Biomedical Engineering Reference
In-Depth Information
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Figure a.6
Classification. of. elements. in. a. distribution. matrix. (◎,☆,◇,▲. represent.
( )
a
mn
i
. with.
;.
ξ
,
ξ
∈
Z
m
n
′
,.
;.and.
′
,.respectively).
ξ
∈
Z
,
ξ
∈
Z
ξ
∈
Z
′ ξ
,
∈
Z
ξ
,
ξ
∈
Z
m
n
m
n
m
n
ProofofLemma4.10
Based.on.the.definition.of.
( )
b
mn
i
.given.in.Equation.(4.60),.consider
L L
−
L L
−
g
g
∑
∑
1
∑
∑
∑
( )
i
( )
i
b
−
b
=
∆ ξ
(
,
I
;
ξ
,
G
)
mn
mn
i
c
,
k
m
c
,
c
,
k
3
(
L L
−
+
1)
n
n
m n
n
g
ξ
∈
Z
ξ
∈
Z
′
c
=
0
c
=
0
k
∉κ
(
c
)
n
n
m
n
n
n
∑
×
∆ ξ
(
,
M
;
ξ
,
M
′
) (
∆ ξ
,
R
;
ξ
,
R
)
i
c
,
k
n
c
,
k
i
c
,
k
n
c
,
k
n
n
n
n
n
n
n
n
ξ
∈
Z
n
∑
−
∆ ξ
(
,
M
;
ξ
,
M
′
) (
∆ ξ
,
R
;
ξ
,
R
)
i
c
,
k
n
c
,
k
i
c
,
k
n
c
,
k
n
n
n
n
n
n
n
n
ξ
∈
Z
′
.
.
n
.
.
(A.18)
where.
(
κ
c
)
=
(
c
,
c
+
L
]
,.and.for.any.particular.
ξ
∈
Z
.(note:.
Z
.can.be.
Z
odd
.or.
n
n
n
g
n
Z
even
),.the.following.three.cases.can.be.obtained:
.
a.. For.any.cutting.position.
c
n
,.which.cuts.some.actual.bits.in.
ξ
,.it.is.
possible.to.flip.the.rightmost.of.those.bits.to.find.a.corresponding.
Z
ξ
∈
′
,. such. that. the. difference. term. in.
Equation. (A.18)
.
is. zero,.
n
′
that.is,
∑
∆ ξ
(
,
M
;
ξ
,
M
′
) (
∆ ξ
,
R
;
ξ
,
R
)
i
c
,
k
n
c
,
k
i
c
,
k
n
c
,
k
n
n
n
n
n
n
n
n
ξ
∈
Z
n
∑
−
∆ ξ
(
,
M
;
ξ
,
M
′
) (
∆ ξ
,
R
;
ξ
,
R
) 0
=
i
c
,
k
n
c
,
k
i
c
,
k
n
c
,
k
n
n
n
n
n
n
n
n
.
ξ
∈
Z
′
n
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