Shadows of Fuzzy Sets -- A Natural Way to Describe 2-D and Multi-D Fuzzy Uncertainty in Linguistic Terms

Fuzzy information processing systems start with expert knowledge which is usually formulated in terms of words from natural language. This knowledge is then usually reformulated in computer-friendly terms of membership functions, and the system transform these input membership functions into the membership functions which describe the result of fuzzy data processing. It is then desirable to translate this fuzzy information back from the computer-friendly membership functions language to the humanfriendly natural language. In a 1-D case, when we are interested in a single quantity y, it is usually easy to describe the resulting membership function by a word from natural language, because most words do describe 1-D case and there are, therefore, so many of them that the corresponding membership functions form a dense set in the class of all possible membership functions. The problem becomes more complicated in 2-D and multi-D cases, when we are interested in several quantities y1 ; : : :..