Edge Detection in Presence of Impulse Noise - Advances in Image and Graphics Technologies - page 11

where u i,j ≡

u ( i,j ) is the approximate solution of (9) at grid point ( i,j ) with
grid size h = 1 as usual. Then applying the Gauss-Seidel iteration to (10) leads
to

2 μ ( C E + C W + C N + C S )+ ʽ u k +1
i,j

=

2 μ C E u i,j +1 + C W u k +1

i− 1 ,j + C S u i +1 ,j + ʾI i,j + ʾg i,j ,

(11)

i,j− 1 + C N u k +1

where

ˈ k ) i,j +1 ,C W =(1 ˈ k ) i,j− 1 ,C N =(1 ˈ k ) i− 1 ,j ,C S =(1 ˈ k ) i +1 ,j . C E =(1 − − − −

And we implement the relaxation method to speed up the iteration (11) by

u k +1

i,j = u i,j −

ˉ 1 r k +1

i,j ,

(12)

where ˉ 1 > 0 is the relaxation factor. Collecting (12), we have the new concrete
scheme:

= u i,j + ˉ 1 ʾI i,j + ʾg i,j +2 μ [ C E u i,j +1 + C W u k +1

+ C S u i +1 ,j ]

+ C N u k +1
i− 1 ,j i,j− 1

u k +1
i,j

1+ ˉ 1 2 μ ( C E + C W + C N + C S )+ ʾ

.

(13)

2.2 Split Bregman method for solving ψ

Now, we will apply the split Bregman method [11] to solve (7). First, we define
ˁ ( ˈ ):= μ
ʩ

ˈ ) 2

2 d x ,

(1 − |∇ u | (14)

and

⊧
⊨ 1 , ˈ> 1 ,

Tr ( ˈ ):= ˈ, 0 ≤ ˈ ≤ 1 , (15)

⊩

0 , ˈ< 0 .

to make sure that ˈ [0 , 1] for every iteration [8]. Here, we introduce an auxiliary

variable d and solve the following problem:

∈

d x + ˁ ( ˈ ) subject to d =( d 1 ,d 2 )= min

ˈ

ʩ | d |

∇ ˈ,

= d 1 + d 2 and ˁ ( ˈ ) is defined in (14). Following the technique in
[11], the split Bregman iteration can be formulated as

where | d |

− b n ) 2 d x , (16) d x + ˁ ( ˈ )+ ʻ

2

( ˈ n +1 , d n +1 )=argmin
ˈ, d

ʩ | d |

( d −∇ ˈ

ʩ

for given b n ,and

b n +1 = b n +( ˈ n +1 − d n +1 ) . ∇ (17)