Digital watermarking in coding/decoding processes with fuzzy relation equations

By normalizing the values of its pixels with respect to the length of the used scale, a gray image can be interpreted as a fuzzy relation R which is divided in submatrices ( possibly square) called blocks. Every block R-B is compressed to a block GB, which in turn is decompressed to a block D-B ( unsigned) >= R-B. Both G(B) and D-B are obtained via fuzzy relation equations with continuous triangular norms in which fuzzy sets with Gaussian membership functions are used as coders. The blocks D-B are recomposed in order to give a fuzzy relation D. We use the Lukasiewicz t-norm and a watermark ( matrix) is embedded in every G(B) with the LSBM ( Least Significant Bit Modification) algorithm by obtaining a block (D) under bar (B), decompressed to a block (D) under bar (B) ( signed). Both (D) under bar (B) and (D) under bar (B) are obtained by using the same fuzzy relation equations. The blocks (D) under bar (B) are recomposed by obtaining the fuzzy relation (D) under bar ( signed). By evaluating the quality of the reconstructed images via the PSNR ( Peak Signal to Noise Ratio) with respect to the original image R, we show that the signed image (D) under bar is very similar to the unsigned image D for low values of the compression rate.