Digital Watermarking Strings with Images Compressed by Fuzzy Relation Equations

Abstract

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 $R_{B}$ is coded to a fuzzy relation $G_{B}$, which in turn is decoded to a fuzzy relation $D_{B}$ (unsigned) whose values are greater than those of $R_{B}$. Both $G_{B}$ and $D_{B}$ 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 $\underline{D}_{B}^{\prime }s$. In this work we use a watermarking method based on the well-known encrypting alphabetic text Vigenère algorithm. Indeed we embed such watermark in every $G_{B}$ with the Least Significant Bit Modification (LSBM) algorithm obtaining a new matrix $G_{B}$, decompressed to a matrix $\underline{D}_{B}$ (signed). Both $\underline{G}_{B}$ and $\underline{D}_{B}$ are deduced with the same fuzzy relation equations used for obtaining ${G}_{B}$ and ${D}_{B}$. The recomposition of the $\underline{D}_{B}^{\prime }s$ gives the image $\underline{D}$ (signed). The quality of the reconstructed images with respect to the original images is measured from the Peak Signal-to-Noise Ratio (PSNR), and we show that $\underline{D}$ is very similar to $D$ for low values of the compression rate. The binary watermark matrix embedded in every $G_{B}$ is variable, thus this method is more secure than another our previous method, where the binary watermark matrix in every $G_{B}$ is constant.