Digital watermarking in coding/decoding processes with fuzzy relation equations

Abstract

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 G B , 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 [InlineMediaObject not available: see fulltext.], decompressed to a block [InlineMediaObject not available: see fulltext.] (signed). Both [InlineMediaObject not available: see fulltext.] and [InlineMediaObject not available: see fulltext.] are obtained by using the same fuzzy relation equations. The blocks [InlineMediaObject not available: see fulltext.] are recomposed by obtaining the fuzzy relation [InlineMediaObject not available: see fulltext.] (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 [InlineMediaObject not available: see fulltext.] is very similar to the unsigned image D for low values of the compression rate.