Alternating projection onto fuzzy convex sets

Alternating projections onto convex sets (POCS) is powerful tool for signal and image restoration. However, if POCS is among three or more nonintersecting convex sets, the result is not unique and POCS is generally not useful. This, however, can be overcome by allowing solutions that are in some sense close to each convex set. Such relaxation can be achieved through fuzzification of the sets into fuzzy convex sets. By performing POCS among the α-cuts of fuzzified sets, good solutions can be obtained. The authors propose morphological dilation as a fuzzification procedure. Examples of signal synthesis and restoration based on fuzzy POCS are given. Fuzzy POCS is illustrated through application to the problems of time-bandwidth-product minimization, signal extrapolation, and solution of simultaneous equations