# Two-Stage Robust Lossless DWI Watermarking Based on Transformer Networks in the Wavelet Domain
## 1 Introduction  
Due to medical imaging technology's continuous development and maturity, digital medical images play an increasingly important role in hospital diagnostic and therapeutic information. Among them, diffusion-weighted imaging (DWI) [1] is a non-invasive imaging technique that can observe the diffusion motion of water molecules within tissues. As a four-dimensional medical image, DWI is widely used to diagnose diseases such as neurology, cardiology, breast cancer, hepatology, and cervical myelopathy. With the continuous rise in medical costs, remote medical care has become an economically efficient and convenient diagnosis and treatment mode, attracting more and more attention and applications. Remote consultations and surgeries have been widely implemented domestically and internationally in hospitals, and a large amount of digital medical imaging data is transmitted on the public network. As medical images contain patient health information and privacy, the direct dissemination of unprotected medical images on the public network can easily be misused and tampered with by unauthorized users, leading to problems such as patient information leakage, medical misdiagnosis, and copyright disputes. In order to effectively protect the copyright information of medical images, digital watermarking technology has become an essential means of protecting the copyright of digital medical images. In medical diagnostics, applying digital watermarking techniques for copyright protection requires meeting specific requirements, including ensuring the integrity of medical image data and the robustness of the watermark. Traditional robust watermarking algorithms [2; 3; 4] can correctly extract watermarked information for copyright authenticationunder various types of image attacks, but they inevitably cause distortion loss to the cover image. Reversible watermarking algorithms [5; 6; 7], on the other hand, focus on lossless embedding and extraction of watermarked information, but they exhibit weaker resistance to attacks and are susceptible to common attack methods. Vulnerable watermarking [8] is capable of tamper detection and localization for protected images, but cannot extract watermarked information for copyright authentication when under attack. For medical images, the distortion caused by robust watermarking methods, the vulnerability of reversible watermarking to various attacks and the low robustness of vulnerable watermarking are unacceptable. This has prompted researchers to develop a new watermarking method that combines robustness and reversibility. In recent years, robust reversible watermarking algorithms have emerged by integrating robust watermarking and reversible techniques. These algorithms achieve lossless recovery of cover images and robustness of watermarks, thereby better fulfilling the requirements for copyright protection of medical images. [PERSON] [9] first proposed a lossless robust watermarking algorithm based on a two-stage watermarking scheme (TSW). In this algorithm, the watermark information is first embedded into the DCT coefficients of the image to generate a robust watermarked image. Then the data required to recover the cover image is used as compensation information and embedded into the watermarked image using a reversible watermarking algorithm. [PERSON] et al. [10] improved the conventional TSW framework by proposing embedding the watermark and compensation information into two independent regions to avoid the negative impact of the second embedding on the first embedding. [PERSON] et al. [6] used recursive jitter modulation to achieve reversible information embedding and provided a reliable solution for image authenticity protection using Slantlet transform and singular value decomposition. [PERSON] et al. [11] designed a new quantization watermarking strategy to embed robust watermarks into the extreme harmonic transform matrix, obtaining the ability to resist geometric distortion.