Different from the traditional watermarking schemes, zero-watermarking schemes are lossless embedding methods, which are applicable to be used in medical, military, remote sensing and other fields requiring high-integrity image copyright protection. However, most of the existing zero-watermarking schemes only provide copyright protection for one image at a time, which has certain limitations. This paper proposes a novel zero-watermarking scheme for protecting the copyright of two similar medical images simultaneously. Firstly, an accurate polar complex exponential transform (APCET) is designed using Gaussian numerical integration (GNI) method, which effectively improves the computation accuracy of polar complex exponential transform (PCET). Then, ternary accurate polar complex exponential transform (TAPCET) is constructed based on ternary number theory and APCET, which describes two similar medical images simultaneously. Finally, a robust zero-watermarking algorithm for two similar medical images is proposed based on TAPCET and chaotic mapping. The experimental results show that the proposed scheme can resist common image processing attacks and geometric attacks, and is superior to other zero-watermarking algorithms, being applicable for the copyright protection of two similar medical images simultaneously.

与传统的水印方案不同，零水印方案是无损嵌入方法，适用于医疗、军事、遥感等需要高完整性图像版权保护的领域。然而，现有的零水印方案大多一次只对一幅图像进行版权保护，具有一定的局限性。本文提出了一种新颖的零水印方案，用于同时保护两幅相似医学图像的版权。首先，利用高斯数值积分(GNI)方法设计了精确极坐标复指数变换(APCET)，有效提高了极坐标复指数变换(PCET)的计算精度。然后，三元精确极坐标复指数变换（TAPCET）是基于三元数论和APCET构建的，它同时描述了两个相似的医学图像。最后，提出了一种基于TAPCET和混沌映射的两幅相似​​医学图像的鲁棒零水印算法。实验结果表明，该方案能够抵抗常见的图像处理攻击和几何攻击，优于其他零水印算法，适用于同时对两幅相似医学图像进行版权保护。