An effective embedding algorithm for blind image watermarking technique based on Hessenberg decomposition

For digital image copyright protection, watermarking techniques are a promising solution and are of interest to many researchers. In watermarking schemes based on matrix transformation, the embedding element and embedding formula play a very important role in maintaining the quality of a watermark image and the robustness of the watermark. In this paper, a blind image watermarking scheme based on Hessenberg decomposition, where the improvement focuses on the embedding element and embedding formula, is proposed. First, the structure of the Hessenberg factorization is analysed to obtain the most suitable embedding element. Accordingly, this is the first time that the element on the second row and the second column of the upper Hessenberg matrix is selected as an embedding element in a Hessenberg-based image watermarking scheme because of its energy concentration and stability. Second, an improved embedding formula is proposed to address the limitations of previous studies. In the proposed formula, constraint conditions are added to limit the change in all blocks, and a scaling factor is applied to guarantee a trade-off between invisibility and robustness. Here, the scaling factor is carefully calculated by repeating various experiments under different image attacks to achieve an optimal value. Therefore, our proposed embedding formula not only minimizes the modification of the host image after embedding but also helps maintain the robustness of the extracted watermark. Third, to increase the security of the proposed scheme, the watermark image is encoded by the Arnold transform before it is embedded into the host image. The experimental results show that the proposed approach defeats the compared methods in terms of invisibility and execution time. Moreover, the proposed scheme can resist most common attacks when the average normalized correlation value is higher than 0.93 and the extracted watermarks are always clearly recognized. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.