Robust and efficient image watermarking via EMD and dimensionality reduction

In the past decades, bi-dimensional empirical mode decomposition (BEMD) algorithms have been developed for the image watermarking task with good robustness and imperceptibility; however, the non-negligible algorithm efficiency has received little attention at the same time. In this paper, we devise a robust and efficient image watermarking algorithm based on 1D empirical mode decomposition (EMD) and dimensional reduction via Hilbert curve. Converting the problem of two-dimensional image watermarking into the problem of one-dimensional signal watermarking promises to enhance the efficiency and the robustness of the proposed algorithm. Specifically, host image is first reduced into one-dimensional signal by the Hilbert curve. Our key insight is a dimensionality reduction strategy based on the Hilbert curve that preserves the spatial local relationship to the greatest extent possible. Second, the one-dimensional signal is segmented into several short signal intervals, and each of them is decomposed into several intrinsic mode functions (IMFs) and a residue by 1D EMD, which is much faster than BEMD. Third, the maximum or minimum points of the first IMF are chosen as the watermark embedding positions. For the watermark image, it is first encrypted by Arnold transform, which improves the security of the algorithm. And then, it is also transformed into one-dimensional signal correspondingly. A repeated embedding strategy is used in the embedding process to improve the algorithm’s robustness. The final watermarked image can be reconstructed by the inverse Hilbert curve transform after integrating the modified first IMF, the remaining IMFs, and the residual. To improve the security and reduce the length of the key, Arnold transform and Huffman coding are adopted. The watermark extraction is the inverse of the embedding process without using the host image and watermark image. Comprehensive experimental results confirm that our new algorithm exhibits good robustness, efficiency, and high imperceptibility. Compared with the actual watermarking algorithms, the newly proposed watermarking algorithm not only reduces the computational expense, but also shows better performance in combating the cropping attacks, Gaussian noise, median filter, image enhancement attacks, etc.