Shape From Shading Models - Biomedical Image Analysis - page 294

where

D 11 = R p i , j + 3

(2) (0) + 1 ,

(5.74)

D 22 = R q i , j + 3

(2) (0) + 1 ,

(4) (0) ,
D = D 11 D 22 − R p i , j R q i , j

(2) (0) + 2

D 33 = 2

and

C 1 = ( E − R ) R p − p i , j

2 N − 2

(3) ( k ) + (1) ( k )) − (2 p i − k , j + p i , j − k ) (2) ( k ) ,

Z i − k , j (

+

k =− 2 N + 2

C 2 = ( E − R ) R q − q i , j

2 N − 2

(3) ( k ) + (1) ( k )) − ( q i − k , j + 2 q i , j − k ) (2) ( k ) ,

Z i , j − k (

+

k =− 2 N + 2

2 N − 2

(3) ( k ) + (1) ( k ))

C 3 =− ( p i − k , j + q i , j − k )(

k =− 2 N + 2

(2) ( k ) + (4) ( k )) .

+ ( Z i − k , j + Z i , j − k )( (5.75)

Finally, we can write the iterative formula

p m + 1
i , j

= p i , j + δ p i , j ,

(5.76)

q m + 1
i , j

= q i , j + δ q i , j ,

z m + 1
i , j

= z i , j + δ z i , j .

We now summarize this method as the follows:

Step 0. Compute 1D connection coefficients and 2D connection coefficients.

Step 1. Compute the set of coefficients (5.75) and (5.74).

Step 2. Compute the set of variations δ p i , j ,δ q i , j , and δ z i , j (5.73).

Step 3. Update the current ( p i , j , q i , j ) and then the current shape reconstruc-
tion Z i , j using Eq. (5.76).

5.4.3 Summary

The wavelet-based method we demonstrated in this section is based on the
approximation of the objective function in V 0 . It should be pointed out that it