Biomedical Engineering Reference
In-Depth Information
Equation.10.11.is.now.in.the.same.form.as.Equation.10.5..he.integral.on.the.right-hand.side.of.Equation.
10.12.has.the.properties.of.an.inner.product.between.the.modes.
X
i
.and.
X
j
..Indeed,.this.inner.product.is.
identical.to.that.deining.the.orthogonality.of.the.Zernike.polynomials..We.can.also.note.that.the.value.
of.α
i,j
.in.Equation.10.12.can.equivalently.be.obtained.by.calculating.the.partial.derivative.of.the.metric:
M
a a
1
∂
∂ ∂
.
∫∫
.
α =
=
π
X X r r
d d
θ
(10.13)
i j
,
i
j
i
j
If.the.modes.
X
i
.are.chosen.appropriately,.then.one.can.ensure.that.the.modes.are.orthonormal,.and.
α
i,j
=.δ
i,j
.with.δ
i,j
.being.the.Kronecker.delta..Equation.10.11.simpliies.to
∑
M
≈ −
a
i
(10.14)
.
1
2
.
i
which.has.the.desired.form.of.Equation.10.6,.where.each.coeicient.can.be.maximized.independently.
of.the.others.
We.have.seen.that,.for.this.sensorless.adaptive.system,.an.eicient.scheme.can.be.derived.if.the.modes.
are. orthogonal. according. to.the.deinition.of. the.inner. product. in. Equation.10.12..he.Zernike.poly-
nomials.form.a.set.of.modes.that.fulils.this.property..However,.one.is.free.to.use.other.sets.of.modes.
that.have.similar.mathematical.properties.but.are.better.suited.to.a.speciic.application..his.approach.
is.outlined.in.Section.10.8.
10.8 Derivation of optimal Modes
he.example.described.i
n.
Section.10.7
.
illustrates.how.the.required.modal.properties.can.be.extracted.
from.a.mathematical.expression.for.the.optimization.metric..In.principle,.this.approach.can.be.applied.
to.more-complex.sensorless.adaptive.systems,.such.as.adaptive.microscopes.using.image-based.quality.
metrics..However,.the.increased.complexity.of.the.mathematics.describing.the.imaging.process.means.
that. simple. expressions. for. the. inner. product. are. not. readily. obtained.. Similarly,. conventional. ana-
lytic.modal.sets,.such.as.the.Zernike.polynomials,.may.not.have.the.required.mathematical.properties..
Generally,. it. is. necessary. to. derive. new. sets. of. modes. for. a. particular. application. to. ensure. optimal.
performance.of.the.sensorless.system..hree.diferent.methods.that.have.been.used.to.obtain.optimum.
modes.in.these.systems—analytical,.numerical,.and.empirical—are.explained.below.
.
1.. Analytical:.In.some.situations,.it.is.possible.to.deine.a.set.of.analytic.functions.that.are.orthogonal.
with. respect. to. the. inner. product.. his. method. has. been. employed,. for. example,. using. Zernike.
modes.in.a.focusing.system.(Booth.2006).(as.described.in
.
Section.10.7
).or.Lukosz.modes.in.a.focus-
ing.system.(Booth.2007b).or.in.an.incoherent.microscope.using.image.low.spatial.frequency.content.
as.the.metric.(Débarre.et.al..2007)..his.analytical.approach.is.of.relatively.limited.application,.as.
there.are.few.sets.of.known.analytic.modes.that.could.be.matched.to.any.particular.adaptive.system.
.
2.. Numerical:. If. the. functional. form. of. the. inner. product. is. known—for. example,. from. the. Taylor.
expansion. or. by. direct. evaluation. of. ∂
M a
i j
—then. the. optimum. orthogonal. modes. can. be.
obtained.numerically..As.a.starting.point,.one.selects.a.suitable.set.of.modes.as.a.basis.set.(e.g.,.a.
subset.of.low-order.Zernike.polynomials)..he.optimum.modes.are.then.constructed.from.the.basis.
set.using.an.orthogonalization.process.based.around.the.inner.product..his.has.been.shown.in.
structured.illumination.and.two-photon.luorescence.microscopes.(Débarre.et.al..2008,.2009).
∂
∂
.
3.. Empirical:.If.the.functional.description.of.the.inner.product.is.not.available,.the.basis.modes.can.be.
orthogonalized.using.an.empirical.process.in.which.the.form.of.the.metric.function.is.determined.
from.image.measurements..A.sequence.of.images.is.acquired.with.diferent.bias.aberrations.applied.
by.the.adaptive.element..he.corresponding.metric.measurements.map.out.the.shape.of.the.metric.
