Co-Volume Level Set Method in Subjective Surface Based Medical Image Segmentation - Biomedical Image Analysis - page 600

or

m ( T ∩ p )
m ( p )

1
|∇ u n − T | .

M p = (11.18)

T ∈ N p

The averaging of the gradients (11.17) has been used in [25, 56], and the ap-
proximation (11.18) is new and we have found it very useful regarding good
convergence properties in solving the linear systems (see below) iteratively for
ε 1. Regularizations of both the approximations of the capacity function are
as follows: either

1
|∇ u n − 1

M p =

(11.19)

| ε

p

or

m ( T ∩ p )
m ( p )

1
|∇ u n − T | ε .

M p = (11.20)

T ∈ N p

Now we can define coefficients, where the ε -regularization is taken into account,
namely,

b n − 1
p

= M p m ( p ) , (11.21)

g T
|∇ u n − 1
T

1
h pq

a n − 1 c pq

pq =

| ε ,

(11.22)

T ∈ E pq

which together with (11.15) and (11.16) give the following.

Fully-discrete semi-implicit co-volume scheme. Let u p , p = 1 ,..., M ,
be given discrete initial values of the segmentation function. Then, for n =
1 ,..., N we look for u p , p = 1 ,..., M, satisfying

b n − p u p + τ a n − 1

pq ( u p − u q ) = b n − p u n − 1

(11.23) .

p

q ∈ C p

Theorem. There exists a unique solution ( u 1 ,..., u n M ) of the scheme (11.23)
for any τ> 0 , ε> 0 and for every n = 1 ,..., N. Moreover, for any τ> 0 , ε> 0
the following stability estimate holds

u p ≤ min
p

u p ≤ max
p

u p ≤ max
p

u p , min

p

1 ≤ n ≤ N . (11.24)

Proof. The system (11.23) can be rewritten in the form

b n − 1
p

a n − 1
pq

u p − τ a n − 1

pq u q = b n − p u n − 1

(11.25) + τ .

p

q ∈ C p q ∈ C p