A Two-Stage Robust Reversible Image Watermarking Scheme Using 2D Lattice QIM With Quantization Strips

Robust reversible watermarking (RRW) against both common signal processing (CSP) and geometrical deformation (GD) remains challenging due to the stringent requirements for high robustness, large embedding capacity, good invisibility, and perfect reversibility. Several existing RRW methods against CSP and GD use conventional one-dimensional (1D) quantization index modulation (QIM) for watermark embedding, limiting their robustness. To address this challenge, this paper develops a 2D lattice QIM with quantization strips (QS-QIM) and incorporates it into a two-stage RRW scheme for grayscale images. Specifically, quantization cell optimization (QCO) first connects lattice points along a chosen direction to form codeword lines and splices the corresponding Voronoi cells into larger decoding regions, thereby improving robustness at the cost of reduced per-carrier embedding capacity. Based on the optimized decoding regions, perpendicular-foot-based quantization (PFQ) takes the perpendicular foot on the corresponding codeword line as the watermarked version to minimize embedding distortion. Since PFQ may result in inconsistent robustness among different watermarked carriers, decoding region equalization (DRE) further transforms the spliced decoding regions into uniform strip-shaped regions, i.e., quantization strips, thereby ensuring consistent robustness for all watermarked carriers. Integrating QCO, PFQ, and DRE gives rise to the proposed QS-QIM. In the robust embedding stage, QS-QIM embeds watermark bits into polar complex exponential transform (PCET) moments; in the reversible embedding stage, prediction error expansion-histogram shifting (PEE-HS) embeds the auxiliary information required for lossless recovery. In case of no attacks, the embedded watermark can be extracted without error and the original image can be recovered losslessly. Extensive experiments show that, with the same watermark length and comparable PSNR, the proposed scheme achieves higher robustness than the compared state-of-the-art RRW methods against CSP (e.g., AWGN, S&P, JPEG, JPEG2000, and mean/median filtering), demonstrates strong robustness against scaling and rotation, achieves lower BER under most tested cropping, translation, non-uniform scaling, and shearing conditions, and exhibits improved resistance under the tested diffusion-based regeneration and deep-compression attacks. These results verify the feasibility and effectiveness of the proposed scheme.