Biomedical Engineering Reference
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and.is.positive.for.the.planes.extending.into.the.sample..his.expression.has.dropped.pre-factors.such.as.
luorophore.concentration.and.cross.section,.which.Leary.and.Mertz.assumed.to.be.constant.
Alternatively,.the.two-photon.luorescence.intensity.at.depth. z .can.be.expressed.as.a.function.of.the.
illumination.laser.optical.transfer.function,.deined.by.Parseval's.theorem:
.
OTF
(
k z
, )
=
PSF
( , )
ρ
z e
i
ρ
.
k
d
2
ρ
.
.(12.2)
where. k
. represents. the. Fourier. form. wave. vector. coordinates.. Following. Leary. and. Mertz,. this.
leads to
OTF
.
W z
2
( )
2
.
F z
( )
=
(
k
, )
z
d k
2
.(12.3)
(
2 π
)
2
From.Schwarz's.inequality.[31],.we.obtain:
.
OTF
φ k
; )
z
OTF
(
k
; )
z
.
.(12.4)
0
where.subscripts.ϕ.and.0.indicate.the.conditions.where.there.are.optical.aberrations.and.no.optical.aber-
rations,.respectively..hus,.we.can.conclude.as.follows:
.
F z
φ ( )
F z
( )
.
.(12.5)
0
Based.on.this.simpliied.view.of.the.efect.of.aberrations.on.the.PSF,.it.is.reasonable.to.expect.that.
optical.aberrations.reduce.the.two-photon.luorescence.signal.generated.at.the.focal.plane..Conversely,.
compensation. of. those. aberrations. should. increase. the. two-photon. luorescence. signal. in. the. focal.
plane..hese.assumptions.ignore.the.possibility.that.aberration-induced.enlargement.of.the.PSF.might.
generate.additional.two-photon.luorescence.signals.from.structures.above.and.below.the.focal.plane..
Nevertheless,. the. analysis. of. Leary. and. Mertz. ofers. some. expectation. that. two-photon. luorescence.
signal.intensity.would.serve.as.a.reasonable.metric.for.control.of.aberrations.
Other.measurement.metrics,.such.as.sharpness,.have.also.been.used.in.wavefront.correction.systems.
to. improve. image. quality.. Muller. used. image. sharpness. as. the. measurement. metric. to. correct. image.
distortion.from.atmosphere.turbulence.in.the.telescope.application.[32]..he.sharpness.has.the.follow-
ing.deinition:
=
.
S
I x y x y
2 ( ,
)d d .
.(12.6)
where. I ( x ,. y ).is.the.intensity.value.at.each.pixel.on.the.imaging.plane..Variables. x .and. y .are.the.coor-
dinates.of.the.pixel..We.believe.the.image.sharpness.is.a.valid.measurement.metric.for.the.AO.SPGD.
application.and.could.be.used.as.an.alternative.or.complimentary.metric.besides.the.intensity.metric.
12.3.3 Application of Stochastic Parallel Gradient Decent
Algorithm with Zernike Polynomial Basis
In.the.approach.using.the.SPGD.algorithm,.the.time.required.for.metric.gradient.estimation.is.inde-
pendent.of.the.number.of.control.channels,.and.the.imaging.aperture.is.ixed.in.the.control.loops..he.
SPGD.algorithm.presents.an.attractive.solution.for.wavefront.aberration.correction.with.a.much.higher.
eiciency.and.faster.convergence.speed..Formal.analysis.of.the.convergence.rate.shows.that.the.parallel.
 
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