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 /spl alpha/-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.<>