Digital watermarking techniques are essential to prevent unauthorized use of images. Since pirated images are often geometrically distorted by operations such as scaling and cropping, accurate synchronization - detecting the embedding position of the watermark - is critical for proper extraction. In particular, cropping changes the origin of the image, making synchronization difficult. However, few existing methods are robust against cropping. To address this issue, we propose a watermarking method that estimates geometric transformations applied to a stego image using a pilot signal, allowing synchronization even after cropping. A grid-shaped pilot signal with distinct horizontal and vertical values is embedded in the image. When the image is transformed, the grid is also distorted. By analyzing this distortion, the transformation matrix can be estimated. Applying the Radon transform to the distorted image allows estimation of the grid angles and intervals. In addition, since the horizontal and vertical grid lines are encoded differently, the grid orientation can be determined, which reduces ambiguity. To validate our method, we performed simulations with anisotropic scaling, rotation, shearing, and cropping. The results show that the proposed method accurately estimates transformation matrices with low error under both single and composite attacks.

翻译：数字水印技术对于防止图像未经授权使用至关重要。由于盗版图像常因缩放和裁剪等操作而发生几何失真，精确同步——即检测水印的嵌入位置——对于正确提取水印极为关键。特别是，裁剪操作会改变图像的坐标原点，使得同步变得困难。然而，现有方法中鲜有能有效抵抗裁剪攻击的。为解决这一问题，我们提出一种水印方法，该方法利用导频信号估计载体图像所经历的几何变换，从而即使在裁剪后也能实现同步。我们将具有不同水平与竖直取值的网格状导频信号嵌入图像中。当图像发生变换时，网格也随之失真。通过分析这种失真，可以估计出变换矩阵。对失真图像应用Radon变换可以估计网格线的角度和间距。此外，由于水平与竖直网格线采用不同编码方式，网格方向得以确定，从而减少了估计的歧义性。为验证所提方法，我们进行了各向异性缩放、旋转、剪切及裁剪的仿真实验。结果表明，该方法在单一攻击和复合攻击下均能以较低误差精确估计变换矩阵。