The fuzzy transformation and its applications in image processing

The spatial and rank (SR) orderings of samples play a critical role in most signal processing algorithms. The recently introduced fuzzy ordering theory generalizes traditional, or crisp, SR ordering concepts and defines the fuzzy (spatial) samples, fuzzy order statistics, fuzzy spatial indexes, and fuzzy ranks. Here, we introduce a more general concept, the fuzzy transformation (FZT), which refers to the mapping of the crisp samples, order statistics, and SR ordering indexes to their fuzzy counterparts. We establish the element invariant and order invariant properties of the FZT. These properties indicate that fuzzy spatial samples and fuzzy order statistics constitute the same set and, under commonly satisfied membership function conditions, the sample rank order is preserved by the FZT. The FZT also possesses clustering and symmetry properties, which are established through analysis of the distributions and expectations of fuzzy samples and fuzzy order statistics. These properties indicate that the FZT incorporates sample diversity into the ordering operation, which can be utilized in the generalization of conventional filters. Here, we establish the fuzzy weighted median (FWM), fuzzy lower-upper-middle (FLUM), and fuzzy identity filters as generalizations of their crisp counterparts. The advantage of the fuzzy generalizations is illustrated in the applications of DCT coded image deblocking, impulse removal, and noisy image sharpening.