Statistical Image Modeling and Processing Using Wavelet-Domain Hidden Markov Models - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 350

j,k,i p S j,k,i

j

N j − 1
k 0 N j − 1

p

w = | w

m

j,k,i ,

= i = 0

2
j,m

σ = .

N j p S j (

m )

(10.16)

1.3 Iteration: Set p = p + 1. If it converges (or p = N p ), then go to

Step 1.4; otherwise, go to Step 1.1.
1.4 Set c

j,k,i by

= 0 and set the elements in

p S j,k,i ( m ) = p ( S j,x, y = m | w

j,x, y ,

) , (10.17)

j

x y

x y w

2

j,x, y p

( S j,x, y = m | w j,x, y , j )

2
j,k,i,m

σ = , (10.18)

( 2 C j + 1 )

2 p S j,k,i (

m )

x y

p ( S j,x, y = m | w j,x, y , v j,x, y = v , j )

p V j,k,i | S j,k,i (v | m ) = .

p S j,k,i ( m )

(10.19)

•

c j,k,i , k, i

Step 2. E step: Given = 0 , 1 , ... ,N j − 1, calculate (Bayes

rule)

p S j,k,i | V j,k,i ,W j,k,i ( m | w j,k,i , v j,k,i = v)

g w j,k,i |

j,k,i,m

p S j,k,i ( m ) p V j,k,i | S j,k,i (v | m ) 0 , σ

=

j,k,i,m .

(10.20)

1 m = 0

g w

p S j,k,i ( m ) p V j,k,i | S j,k,i (v | m ) | 0 , σ

j,k,i

•

c + 1

Step 3. M step: Compute the elements of

j,k,i , k, i

= 0 , 1 , ... ,N j − 1,

by

p S j,k,i ( m ) = p S j,x, y | V j,x, y ,W j,x, y ( m | w j,x, y , v j,x, y ) , (10.21)

x y

x y w

j,x, y p S j,x, y | V j,x, y ,W j,x, y (

m | w

j,x, y ,

v )

j,x, y

j,k,i,m

σ = ,

( 2 C j + 1 )

2 p S j,k,i (

m )

(10.22)

x y

p S j,x, y | V j,x, y ,W j,x, y ( m | w j,x, y , v j,x, y = v)

p V j,k,i | S j,k,i (v | m ) = .

p S j,k,i ( m )

(10.23)

•

Step 4. Iteration: Set c = c + 1. If it converges (or c = N c ), then stop;

otherwise, go to Step 2.