Abstract Polar harmonic Fourier moments (PHFMs) are very effective global image descriptors and can be used in many areas of digital image processing. However, the numerical integration error of traditional computation methods significantly affects its computation accuracy. This paper proposed an accurate PHFMs computation method based on Gaussian numerical integration (GNI), which effectively mitigated the numerical integration error, and proposed a novel watermarking algorithm resistant to geometric attacks based on accurate PHFMs and chaotic mapping. This algorithm improves its robustness against geometric attacks by relying on the geometric invariance of accurate PHFMs and scrambles the watermark image by relying on the sensitivity of the Tent chaotic system to the initial value, enhancing its security. The experimental results show that the proposed algorithm can effectively resist geometric attacks, such as scaling, rotation, length-width ratio change and flipping, and is superior to other watermarking algorithms.

中文翻译：

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使用不变的精确极谐波傅里叶矩和混沌映射的鲁棒图像水印

摘要 极谐波傅里叶矩（PHFM）是非常有效的全局图像描述符，可用于数字图像处理的许多领域。然而，传统计算方法的数值积分误差极大地影响了其计算精度。本文提出了一种基于高斯数值积分(GNI)的精确PHFMs计算方法，有效地减轻了数值积分误差，并提出了一种基于精确PHFMs和混沌映射的抗几何攻击的新型水印算法。该算法依靠精确PHFM的几何不变性提高了其对几何攻击的鲁棒性，并依靠Tent混沌系统对初始值的敏感性对水印图像进行加扰，增强了其安全性。