Iterative Optimization in Inverse Problems

164 Iterative Optimization in Inverse Problems

11.7 The MART and EMART Algorithms

The MART algorithm has the iterative step

x

k

j

= x

k−1

j

(y

i

/(Px

k−1

)

i

)

P

ij

m

−1

i

, (11.17)

where i =(k −1)(mod I)+1and

m

i

=max{P

ij

|j =1, 2, ..., J}. (11.18)

When there are nonnegative solutions of the system y = Px, the sequence

{x

k

} converges to the solution x that minimizes KL(x, x

0

) [50, 51, 52]. We

can express the MART in terms of the weighted KL projections T

i

(x

k−1

);

x

k

j

=(x

k−1

j

)

1−P

ij

m

−1

i

(T

i

(x

k−1

)

j

)

P

ij

m

−1

i

. (11.19)

We see then that the iterative step of the MART is a relaxed weighted KL

projection onto H

i

, and a weighted geometric mean of the current x

k

j

and

T

i

(x

k−1

)

j

. The expression ...