Biomedical Engineering Reference
In-Depth Information
Most.phase-retrieval.algorithms.work.by.starting.with.an.initial.guess.for.the.ield..he.ield.is.then.
propagated. to. each. plane. where. the. intensity. was. measured.. At. each. measurement. plane,. the. ield.
amplitude. is. replaced. by. the. square. root. of. the. measured. intensity.. he. process. is. iterated. until. the.
ield.is.consistent.with.all.the.measurements..In.the.original.phase-retrieval.algorithm,.the.phase.was.
.calculated.from.the.intensity.measurements.in.the.focal.and.pupil.planes.of.a.low-NA.system.yielding.
the.phase.in.the.pupil.plane.(Gerchberg.and.Saxton.1972)..he.algorithm.of.Gerchberg.and.Saxton.was.
then.improved.upon.and.applied.to.astronomy.by.Fienup.(1978).
Phase.retrieval.has.been.demonstrated.for.high-NA.microscopes.(Hanser.et.al.,.2003,.2004)..In.this.
case,.the.phase.in.the.back.pupil.plane.of.the.microscope.objective.is.determined.from.multiple.images.
around.the.focal.plane..Hanser.et.al..used.a.luorescent.bead.with.a.diameter.smaller.than.the.micro-
scope.resolution.to.serve.as.an.efectively.coherent.point.source..Taking.images.through.focus.yields.the.
3D.PSF.(Hiraoka.et.al..1990).
In. the. modiied. Gerchberg-Saxton. algorithm. used. by. Hanser. et. al.,. the. amplitude. of. the. ield. is.
replaced.by.the.measured.data.at.each.measurement.plane,.which.minimizes.the.diference.in.amplitude.
between. the. retrieved. ield. and. the. measurements.. his. algorithm. starts. with. an. initial. guess. for. the.
pupil.function.(the.wavefront.in.the.back.pupil.plane),.
P k k
x
(
)
=
1
.within.the.back.pupil.plane.aperture..
,
y
he.PSF.is.then.calculated.through.focus:
(
)
=
(
)
2
.
.
PSF
x y z
,
,
A x y
,
.(15.8)
i
i
(
)
=
(
)
(
(
)
)
(
)
∫∫
.
A x y
,
P k k
,
exp
iz
k k
,
exp
ik x ik
y dk dk
x
.
.(15.9)
γ
+
i
x
y
i
x
y
x
y
y
2
2
π
λ
n
−
λ
π
k
n
(
)
=
.
γ
k k
,
1
.
.(15.10)
x
y
2
At. each. measurement. plane,.
z
i
,. the. amplitude. of. the. calculated. light. ield,.
A x y
)
,. is. replaced. by.
(
,
i
(
)
.where.
I
.is.the.measured.PSF..his.can.be.expressed.as
I x y z
i
,
,
(
)
I x y z
A x y
,
,
(
)
=
(
)
.
i
.
A x y
,
A x y
,
.(15.11)
(
)
i
i
,
i
he.pupil.function.is.then.calculated.as
(
)
=
1
(
(
)
)
(
)
(
)
∑
∫∫
.
P k k
,
A x y
,
exp
−
iz
γ
k k
,
exp
−
ik
x ik y
−
d d ,
.
x y
.(15.12)
x
y
i
i
x
y
x
y
N
i
and.the.process.is.iterated.
Here.we.use.a.modiication.of.this.algorithm.(Deming,.2007).that.minimizes.the.diference.in.inten-
sity.between.the.retrieved.ield.and.the.measurements..his.equivalently.minimizes.the.relative.entropy.
between.the.measured.3D.PSF.and.the.PSF.calculated.from.the.retrieved.ield..Here.we.refer.to.Deming's.
method.as.the.MRE.(for.minimize.relative.entropy).method..In.efect,.the.only.diference.between.these.
two.methods.is.to.replace.Equation.15.10.by
(
)
I x y z
A x y
,
,
(
)
=
(
)
i
,
,
.
A x y
A x y
.
.(15.13)
i
2
i
(
)
,
i
At. each. iteration,. the. diference. between. the. calculated. PSF. and. the. measured. PSF. will. always.
decrease,.but.the.error.in.the.wavefront.will.start.to.increase.ater.a.certain.number.of.iterations.as.the.
wavefront.is.modiied.to.it.the.noise.in.the.PSF.(Deming.2007).
