This paper extends the traditional discrete fractional Krawtchouk transform (DFrKT) with quaternion theory to quaternion discrete fractional Krawtchouk transform (QDFrKT) for the purpose of color image processing. Also, the implementation of QDFrKT is given from two points of view, i.e. direct computation and calculation with quaternion symplectic form. The detailed mathematical derivation together with definition and some properties of quaternion algebra are provided. In order to evaluate the applicability of the proposed QDFrKT method, two applications are investigated, including color image encryption and color image watermarking. For color image encryption, experiments are conducted on key sensitivity, histogram analysis, and noise attacks; while for color image watermarking, experiments are conducted on watermarking invisibility and the robustness on attacks of media filter, average filter, Gaussian filter, and geometric attacks. Moreover, with the imperfect decryption property of the encryption algorithm, a potential application of the encryption strategy for designing watermarking algorithm is discussed. The promising experiment results demonstrate the effectiveness and efficiency of our proposed QDFrKT based color image encryption and watermarking techniques.

本文将传统的离散分数阶 Krawtchouk 变换 (DFrKT) 与四元数理论扩展为四元数离散分数阶 Krawtchouk 变换 (QDFrKT)，以用于彩色图像处理。同时，从直接计算和四元数辛形式计算两个角度给出了QDFrKT的实现。提供了详细的数学推导以及四元数代数的定义和一些性质。为了评估所提出的 QDFrKT 方法的适用性，研究了两种应用，包括彩色图像加密和彩色图像水印。对于彩色图像加密，进行了密钥敏感度、直方图分析和噪声攻击的实验；而对于彩色图像水印，对水印不可见性和对媒体过滤器、平均过滤器、高斯过滤器和几何攻击攻击的鲁棒性进行了实验。此外，针对加密算法不完善的解密特性，讨论了加密策略在设计水印算法中的潜在应用。有希望的实验结果证明了我们提出的基于 QDFrKT 的彩色图像加密和水印技术的有效性和效率。