pair of bright dots for each mode to be sensed. A normalized difference signal between the intensities of the two dots is proportional to the amplitude of the sensed Zernike mode. In our method the number of holograms to be multiplexed is decreased, thereby reducing the modal cross talk significantly. We validated the proposed method through simulation studies for several cases. The simulation results demonstrate simultaneous wavefront detection of lower-order Zernike modes with a resolution better than lambda/50 for the wide measurement range of +/-3.5lambda with much reduced cross talk at high speed.Applied Optics 11/2009; 48(33):6458-65. · 1.69 Impact Factor
Adaptive optics implementation with a Fourier
Oded Glazer, Erez N. Ribak, and Leonid Mirkin
Adaptive optics takes its servo feedback error cue from the wavefront sensor. The common Hartmann–
Shack spot grid, representing the wavefront slopes, is usually analyzed by finding the spot centroids. In
a novel application, we used the Fourier decomposition of the spot pattern to find deviations from grid
regularity. This decomposition was performed either in the Fourier domain or in the image domain, as
a demodulation of the grid of spots. We analyzed the system, built a control loop for it, and tested it
thoroughly. This allowed us to close the loop on wavefront errors caused by turbulence in the optical
system.© 2007 Optical Society of America
010.1080, 010.7350, 100.2650, 100.5070.
Adaptive optics deals with sensing of a distorted
wavefront and its correction. There is a variety of
applications for adaptive optics, for instance in astro-
nomical observations where we want to discern as-
tronomical objects. The optical signal passes through
free space without distortions, but once it encounters
the earth’s atmosphere the beam is affected and ab-
errated. There is a need to compensate for the refrac-
tive index variation produced by thermal air streams
that make the atmosphere turbulent.1Adaptive op-
tics is also being used for military and ophthalmic
Wavefront correction is achieved by applying a se-
ries of calculated reconstruction commands on the
phase modulators. The phase modulator is a device
beam by adding a refractive or reflective delay, in our
case using a membrane deformable mirror. The force
on the membrane is affected by a number of parallel
actuating capacitors to which voltage is applied. The
data for the reconstruction is based on a Hartmann–
Shack wavefront sensor which is essentially a lenslet
array, sampling the wavefront slope. This correction
process must be achieved within a few milliseconds in
order to improve the image quality before the aberra-
tions change significantly. In the future, with the ad-
vent of sensitive and low-noise detectors, with the
development of larger aperture mirror telescopes with
thousands of actuators, large deformable secondary
mirrors, and huge telescope structures we need to de-
velop faster real-time compensating algorithms to
meet these demands, and more efficient schemes for
phase compensation must be built. This work is in-
tended as a step in this direction.
Currently adaptive optics systems use centroid
method reconstructors to convert gradient measure-
ments to wavefront phase estimates, where local
slopes move beams from their nominal positions and
correction is made according to this movement.3Most
works have utilized a least-squares control algorithm
and predetection compensation to minimize the wave-
front error in an optical system.4–7The purpose of
this work was to test for what is believed to be the
first time the implementation of two alternative
approaches: Fourier demodulation8and direct demo-
dulation of the spot pattern9in order to find the
wavefront slope.10This is multiplied, as in the con-
matrix to produce a vector of mirror commands.
In general, the reconstruction process involves two
consecutive stages: (1) calculation of the wavefront
slopes using the Hartmann spots, or the wavefront
and (2) calculation of the wavefront itself or the mir-
ror positions, if the deformable mirror is of the piston
type. For other phase modulators, such as the bi-
morph mirror, stage (2) provides the mirror com-
The authors are with Technion, Israel Institute of Technology,
Haifa 32000, Israel. O. Glazer and L. Mirkin are with the Faculty of
Mechanical Engineering. E. Ribak (firstname.lastname@example.org) is
with the Faculty of Physics.
Received 13 April 2006; revised 3 October 2006; accepted 3 Oc-
tober 2006; posted 5 October 2006 (Doc. ID 69931).
© 2007 Optical Society of America
1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS1
mands. Until now, Fourier analysis was only applied
in the second stage for integration of the slopes,11,12
whereas now we test its real-time employment in the
first stage as well. A full Fourier approach is also
possible for well-behaved wavefronts where both
stages are performed in the Fourier domain: The
Hartmann data are transformed, the wavefront gra-
dient isolated and integrated, and the phase is ob-
tained by an inverse transform. This approach was
applied for wavefront calculation,13but still has to be
tested for adaptive optics. Its obvious advantage is
the absence of the laborious multiplication by the
reconstructor matrix, prohibitively large for extended
systems. Since our mirror has a polar and not a Car-
tesian symmetry, we did not employ this option.
We start our description with the introduction and
investigation of the techniques we used for the con-
trol scheme and also define a discrete formalism for
the deformable mirror in the linear-discrete regime.
Next we turn to the optical setup and the other main
topic of this paper, detailing the sensing analysis
methods that we applied.
Our adaptive optical system is basically a laser beam,
which gets distorted in the optical system, reflects off
a deformable mirror corrector, and is measured in
a wavefront sensor. Our goal here is to flatten the
wavefront according to a premeasured reference. The
wavefront is measured through an image processing
algorithm described below. The control of the wave-
front is thus actually performed in terms of its
2 ? 25 ? 50 slope components, which are compared
with the corresponding slopes of the reference wave-
front. The corrections of the wavefront are achieved
by the use of a deformable mirror, the shape of which
can be affected by an array of 37 electromechanical
actuators, which, in turn, are controlled through in-
put voltage V. This system can be presented by the
block diagram in Fig. 1.
Note that although the electromechanical actua-
tors are nonlinear, this nonlinearity is known: The
stroke is proportional to the square of the input volt-
age. Thus we can linearize the actuators’ responses
by inverting (precompensating) the actuator nonlin-
earity in the control law. Therefore we analyze the
system assuming that the actuation part is linear
(the resulting controller is always complemented by
the inverse of the actuation nonlinearity). Also, the
dynamics and settling time of the actuators are much
faster than the achievable controller bandwidth,
which is limited by the computational time required
to perform the image processing step. Thus we can
safely neglect all dynamical effects in the actuators
and absorb the first two blocks in Fig. 1 into the
controller. Consequently, the controller design below
will address only the last two blocks, the wavefront
sensor and the signal processing unit. These blocks
will be united into a single, static system, H (the
Hartmann–Shack sensor), which is a 50 ? 37 recon-
Flattening the wavefront is complicated by the
presence of disturbances and unmodeled effects and
nonlinearities, such as residual mirror hysteresis,
membrane response to sound and vibrations, and
thermal effects. To cope with these effects and com-
pensate them, a feedback control law shall be used.
The most important property of feedback is its ability
to cope with modeling errors, external disturbances,
In our system, the feedback control can be imple-
mented by assigning the input voltage to the actua-
tors on the basis of the mismatch between the actual
and desired wavefronts, w and r. An important point
here is that we cannot generate any wavefront as we
have only 37 actuators to control 50 variables (an
underactuated system). We therefore can track ref-
erences from the 37-dimensional image space of H
only. This can be interpreted as imposing the con-
straint r ? Hraon the reference signal r, where rais
arbitrary. To simplify the design of the feedback con-
troller, or more precisely, to decouple the control loop,
we premultiply the tracking error, r ? w, by the
pseudoinverse of H (Ref. 14) H#? ?H?H??1H?, before
processing it by the controller. Since H#?Hra? w?
? ra? H#w, this is equivalent to tracking raby a
filtered wave front H#w.
Thus the data from the Hartmann–Shack sensor
are multiplied by H#, compared with the chosen ref-
erence ra, and this difference is processed by a digital
controller C1?z? acting at the sampling rate of 30 Hz,
as dictated by the video rate (Fig. 2). Signal d there
reflects external disturbances and unmodeled effects.
Signal n reflects the measurements noise, which is
brought about by the measurement device, namely,
the noise and the background signal in the CCD sen-
sor. Our goal here is to design C1?z?, which stabilizes
the closed-loop system and makes the tracking error
e ? ra? H#w small despite the presence of d (which
is typically relatively slow) and n (which is typically
The system in Fig. 2 is not yet determined as we do
not know the relation between the control variable u
and the processed wavefront w, the wavefront slopes
or their Fourier low frequencies. Thus the first step of
Calibration (Identification of H)
Fig. 1. The plant (controlled process) and the sensor.
Fig. 2.Block diagram of the feedback control system.
2APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007
the proposed scheme is to identify this relation, i.e.,
the operator H.
The operator H is constructed in the calibration
process. We identify it by giving the maximal voltage
to each separate actuator, one at a time, and the
wavefront of each of these pokes is measured by cal-
culating the Fourier demodulation of the spot pat-
tern. The maximal voltage for each poke is chosen to
reduce errors attributable to the presence of the mea-
surement noise. Each step of this procedure immedi-
ately yields the column of H corresponding to the
exciting actuator. Notice that we simply use Fourier
modes, namely the Fourier components of the gradi-
ent of the wavefront, in contrast with other processes
employing gradients of Zernike modes, and in other
cases atmospheric modes.
It is readily verified that the closed-loop tracking er-
ror e ? ra? y satisfies
where S?z? ? ?I ? C1?z???1and T?z? ? ?1 ?
C1?z???1C1?z? ? I ? S?z? are the sensitivity and com-
plementary sensitivity transfer matrices, respectively.
In the choice of the controller C1?z? the following re-
quirements were taken into account:
Simplicity: Since the feedback control law is to
be implemented in real time, we cannot afford com-
plicated (i.e., high-order) controllers, the implemen-
tation of which would be numerically demanding.
Set-point tracking: Since the reference vector rz
is assumed constant, we require a zero steady state
error, limt→?ez?t? ? 0, for all constant rzand dz. This
amounts to requiring that the static gain of the sen-
sitivity transfer matrix be zero, S?1? ? 0, or, equiva-
lently, that the static gain of the controller, C1?1?, be
Low sensitivity to measurement noise: The
measurement noise nzis typically fast, so to prevent
its amplification by the feedback loop we require that
the magnitude of the complementary sensitivity fre-
quency response, T?ej??, ? ? ???, ??, at high frequen-
cies be low (smaller than 1). This requirement can be
cast as |T??1?| ? ?, for some ? ? 1, i.e., as a con-
straint on |T?z?| at the highest frequency.
The first two requirements suggest the following
digital decentralized proportional-integral (PI) form
of the controller:
where kpis the proportional gain, and kais the accu-
mulator (the presence of the integrator term guaran-
tees that the static gain of this controller is infinite).
Note that C1?z? has a pole at the origin to reflect
inevitable computational delay.
Some lengthy, albeit straightforward, algebra yields
then that the third requirements, combined with the
standard closed-loop stability requirement, imposes
the following constraints on the controller parameters:
1??, if ??1
The gray areas in Fig. 3 show admissible pairs ?kp, ka?
for several values of ?.
Note that the stability conditions are actually re-
covered as ? → ?. As ? decreases, the admissible
areas shrink. The requirement of a small high-
frequency gain is especially restrictive in the first
quadrant, where the proportional gain cannot be in-
creased much. Since a low proportional gain gives
rise to slow transient response, this essentially im-
plies that negative controller coefficients are prefer-
able. Indeed, in the third quadrant we can still use a
high proportional gain, even if ? is small.
To test the system and control, we built a simple
optical system (Fig. 4). The measurement was per-
formed by using a lenslet array, and compensation
was produced by a membrane mirror (OKO Technol-
ogies 37-channel electrostatic mirror). This was a
part from a package from Adaptive Optics Associates,
Inc. (AOA), which also included a Tk?Tcl language
program for Hartmann–Shack wavefront sensing
and for adaptive optics closed-loop control and diag-
nostic graphics. We first blocked the deformable mir-
ror arm and took a reference image for calibration of
the wavefront sensor itself. Then we blocked the ref-
erence arm and measured the aberrations in the
?kp, ka? plane.
Stability and high-frequency performance areas in the
1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS3
other arm. To calibrate the system, we poked each
mirror element in turn and measured the wavefront.
Alternatively, we poked a whole mode (a combination
of elements) to find its response. We inverted the
responses by using singular value decomposition, to
construct a matrix response to any given wavefront
through application of voltage to the relevant actua-
tors. After testing the system by correcting for static
errors, we moved on to dynamic correction: A hair
dryer was operated at lower voltage to create weak
turbulence across the optical path in front of the
deformable mirror. In all cases full correction was
The common method today for constructing the wave-
front phase distortion is by sampling its local slopes
using a lenslet array and applying a digital closed-
loop process. The slopes are measured at the focal
plane of the lenslet array as foci shifts. A centroiding
algorithm3finds the location of each focal spot, from
which the whole map of slopes is constructed, to serve
as inputs for the control loop.
To calculate the local shifts of the foci we also used
two new methods: First, we performed a Fourier de-
modulation of the Hartmann–Shack pattern.8This
method extracts the essential data of the wavefront
behavior, namely the shifts of focal spots, by consid-
ering them as modulation of the full regular spot
pattern. Its disadvantage is the requirement for a full
Fourier transform at all possible frequencies, even
irrelevant ones. Second, we conducted a direct de-
modulation of the foci frequency only, without revert-
ing to Fourier transforms. This helped us to reduce
the processing times.9
For the first method, we used a 2D fast Fourier
transform (FFT) to obtain the slope of the wavefront
phase.1,15That slope appears in the complex side
lobes of the transform, where these side lobes corre-
spond to the lenslet period. One can try to retrieve it
from the real, imaginary, amplitude, phase, or her-
mitian parts of the transform.8,13While the hermi-
tian option is theoretically the best, it requires an
integer frequency of spots, which was not possible
for us, or Fourier plane interpolation, not available
through the given software. Instead, we chose exper-
imentally the most suitable part of the transform
by realizing that it should have the highest rms
response to the same set of input wavefront aberra-
tions. The 200-odd Hartmann–Shack spots were im-
agedwithaCCDcameraholding480 ? 640 pixels.To
avoid aliasing we removed the side columns and
added empty rows to create a 512 ? 512 pixel array.
Not surprisingly, the amplitude part of the transform
yielded the highest response to the aberrations. Un-
fortunately, the loss of phase information inhibited
their use, and indeed an adaptive optics control loop
became unstable. Instead, we chose the imaginary
part of the transform, which proved to be much more
reliable and stable.
To build the reconstruction matrix, we tried a num-
ber of schemes, and settled on a poke reconstructor:
The response of the wavefront to a constant voltage
on each actuator resulted in a different transform
pattern. We took the imaginary part of the two side
lobes corresponding to the x and y slopes in the trans-
form and stacked them one on top of the other for
display purposes (Fig. 5), or staggered them into a
linear array as a column input. Each poked actuator
gave such a column, and their combination provided
the response of the system. That response matrix has
the width of the number of actuators (37), and the
height of the combined number of pixels in the two
side lobes (50). Each such matrix element contained
the imaginary part of the Fourier component of the
wavefront slope. From it was subtracted the refer-
ence response matrix without any voltage applied to
the actuators (the calibration frame). The arrange-
ment of the data in the slope vectors is not important
but should be kept consistent between calibration
current measurement of the wavefront (less a reference array).
x and the y slopes. On top is a square of 25 pixels of the 0, 1 lobe,
below it the square of the 1, 0 lobe.
(Color online) Selected part of the Fourier transform of the
from the bottom right and splits (in beam splitter BS2) into either
a reference channel or a deformable mirror channel. Light from
the unblocked arm returns into the wavefront sensor (WFS:
Hartmann–Shack lenslet array and camera). An image of the cor-
rected laser beam is split (in beam splitter BS1) into another
camera. Turbulence is added between the lenses in front of the
The optical system. A collimated, wide laser beam enters
4 APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007
and operation. Next we used a singular value decom-
position (SVD) algorithm to determine the pseudoin-
verse of the response matrix. Thresholding removed
the weaker eigenvalues of the inverse. Thus the pro-
cedure to construct a reconstructor matrix is the
same as for centroiding, except that the Fourier
modes of the slopes are used and not the slopes them-
In running the adaptive optics loop, each Hartmann–
Shack frame is Fourier transformed, and the same
vector is formed from its two orthogonal side lobes
(again, the imaginary values of 25 elements around
the main frequency in each direction). This vector is
next mirror commands for the controller.
Time is a critical factor in a fast-changing system,
and thus a minimum of CPU cycles is required to
close the adaptive optics loop. Even though the Fou-
rier transform is very efficient, we have just seen that
we may use a very small number of Fourier compo-
nents out of the many calculated. For example, we
were able to close the loop with 25 x-lobe and
25 y-lobe Fourier components, out of the 5122?
262, 144 in the full frame. It makes more sense to
the frequency k0of the Hartmann spots, we multiply
by ejk0x, and integrate by smoothing the complex
result. Smoothing can be achieved by convolution
with a kernel the width of the Hartmann pitch,
which is too slow for a real-time process. Instead,
we shifted and added the complex array to half its
values, and then again at half the pitch.9This is
equivalent to finding only the k0harmonic in one
direction. In a temporal sense this is similar to super-
heterodyne or locked-in detection. Finally we ex-
tracted the phase of the smoothed array. This phase
is the x component of the slope. The process was
repeated for the y direction. The results are shown in
The analysis is performed in the image domain and
hence is very fast.9As the resulting number of points
(slope components) is equal to the number of cam-
era pixels inside the aperture, much larger than the
number of actuators, coarser smoothing is possible.
Following the smoothing and phase extraction, we
simply sampled regularly the aperture for the slope
(once for each direction) in 2 ? 25 points as an input
for the control loop. From this point on we proceeded
exactly as in the centroiding scheme: The phases from
these 50 samples made a single slope vector, used to
create the response matrix for each of the 37 actua-
tors. SVD created the reconstructor matrix. Then each
new Hartmann–Shack pattern was twice multiplied,
smoothed, and down sampled to obtain the same
slopes vector, to be used in the mirror control loop.
to their final values within five iterations for kp ? ?0.3 and
ka? ?0.3. Insert on right: cuts through the laser input images for
three iterations, when they reduce to diffraction size (bottom
curve). To better see the width, the camera was overexposed, and
the top of diffraction images is saturated.
(Color online) Screen shot of the first six modes converging
diameter). The Hartmann pattern is grabbed, multiplied by the exponential of its base frequency, smoothed, and its phase extracted.
The slope is sampled at 25 spots inside this pattern (left panel). After subtraction of the reference image (flat surface) we get the control
signal input (right panel).
(Color online) Hartmann pattern filtered by smoothing: y-slope response to a single actuator (0.3 ?m stroke, 15 mm mirror
1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS5
were able to close the control loop in 80 ms (30 Hz
frame rate) in the classical centroid approach. In the
next stage we tried the Fourier and smoothing phase
extraction methods. The inefficiency of the FFT
algorithm and the shear frame size made the pro-
cess much slower, closing the loop in approximately
600 ms. At this rate we were still able to correct
aberrations due to slow turbulence (Fig. 7). No at-
tempt was made to subsample or bin down the frame,
or locate a more efficient FFT or similar algorithm.11
The reason is that the AOA software package is no
longer supported, and the inclusion of new C subrou-
tines is not possible. As a result, the smoothing algo-
rithm control loop exceeded 2500 ms. Its application
in C or MATLAB was, on the other hand, much faster
then the previous methods.9Using a different com-
puter, it was applied successfully to a living human
We also verified the control schemes for different
parameters. By changing the hair dryer speed we
induced weak and strong turbulence changes, which
translated into corresponding temporal and spatial
aberration scales. Indeed we found that our model led
to areas of divergence and uncontrolled oscillations
and areas of stable operation, exactly matching our
model. We tested many points in the ?kp, ka? plane,
and we show one example in Fig. 8. Future work, on
a more advanced system, will test our theory on the
Timing and Control
propagation of noise17and further validate the con-
We operated what we believe to be the first adaptive
optics control system utilizing Fourier phase extrac-
tion. In one realization we used a full Fourier trans-
form of the Hartmann spots, and extricated the
phases (namely the wavefront slopes) from the or-
thogonal side lobes as control inputs. We were able to
close the adaptive optics loop in acceptable time,
which can be further reduced by improved FFT algo-
rithms or hardware. Fourier modes might be better
descriptors of turbulence-created phase and of de-
formable mirrors with Cartesian or hexagonally tiled
actuators. For very large systems they should be the
natural choice. Add to that the advantage that strong
aberrations are better tolerated in Fourier analysis,
as there is no need to contain the Hartmann–Shack
spots within the designated lenslet areas. Thus the
lenslets can have longer focal lengths, and increase
the phase sensitivity. Yet another advantage lies in
taking the first side lobes: High-frequency noise is
filtered out, making the use of a hardware aperture12
All these advantages exist also in the second real-
ization, where we demodulated the same Hartmann
grid in the image domain, and sampled it inside the
deformable mirror. These sample points were also
used to close the loop and correct the wavefront for
a screen shot of the first six modes and their convergence under mild turbulence for kp? ?0.25 and ka? ?0.4.
(Color online) Comparison between the simulation of the control system response (right panel) and the experiment (left panel),
6 APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007
different turbulence conditions. Our control system
functioned as predicted and was able to follow and
immediately correct even strong cases of turbulence.
In a separate experiment16we were able to close the
loop on a living eye using the same method.
There is an advantage to sampling the phase of the
interpolated wavefront: Sample locations can be ju-
diciously chosen according to criteria other than the
given lenslet positions. For example, their density
and spread can be guided by the turbulence spectrum
and by the wind: Sampling can be denser in the up-
wind direction to improve prediction.18In ocular
adaptive optics, spherical aberration can be domi-
nant, which also calls for more sampling around the
pupil periphery. Other factors can be the aperture
shape or boundary and the positions of the actuators.
It might be advantageous to sample where the slopes
of the influence functions of the actuators are maxi-
mal and the sensitivity is better. Sampling can also
be optimized experimentally as in the SVD process.
We intend to further investigate these points since it
might be possible to minimize or even avoid the ma-
trix multiplication in the control loop altogether if the
right sampling choice is taken.
Parts of this work were supported by the Israeli
Ministry of Science, and in conjunction with the EU
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