Iterative Optimization in Inverse Problems

16 Iterative Optimization in Inverse Problems

x

1

x

2

x

6

x

7

x

12

x

36

x

31

x

25

FIGURE 1.4: Line integrals through a discretized object.

With i = k(mod I) + 1, the iterative step of the ART algorithm is

x

k+1

j

= x

k

j

+

1

|L

i

|

(b

i

− (Ax

k

)

i

), (1.19)

for j in L

i

,and

x

k+1

j

= x

k

j

, (1.20)

if j is not in L

i

. In each step of ART, we take the error, b

i

− (Ax

k

)

i

,

associated with the current x

k

and the i-th equation, and distribute it

equally over each of the pixels that intersects L

i

.

This model is too simple; we are assuming that if the line segment in-

tersects a pixel, then the entire amount of attenuating material within that

pixel affects the x-ray strength. A somewhat more sophisticated ...