Edge Detection in Presence of Impulse Noise - Advances in Image and Graphics Technologies - page 12

It is equivalent to write (16)-(17) with the form of d =( d 1 ,d 2 )and b =( b 1 ,b 2 ).

d 1 + d 2 d x

( ˈ n +1 ,d n +1
1

,d n +1
2

)=arg min
ˈ,d 1 ,d 2

ʩ

+ μ

2 d x + ʻ
2

ˈ ) 2 b 1 ) 2 d x

(1 − |∇ u | ( d 1 − ∂ x ˈ −

(18)

ʩ ʩ

b 2 ) 2 d x , + ʻ

2

( d 2 − ∂ y ˈ −

ʩ

and

= b 1 + ∂ x ˈ n +1

, n +1
2

= b 2 + ∂ y ˈ n +1

.

b n +1
1

d n +1
1

d n +1
2

− −

The Euler-Lagrange equation of (18) for ˈ with fixed d 1 and d 2 is

2 ( ˈ

b 1 ) b 2 )=0 ,

− ʻʔˈ +2 μ |∇ u | − 1) − ʻ∂ x ( d 1 − − ʻ∂ y ( d 2 −

with the Neumann boundary conditions ∂ˈ

∂ n =0 . Applying the same technique
in Subsection 2.1, we need to solve the equations about ˈ :

i,j = ˈ k + ˉ 2 F i,j + ʻ ( ˈ i +1 ,j + ˈ k +1

i,j− 1 + ˈ i,j +1 ) i− 1 ,j + ˈ k +1

ˈ k +1 , (19)

u ) k +1
i,j

1+ ˉ 2 (2 μ

| ∇ |

2 +4 ʻ )

(

where ˉ 2 > 0 is the relaxation factor and

2 + ʻ∂ x ( d 1 −

b 1 )+ ʻ∂ y ( d 2 − b 2 ) . F := 2 μ |∇ u |

For fixed ˈ , the optimality condition of (18) with respect to d 1 and d 2 gives
d 1 =max h

ʻ , 0 h 1 h , 2 =max h ʻ , 0 h 2

1 1

− −

h ,

(20)

where

h 1 = ∂ x ˈ + b 1 , 2 = ∂ y ˈ + b 2 , h = h 1 + h 2 .
Now, we summarize the split Bregman (SB) algorithm as follows.

Split Bregman Algorithm

1. Initialization: set d 1 = d 2 = b 1 = b 2 =0 , and input ˈ 0 ,u 0 ,μ,ʽ,ˉ 1 ,ˉ 2 ,ʻ.
2. For n =0 , 1 ,

,
(i) Update u n +1 using the iteration scheme (13) with initial value for iter-
ation: u n and ˈ n (in place of ˈ );
(ii) Update g n by

···

n max 0 , 1

;

1

g n = | u − I | −

ʾ | ( u − I ) n |