# Blind Watermarking of Color Medical Images Using Hadamard Transform and Fractional-Order Moments
## 2 Related Work
This method is successful in decreasing the computational time compared with other similar classical quaternion methods. This method achieves a high robustness level against more common attacks. [PERSON] and [PERSON] [28] proposed a watermarking method of the quaternion type using radial substituted Chebyshev moments (QRSCMs) in a holistic way to deal with color images. This method successfully overcame the limitations of other similar methods, such as low-level accuracy, low-speed computation, and instability changes over the coefficient matrix. [PERSON] and [PERSON] [29] introduced an image watermarking method based on fast and accurate QLFMs that first scrambled the binary watermark using Arnold transformation and then embedded this scrambled watermark in the quantized version of selected QLFMs using dither modulation. [PERSON] et al. [30] introduced a watermarking method on color images using Hadamard transformation and Zernike moments of a quaternion-type; quaternion Hadamard transform (QHT) is holistically mutated. Then the transformed components were divided into 8 \(\times\) 8 non-overlapping blocks to apply Schur decomposition, with the watermark embedded on a selected block. The scheme proved its ability to overcome many counterattacks and its robustness against most common geometric attacks. In addition, the scheme lacks evaluation over combined attacks. [PERSON] et al. [31] introduced a scheme on color images using QPHT block magnitudes and Bessel Key form distribution (BKF); the binary watermark was embedded in the amplitudes of the selected precise moment of the cover images blocks. This scheme introduced high imperceptibility compared with similar methods; however, it is still sensitive to most common geometric attacks. [PERSON] et al. [32] introduced a method for copyrights provisions on medical images using QPHFMs. Computing QPHFMs generate the key image to the input CT images to reconstruct the features image, shuffling binary bit pixels using a chaotic system and then applying XOR between the generated feature image and the binary sequence image. [PERSON] et al. [33] proposed a blind color image watermarking scheme depending on optimal selective color channel bits embedding in two middle-frequency bands of quantized coefficients. [PERSON] et al. [34] introduced a robust method on medical images based on QPHFMs using a sped-up robust feature (SURF) to generate multiple zero-watermarking systems. The method is superior in robustness to different complex attacks. [PERSON] et al. [35] introduced a Null-watermarking method on color medical images based on precise quaternion polar harmonic Fourier moments (PQPHFMs) and chaotic maps principles. Similarly, [PERSON] et al. [36] introduced an image watermarking scheme based on FrCMMs and SVD for copyrights provisions. [PERSON] et al. [37] introduced a fractional-order Krawtchouk watermark scheme depending on the decomposition of eigenvalues. This scheme is successful at achieving high robustness compared with other classical integer-order Krawtchouk moments. [PERSON] et al. [38] derived a new fractional order of Jacobi-Fourier moments using mixed low-order moment attributes to increase image descriptor representation and robustness. [PERSON] et al. [39] derived FrDCMs by using the decomposition eigenvalues of the kernel matrix and applied it to a proposed watermarking scheme. [PERSON] et al. [40] proposed a new zero-watermarking method for color images based on accurate computation of multi-channel fractional Legendre-Fourier moments (MfrLFMs) and Arnold encryption to generate a robust watermarking system. [PERSON] et al. [41] proposed a new multi-channel fractional exponent moment order (MfrEM) to construct a robust color image watermarking method. In [42], the authors proposed a semi-blind watermarking method based on combined DWT-CT and Schur SVD decompositions; the method attained a high visual imperceptibility and robustness.