Copyright protection of medical images is a vital goal in the era of smart healthcare
systems. In recent telemedicine applications, medical images are sensed using medical imaging
devices and transmitted to remote places for screening by physicians and specialists. During their
transmission, the medical images could be tampered with by intruders. Traditional watermarking
methods embed the information in the host images to protect the copyright of medical images.
The embedding destroys the original image and cannot be applied efficiently to images used in
medicine that require high integrity. Robust zero-watermarking methods are preferable over other
watermarking algorithms in medical image security due to their outstanding performance. Most
existing methods are presented based on moments and moment invariants, which have become a
prominent method for zero-watermarking due to their favorable image description capabilities and
geometric invariance. Although moment-based zero-watermarking can be an effective approach to
image copyright protection, several present approaches cannot effectively resist geometric attacks,
and others have a low resistance to large-scale attacks. Besides these issues, most of these algorithms
rely on traditional moment computation, which suffers from numerical error accumulation, leading to
numerical instabilities, and time consumption and affecting the performance of these moment-based
zero-watermarking techniques. In this paper, we derived multi-channel Gaussian–Hermite moments
of fractional-order (MFrGHMs) to solve the problems. Then we used a kernel-based method for the
highly accurate computation of MFrGHMs to solve the computation issue. Then, we constructed
image features that are accurate and robust. Finally, we presented a new zero-watermarking scheme
for color medical images using accurate MFrGHMs and 1D Chebyshev chaotic features to achieve
lossless copyright protection of the color medical images. We performed experiments where their
outcomes ensure the robustness of the proposed zero-watermarking algorithms against various
attacks. The proposed zero-watermarking algorithm achieves a good balance between robustness and
imperceptibility. Compared with similar existing algorithms, the proposed algorithm has superior
robustness, security, and time computation.

Volumetric images have a three-dimensional (3D) view, in which viewers can examine their characteristics from any angle. The more accurate the digital representation of volumetric images, the more precise and valuable the assessment of what these images represent. The representation of volumetric images is a significant area of study in pattern recognition and computer vision. Recently, volumetric image analysis using orthogonal moments with fractional order has opened up a new study pathway, which has led scholars to discover many real-life applications through investigating efficient algorithms to represent the features of 3D images. In this study, a new set of 3D shifted fractional-order Gegenbauer moments (FrGMs) for volumetric image representation is proposed. First, a mathematical description of the shifted Gegenbauer moments for 3D images is presented. Second, a fast, highly accurate method for calculating the fractional-order shifted Gegenbauer moments of 3D images is introduced. Finally, the efficiency of the proposed FrGMs is evaluated through various suitable experiments and compared with existing methods in terms of the reconstruction of 3D images, the invariability property, sensitivity to noise, and computation time. The experimental results clearly show that FrGMs outperform existing related algorithms