Iterative Optimization in Inverse Problems

90 Iterative Optimization in Inverse Problems

and

F = {b ∈ B|d(b, A)=d(B,A)};

assume that both E and F are not empty. The displacement vector is v =

P

K

(0),whereK is the closure of the set B − A. For any transformation

T : R

N

→ R

N

,denotebyFix(T ) the set of all x ∈ R

N

such that Tx = x.

Prove the following:

(a) v

2

= d(A, B);

(b) E + v = F ;

(c) E = Fix(P

A

P

B

)=A ∩ (B − v);

(d) F = Fix(P

B

P

A

)=B ∩ (A + v);

(e) P

B

e = P

F

e = e + v, for all e ∈ E;

(f) P

A

f = P

E

f = f − v,for all f ∈ F .

Ex. 6.14 Prove Corollary 6.1.

Ex. 6.15 Prove Corollary 6.2.

Ex. 6.16 Prove Corollary 6.4.