Digital watermarking strings in image compression processes with fuzzy relation equations

A gray image is seen as a fuzzy relation R if its pixels are normalized with respect to the length of the used scale. This relation R is divided in submatrices defined as blocks and each block RB is coded to a fuzzy relation GB, which in turn is decoded to a fuzzy relation DB (unsigned) whose values are greater than those ones of RB. Both GB and DB are obtained via fuzzy relation equations with continuous triangular norms (in particular, here we use the Lukasiewicz t-norm) and the involved fuzzy sets (coders) are Gaussian membership functions. Let D be the image obtained from the recomposition of the D′Bs. In this work we use a watermarking method based on the well known encrypting alphabetic text Vigenere algorithm. Indeed we embed such watermark in every GB with the LSBM (Least Significant Bit Modification) algorithm obtaining a new matrix GB, decompressed to a matrix DB (signed). Both GB and DB are deduced with the same fuzzy relation equations used for obtaining GB and DB. The recomposition of the D′Bs gives the image D (signed). The quality of the reconstructed images with respect to the original images is measured from PSNR (Peak Signal to Noise Ratio) and we show that D is very similar to D for low values of the compression rate. The binary watermark matrix embedded in every GB is variable, thus this method is more secure than another our previous method where the binary watermark matrix in every GB is constant.