Fuzzy data granulation and relational compression

The study concentrates on fuzzy relational calculus and views it as a basis of data granulation and data compression. In this setting, data and images, in particular, are represented as fuzzy relations. We investigate fuzzy relational equations as a vehicle of data compression. It is shown that both compression and decompression (reconstruction) phases are closely linked with the way in which fuzzy relational equations are usually being formulated and solved. The underlying findings that are encountered in the theory of these equations are easily accommodated as an important backbone of any relational compression. The character of the solutions to the equations make them ideal for reconstruction purposes, as they specify the extremal elements of the solution set and in such a way help establish some envelopes of the original images under compression. The flexibility of the conceptual and algorithmic framework arising there is also discussed. Numerical examples provide a suitable illustrative material emphasizing the main features of the compression mechanisms