Iterative Optimization in Inverse Problems

The SMART and EMML Algorithms 169

shifted distance KL(v − x, v −z) applies only to those vectors x and z for

which x

j

≤ v

j

and z

j

≤ v

j

.Foru

j

≤ v

j

, the combined distance

KL(x −u, z − u)+KL(v − x, v − z)

is restricted to those x and z whose entries x

j

and z

j

lie in the interval

[u

j

,v

j

]. Our objective is to mimic the derivation of the SMART, EMML

and RBI methods, replacing KL distances with shifted KL distances, to

obtain algorithms that enforce the constraints u

j

≤ x

j

≤ v

j

,foreach

j. The algorithms that result are the ABMART and ABEMML block-

iterative methods. These algorithms were originally presented in [53], in

which the vectors u and v were called a and