The fractional-order functions show better performance than their corresponding integer-order functions in various image processing
applications. In this paper, the authors propose a novel utilization of fractional-order chaotic systems in color image
encryption. The 4D hyperchaotic Chen system of fractional-order combined with the Fibonacci Q-matrix. The proposed
encryption algorithm consists of three steps: in step#1, the input image decomposed into the primary color channels, R, G,
& B. The confusion and diffusion operations are performed for each channel independently. In step#2, the 4D hyperchaotic
Chen system of fractional orders generates random numbers to permit pixel positions. In step#3, we split the permitted image
into 2 × 2 blocks where the Fibonacci Q-matrix diffused each of them. Experiments performed where the obtained results
ensure the efficiency of the proposed encryption algorithm and its ability to resist attacks.

Robust watermarking is an effective method and a promising solution for securing and
protecting the copyright of digital images. Moments and moment invariants have become
popular tools for robust watermarking due to their geometric invariance and favorable
capability of image description. Many moments-based robust watermarking schemes
have been proposed. However, there is a challenging problem of these schemes that
should be addressed. One of these problems is to improve both imperceptibility and
robustness. In contrast, the other problem, most of these schemes used inefficient,
traditional computation methods of the moments, resulting in an inaccurate and inefficient
performance of the watermarking schemes. To overcome these challenges, in this paper,
we propose a novel robust color image-watermarking algorithm based on new multiple
fractional multi-channel orthogonal moments, fractional-order exponent moments
(MFrEMs), fractional-order polar harmonic transforms (MFrPHTs), and fractional-order
radial harmonic Fourier moments (MFrRHFMs). Firstly, highly accurate fractional new
multi-channel orthogonal moments are computed for the host color images. Then, more
stable and accurate coefficients of fractional new multi-channel orthogonal moments are
selected. Finally, a robust color image watermarking approach for multiple watermarks
images is proposed based on MFrEMs, MFrPHTs, and MFrRHFMs using a 1D Sine
chaotic map. The experimental results demonstrate that the proposed approach provides
robustness against various attacks and better imperceptibility than the existing methods.

A novel cryptosystem of color images is proposed. We defined a multi-channel orthogonal Gegenbauer moments with fractional order (FrMGMs) in Cartesian coordinates and then combined the FrMGMs with the 2D logistic-sine map. The proposed color image encryption method has three main steps in this process. The first step is confusion, where image pixels’ values are changed based on a 2D logistic-sine map to reconstruct the scrambled image. In the second step, the key is generated, which is used in encryption. The key is produced by combining the logistic map and the multi-channel Gegenbauer moments with fractional order. We used the original image to set the logistic map’s initial condition. In the third step, the scrambled image is diffused using the key to obtain an encrypted image. The experiments’ results and the security analysis show that the proposed technique has an extensive keyspace, high key sensitivity, and good encryption effect and can be judged robust against common security attacks.

Orthogonal moments (OMs) are used to extract features from color images. OMs with frac- tional orders are better than the OMs with integer orders due to their ability to extract fine features. This paper defined novel quaternion orthogonal shifted Gegenbauer moments (FrQSGMs) of fractional orders for color image analysis and recognition. Since both shifted Gegenbauer polynomials and the input digital images are defined in the domain [0, 1] ×[0, 1], the proposed FrQSGMs did not need any image mapping or image interpolation. The invariance to geometric transformations of the proposed FrQSGMs is derived by express- ing these moments in geometric moment invariants of fractional order. We conduct var- ious experiments to test the accuracy, invariance to RST, sensitivity to noise, recognition of similar color images, and computational times. The proposed descriptors outperformed the existing orthogonal moments with fractional orders.

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