Fuzzy and extra crisp alternating projection onto convex sets (POCS)

Alternating projections onto convex sets (POCS) is a powerful tool for signal and image restoration and synthesis. Convex sets of signals obeying desired constraints are first specified. Then, by repeated projection onto these sets, convergence is to a signal obeying all desired constraints. The method assumes, however, that there is a nonempty intersection of the sets. If the intersection is empty, the result of POCS is not unique and, if the sets are not `close´, generally considered to be of little use. To construct sets that are closer, one or more of the convex sets is fuzzified. The α-cuts of the fuzzified sets, also convex, will eventually result in constraints with a nonempty intersection. Using a fuzzification of the convex constraint set allows approximate satisfaction of inconsistent constraints. Example applications are presented for computer tomography and optical diffraction synthesis