Fast solving method of fuzzy relational equation and its application to lossy image compression/reconstruction

A fast solving method of the greatest solution for max continuous t-norm composite fuzzy relational equation of the type G(i,j) = (R ^T□A _i) ^T□B _j, i = 1,2,···,I,j = 1,2,···,J, where A _i∈F(X) X = {x ₁,x ₂,···,x _M}, B _j∈F(Y) Y = {y ₁,y ₂,···,y _N}, R∈F(X×Y), and □: max continuous t-norm composition, is proposed. It decreases the computation time IJMN(L+T+P) to JM(I+N)(L+P), where L, T, and P denote the computation time of min, t-norm, and relative pseudocomplement operations, respectively, by simplifying the conventional reconstruction equation based on the properties of t-norm and relative pseudocomplement. The method is applied to a lossy image compression and reconstruction problem, where it is confirmed that the computation time of the reconstructed image is decreased to 1/335.6 the compression rate being 0.0351, and it achieves almost equivalent performance for the conventional lossy image compression methods based on discrete cosine transform and vector quantization.