Biomedical Engineering Reference
In-Depth Information
b
n
γ
b
m
M
b
m
b
n
FIGuRE 10.6
he. shape. of. the. multidimensional. metric. ellipsoid. can. be. determined. by. performing. measure-
ments.with.diferent.test.aberrations.applied.(
let
.image).
10.10 orthogonalization of Modes
In.principle,.the.set.of.new.aberration.modes.is.derived.through.orthogonalization.of.the.basis.modes..
his.process.uses.a.particular.inner.product.that.depends.upon.the.nature.of.the.imaging.system.and.
the.choice.of.optimization.metric..In.practice,.one.obtains.values.of.the.coeicients.α
i,j
.of.Equation.10.5.
either.through.direct.calculation.of.the.inner.product.or.indirectly.through.empirical.measurements,.as.
explained.in
.
Section.10.9.
.
Once.the.coeicients.α
i,j
.have.been.obtained,.the.orthogonalization.process.is.
facilitated.though.a.matrix.formulation..We.outline.this.process.as.follows..First,.we.deine.a.matrix.
A
.
whose.elements.are.the.values.of.α
i,j
..Equation.10.5.can.then.be.expressed.in.the.convenient.form
M M
≈
−
T
a A a
.
.
(10.16)
.
0
where.the.vector.
a
.consists.of.the.aberration.coeicients.
a
i
..We.then.derive.an.alternative.representation.
by.diagonalizing.
A
.using.standard.eigenvector/eigenvalue.decomposition:
A V B V
=
T
(10.17)
.
.
where.
B
.is.a.diagonal.matrix.whose.on-diagonal.entries.β
i
.are.the.eigenvalues.of.
A
..he.columns.of.
V
.are.
the.corresponding.eigenvectors..he.optimization.metric.then.becomes:
M M
≈
−
a V B V a
=
M
−
d B d
.
(10.18)
.
T
T
T
0
0
where.
d
.=.
V
T
.
a
..Denoting.the.elements.of.the.vector.
d
.as.
d
i
,.we.can.show.that.Equation.10.18.is.now.
equivalent.to.the.desired.form.of.Equation.10.6,.where
∑
M M
≈
−
β
d
(10.19)
.
i
.
2
i
0
i
he.values.of.
d
i
.are.the.coeicients.of.the.aberration.expansion.in.terms.of.the.optimal.modes.
Y
i
(
r
,θ),.
where.
∑
∑
Φ θ =
( , )
r
a X r
( , )
θ =
d Y r
( , )
θ
.. he. new. modes.
Y
i
. can. themselves. be. calculated. from. the.
i
i
i
i
i
i
basis.modes.as
∑
.
Y r
( , )
θ =
V X r
( , )
θ
.
(10.20)
i
i j
,
j
j
