Advanced Segmentation Techniques - Biomedical Image Analysis - page 499

setting them to zero results in two necessary but not sufficient conditions for
J m to be at a local extrema. In the following subsections, we will derive these
three conditions.

9.4.3.1 Membership Evaluation

The constrained optimization in Eq. 9.26 will be solved using one Lagrange
multiplier

u ik D ik + N R u ik γ i

c N c

F m = 1 − u ik (9.27)

+ λ ,

i = 1 k = 1 i = 1

2 . Taking the derivative of
F m w.r.t. u ik and setting the result to zero, we have, for p > 1,
δ F m

and γ i = x r ∈ N k || x r − v i ||

2

where D ik =|| x k − v i ||

ik D ik + α p

δ u ik = pu p − 1 N R u ik γ i − λ

u ik = u ik = 0 .

(9.28)

Solving for u ik , we have

1

p − 1

λ
p ( D ik + N R γ i )

u ik =

(9.29)

.

Since c j = 1 u jk = 1 ∀ k ,

1

c p − 1

λ
p ( D jk + N R γ j )

= 1 (9.30)

j = 1

or

p

(9.31) λ =

1 p − 1

c j = 1

1

p −

1
( D jk + N R γ j )

Substituting into Eq. 9.29, the zero-gradient condition for the membership esti-
mator can be rewritten as

1
c j = 1 D ik + N R γ i

u ik =

p − 1 .

(9.32)

1

D jk + N R γ j