Biomedical Engineering Reference
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should.be.imaged.into.an.area.of.2.×.2.pixels.to.acquire.the.most.information.without.losing.precision.
in.centroiding..One.of.the.advantages.of.using.the.cross-correlation.algorithm.is.that.the.computation.
can.be.done.in.the.Fourier.domain,.taking.advantage.of.the.Fast.Fourier.Transform.(FFT).algorithm.
available.to.improve.the.speed.of.the.AO.system..Although.the.reference.function.can.be.modeled.as.
a.Gaussian.spot,.it.can.be.shown.that.using.a.real-time.Hartmann-spot.measurement.from.the.SHWF.
sensor. images. can. improve. the. accuracy. of. the. centroiding. algorithm. (homas. et. al.. 2006).. his. is.
mainly.due.to.the.amount.of.information.aforded.by.the.real-time.spot.measurement.compared.to.that.
of.a.Gaussian.image..he.inal.step.needed.to.measure.the.amount.of.slope.at.each.subaperture.using.the.
cross-correlation.algorithm.is.to.ind.the.maximum.for.each.subaperture.(homas.et.al..2006).
17.1.4 reconstruction
here.are.various.ways.of.estimating.a.wavefront.from.the.Hartmann.slopes.(Hardy.1998;.Gavel.2003)..
Two.essential.pieces.of.information.are.needed.for.this:.(1).the.phase.diference.(slope.measurements.
times.subaperture.size).from.each.subaperture.and.(2).the.geometrical.layout.of.the.subapertures..he.
wavefront. can. then. be. calculated. by. relating. the. slope. measurement. to. the. phases. at. the. edge. of. the.
subaperture.in.the.correct.geometrical.order..A.method.for.directly.obtaining.the.deformable.mirror.
(DM).commands.from.wavefront.sensor.measurements.is.described.by.Tyson.(1998)..First.a.mask.with.
the. subapertures. must. be. created;. this. will. generate. the. geometric. layout. of. the. subapertures. in. the.
aperture..he.next.step.is.to.measure.and.record.the.response.of.all.the.subaperture.slope.changes.while.
actuating.each.actuator..he.results.obtained.are.set.of.linear.equations.that.show.the.response.of.the.
wavefront. sensor. for. each. actuator. command. known. as. the. poke. matrix. (also. known. as. the. actuator.
inluence.matrix)..he.DM.commands.can.then.be.obtained.by.solving.the.equation
(17.4)
.
.
s Av
=
where.
s
.is.an.
n
-size.vector.obtained.from.the.SHWF.sensor.slope.measurements,.
v
.is.an.
m
-size.vector.
with.the.DM.actuator.commands,.and.
A
.is.an.
n
.×.
m
-sized.poke.matrix..In.the.linear.approximation,.
Equation.17.4.can.be.pseudo-inverted.to.obtain.an.estimate.of.the.DM.command.matrix..Note.that.the.
DMs.are.nonlinear.devices,.but.the.matrix.given.in.Equation.17.4.performs.well.in.a.closed-loop.system,.
as.only.very.small.voltage.changes.occur,.thus.reducing.the.nonlinear.efects..here.are.various.methods.
for.inverting.the.matrix.
A
,.including.singular.value.decomposition.(SVD)..he.advantage.of.using.SVD.
is.that.the.mode.space.can.be.directly.calculated..he.noisier.modes,.and.all.the.null.space.modes.by.
default,.can.then.be.removed.by.setting.a.threshold.on.the.singular.value.space.(Gavel.2003).
Figure.17.4a
.
shows.a.poke.matrix.obtained.by.using.the.method.described.here..he.process.begins.
by.poking.an.actuator.to.a.predetermined.voltage.(
V
);.each.of.the.subapertures'.slope.changes.is.then.
recorded..his.process.is.repeated.100.times.for.each.actuator,.and.the.recorded.data.are.then.averaged.
to.reduce.the.efect.of.noise..he.slope.changes.are.determined.using.the.cross-correlation.centroiding.
algorithm.. Further. conditioning. of. the. poke. matrix. is. performed. by. thresholding. the. data. to. 20%. of.
the.maximum.slope.changes.measured.for.all.actuators..For.each.actuator.poke,.there.will.be.an.area.in.
the.SHWF.sensor.that.will.show.stronger.slope.changes.(i.e.,.an.inluence.function)..In.AO,.a.15-20%.
inluence.function.between.actuators.is.usually.considered.good.as.this.allows.for.high.spatial.deforma-
tions.to.be.well.reproduced.by.the.DM..he.thresholding.step.mentioned.earlier.essentially.windows.the.
slope.measurements.to.an.area.near.the.center.of.the.actuator.poke.with.a.20%.slope.inluence.matrix..
Considering. a. larger. area. can. introduce. higher. spatial. frequencies. that. are. dominated. by. noise,. thus.
introducing.noisier.modes.into.our.singular-value.space.
Figure.17.4b
.
shows.the.singular-value.pseudo.inverse.of.the.poke.matrix.shown.in.
Figure.17.4a
.
.he.
pseudo.inverse.has.the.singular-value.space.shown.in.
Figure.17.4c
,
.which.has.been.regularized.to.remove.
the.singular-value.modes.that.are.lower.than.15%.of.the.maximum.mode.as.described.by.Gavel.(2003)..
By.multiplying.the.poke.matrix.in.
Figure.17.4a
.
and.its.pseudo.inverse.in.
Figure.17.4b
,.the.actuator.space.
