Fractional Krawtchouk Transform is a generalization of the Krawtchouk transforms which has two fractional orders. By adjusting these fractional orders in the weighted two dimensional Krawtchouk polynomials, local image features can be located. This paper proposes a robust image watermarking method using FrKT with firefly and cuckoo search optimization algorithms. The frequency domain image is obtained by applying FrKT for the input image blocks. The optimal fractional parameters of the transform improve the imperceptibility of the secret data in the host images. The fractional parameter selection for the image transformation is performed by Firefly optimization algorithm. Also, the optimal location in each block to hide the secret data is identified by the cuckoo search algorithm. The histogram shifting technique is used to embed the secret data in the optimal locations due to its less computational complexity. The parameters like Peak Signal to Noise Ratio, Normalized Correlation Coefficient, Structural SIMilarity index, Bit Error Rate are used for comparison of the proposed method using the optimization algorithms. The experimental results of the proposed method FrKT with combination of both Firefly and cuckoo search optimization shows better quality of the watermark, robustness and imperceptibility against various attacks. It can be concluded that the proposed FrKT + CS + FA provides an average of 0.92 for most of the attacks that prove the robustness of the proposed scheme.

分数Krawtchouk变换是Krawtchouk变换的推广，具有两个分数阶。通过在加权的二维Krawtchouk多项式中调整这些分数阶，可以定位局部图像特征。提出了一种结合萤火虫和杜鹃搜索优化算法的FrKT鲁棒图像水印方法。通过对输入图像块应用FrKT获得频域图像。变换的最佳分数参数提高了主机图像中秘密数据的不可感知性。通过Firefly优化算法执行图像变换的分数参数选择。而且，通过布谷鸟搜索算法可以识别每个块中隐藏机密数据的最佳位置。直方图移位技术由于其计算复杂度较低而用于将秘密数据嵌入最佳位置。使用诸如优化算法的峰值信噪比，归一化相关系数，结构相似性指数，误码率等参数对所提出的方法进行比较。所提出的方法FrKT与萤火虫和布谷鸟搜索优化相结合的实验结果表明，水印的质量更好，鲁棒性和对各种攻击的不易察觉性。可以得出结论，建议的FrKT + CS + FA为大多数攻击提供了0.92的平均值，证明了建议方案的鲁棒性。使用结构相似指数，误码率对使用优化算法的方法进行比较。所提出的方法FrKT与萤火虫和布谷鸟搜索优化相结合的实验结果表明，水印的质量更好，鲁棒性和对各种攻击的不易察觉性。可以得出结论，建议的FrKT + CS + FA为大多数攻击提供了0.92的平均值，证明了建议方案的鲁棒性。使用结构相似指数，误码率对使用优化算法的方法进行比较。所提出的方法FrKT与萤火虫和杜鹃搜索优化相结合的实验结果表明，水印的质量更好，鲁棒性和对各种攻击的不易察觉性。可以得出结论，建议的FrKT + CS + FA为大多数攻击提供了0.92的平均值，证明了建议方案的鲁棒性。