Computer Standards & Interfaces 97 (2026) 104115 Contents lists available at ScienceDirect Computer Standards & Interfaces journal homepage: www.elsevier.com/locate/csi Robust zero-watermarking method for multi-medical images based on Chebyshev–Fourier moments and Contourlet-FFT Xinhui Lu a , Guangyun Yang a , Yu Lu a , Xiangguang Xiong a,b ,∗ a School of Big Data and Computer Science, Guizhou Normal University, Guiyang 550025, China b Guizhou Provincial Specialized Key Laboratory of Information Security Technology in Higher Education Institutions, Guiyang, 550025, China ARTICLE INFO ABSTRACT Keywords: Classical robust watermarking methods embed secret data into a cover image designed to protect its copyright. Zero-watermarking However, they suffer from the problem of balancing imperceptibility and robustness. To address this issue, the Lorenz chaotic system impact of conventional attacks on the stability of feature vectors extracted from the cover image is examined. Chebyshev–Fourier moments Accordingly, we proposed a zero-watermarking method with high attack resistance for multi-medical images Contourlet transform by employing Contourlet transform (CT), Chebyshev–Fourier moments (CHFMs), and fast Fourier transform Fast Fourier transform (FFT). First, each medical image is normalized separately, and the normalized images are fused using a dual- tree complex wavelet transform-based method. Second, the effective region is extracted and subjected to the CT. The CHFMs of the low-frequency sub-bands are calculated, and the FFT is performed on the generated amplitude sequence to construct a feature matrix. A feature image is generated by combining the magnitude of each feature value with the overall mean. Finally, the copyrighted image is encrypted using the Lorenz chaotic system and Fibonacci Q-matrix, after which an exclusive-OR operation is applied between the generated feature image and the encrypted copyrighted image to produce a zero-watermarking signal. The results show that the proposed method exhibits excellent resistance to attack with a normalized correlation coefficient of up to 0.994 between the extracted image and the original copyrighted one. Furthermore, the average anti- attack performance of our proposed method is approximately 2% higher compared to similar existing methods, indicating that our proposed method is highly resistant to conventional, geometric, and combinatorial attacks. 1. Introduction ensuring the robustness and imperceptibility of traditional embedded watermarking techniques, making it valuable for essential applications Steganography is a widely used technique for covertly embedding in many fields, such as multimedia data management. secret data within multimedia covers, aiming to ensure undetectability With the different domains used in constructing feature images, and robustness. By effectively concealing data presence, it enhances se- curity and privacy, with broad applications across various fields [1–4]. there are three categories of zero-watermarking techniques. The first Unlike steganography, which has the primary purpose of concealing type comprises spatial-domain-based zero-watermarking methods [10– the existence of data, robust digital watermarking techniques [5–8] 15]. Yang et al. [10] suggested a zero-watermarking method that aim to confirm copyright ownership by embedding specific secret data uses the center pixels of different channels of the cover image as the in the protected object. However, because of the strategy used to center of the circle, whereby the pixels covered by rings with different embed the secret data into the cover, increasing the strength of the radii and widths constitute the feature image. After that, the final embedding degrades the quality of the cover, thus damaging cover zero-watermarking signal is generated by executing an exclusive-OR integrity. To address the limitations of traditional watermarking meth- operation on the encrypted copyrighted and feature images. Chang ods, Wen et al. [9] introduced zero-watermarking. Unlike conventional et al. [11] proposed a method using secret sharing exhibiting strong approaches, this technique preserves the original image by generating robustness and security. Chang et al. [12] used a Sobel operator to authentication data from stable image features rather than altering extract the texture and edge features of a cover image to construct a pixel values, ensuring both integrity and copyright protection. As a robust zero-watermarking signal. Zou et al. [13] proposed a similarity result, the zero-watermarking technique can effectively balance the retrieval method with good resistance to attack. These methods process contradiction between reducing the original cover image’s quality and ∗ Corresponding author at: School of Big Data and Computer Science, Guizhou Normal University, Guiyang 550025, China. E-mail address: xxg0851@163.com (X. Xiong). https://doi.org/10.1016/j.csi.2025.104115 Received 3 April 2025; Received in revised form 12 November 2025; Accepted 8 December 2025 Available online 8 December 2025 0920-5489/© 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies. X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 the cover image directly in the pixel domain, offering implementation (2) The protection cost of multiple medical images was reduced by advantages of simplicity and intuitiveness. a fusion operation using a dual-tree complex wavelet transform-based The second type of method is frequency-domain-based [16–22]. method. Yang et al. [16] proposed a method that was based on the non- (3) The CT and CHFMs were employed to construct a zero- subsampled Shearlet transform and Schur decomposition, which watermarking signal to address the problem that existing methods are achieved better anti-attack performance. Huang et al. [17] extracted only resistant to limited attacks. low-frequency sub-bands (LSs) using a dual-tree complex wavelet trans- (4) The Lorenz chaotic system and Fibonacci Q-matrix were utilized form (DTCWT), partitioned the LSs, and used Hessenberg decomposi- to encrypt the copyrighted image to heighten the proposed method’s tion to yield a robust signal. Lu et al. [18] proposed fusing the cover security. images with the gray-weighted averaging fusion method, generating The remainder of the paper is organized as follows: Section 2 a robust zero-watermarking signal using the fast finite Shearlet trans- presents the basic theory, including the Lorenz chaotic system and Fi- form and Schur decomposition. Wu et al. [20] presented a robust bonacci Q-matrix, image normalization technique, image fusion method scheme for constructing a zero-watermarking signal to encrypt medical using DTCWT, CHFMs, CT, and FFT. Section 3 analyzes the effect of images using the Contourlet transform (CT). These methods generate conventional attacks on the stability of feature vectors extracted from robust zero-watermarking signals by transforming the cover image cover images. Section 4 describes the key steps of the copyrighted im- from the spatial to the frequency domain, leveraging frequency-domain age encryption, zero-watermarking signal construction, and detection. properties that offer enhanced resistance against non-geometric attacks. Section 5 presents the attack resistance of our method and evaluates Although the above methods are effective against conventional its superiority by comparing it with similar ones. The final section image processing attacks, they are not against large-scale geometric attacks such as rotation, scaling, and cropping, because the features ex- concludes this paper. tracted by these methods are not geometrically invariant. Therefore, to enhance the resilience of zero-watermarking methods against geometric 2. Basic theory attacks, some scholars have proposed the use of continuous orthogonal moments that possess stability and geometric invariance [23–27], to 2.1. Lorenz chaotic system optimize the construction and verification of zero-watermarking sig- nals. This falls into the third category of zero-watermarking methods. The Lorenz chaotic system is a nonlinear dynamic system discovered Bessel–Fourier moments (BFMs) [23] are among the most representa- by the American meteorologist Edward Norton Lorenz in 1963 during tive continuous orthogonal moments. Their radial polynomials are con- his research on weather changes. The system models atmospheric con- sidered to be feature functions with good orthogonality and are widely vective motion using three-dimensional ordinary differential equations, used in the field of pattern recognition. Gao et al. [24] proposed a ro- generating high-quality chaotic sequences free from short-cycle ef- bust method using BFMs. This method first normalizes the cover image fects. Its unpredictability and randomness make it particularly suitable to get translation and scaling invariance. Then it computes the BFMs for image encryption applications. The Lorenz chaotic system [37] is of the normalized image to construct a zero-watermarking signal us- represented as follows: ing moment–rotation invariance. Subsequently, neural networks were introduced into watermarking techniques to comprehensively improve ⎧𝑥̇ = 𝑎(y − 𝑥) ⎪ their adaptability and robustness in the face of complex and changing ⎨𝑦̇ = 𝑥(𝑐 − 𝑧) − 𝑦 (1) image processing and geometric attacks [28–36]. Gong et al. [28] ⎪ ⎩𝑧̇ = 𝑥𝑦 − 𝑏𝑧 proposed a robust medical image zero-watermarking method based on a residual DenseNet. He et al. [29] proposed a robust image method where 𝑎, 𝑏, and 𝑐 denote the three constants of the Lorenz chaotic based on shrinkage and a redundant feature elimination network. Such system, and 𝑥, 𝑦, and 𝑧 represent its three state variables. The Lorenz methods provide a higher level of understanding and protection of chaotic system produces a butterfly-shaped chaotic attractor as dis- image content using the superb feature extraction capability of Neural played in Fig. 1(a) when 𝑎 = 10, 𝑏 = 83 , 𝑐 = 28, and (𝑥0 , 𝑦0 , 𝑧0 ) = networks. However, Neural network-based zero-watermarking methods (0.1, 0.1, 0.1). In Fig. 1(a), the system is bounded, stochastic, and non- face multiple challenges, including substantial training data require- periodic. Fig. 1(b) and (c) illustrate the bifurcation and Lyapunov ments, high computational complexity, limited interpretability, and exponential plots under variations in parameter 𝑐. susceptibility to adversarial attacks. All of the above methods satisfy the basic requirements of digital 2.2. Fibonacci Q-matrix watermarking technology. However, most of these methods lack a strong anti-attack ability to resist diverse attacks, with poor perfor- To enhance encryption security and reliability, the researchers uti- mance against geometric and combinatorial attacks. Additionally, the lize properties of the Fibonacci sequence in their method design, signifi- costs of centralized protection and the occupation of storage space for cantly improving protection capabilities for enhanced privacy preserva- multiple images are relatively high. To address these issues, a zero- tion and information security. The recurrence formula for the Fibonacci watermarking method that combines CT, Chebyshev–Fourier moments sequence [38] is calculated as follows: (CHFMs), and fast Fourier transform (FFT) is proposed. This approach leverages the directional selectivity and sparsity of CT, the orthogonal- 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 , 𝑛 > 2 (2) ity and rotational invariance properties of CHFMs, and the computa- tionally efficient and numerically stable properties of FFT. Compared where 𝐹1 = 𝐹2 = 1, 𝐹𝑛 denotes the 𝑛th Fibonacci number. with zero-watermarking methods that only use frequency domain or The Fibonacci Q-matrix is constructed using Fibonacci numbers. It orthogonal moments, this method enhances robustness against geomet- is usually represented as a 2 × 2 matrix as follows: ric and combinatorial attacks by combining CT and CHFMs, which [ ] 1 1 fully utilize the multi-scale features of CT and the geometric invari- 𝑄= (3) 1 0 ance of CHFMs. Additionally, the method adopts DTCWT-based fusion for efficient multi-image protection and storage reduction. The main The corresponding inverse matrix 𝑄−1 of the Q-matrix is defined as: contributions are as follows: (1) The effects of conventional attacks on the stability of feature [ ] 1 −1 vectors extracted from cover images were analyzed. The results indicate 𝑄−1 = (4) −1 0 that the extracted feature vectors are highly resistant to attacks. 2 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Fig. 1. Chaotic attractor diagram, bifurcation diagram, and Lyapunov exponential spectrum of the Lorenz system. Fig. 2. Experiment results of image normalization. The 𝑛th power of the Q-matrix is defined as follows: 2.4. Image fusion using DTCWT [ ] 𝐹 𝐹𝑛 𝑄𝑛 = 𝑛+1 (5) The dual-tree complex wavelet transform (DTCWT) is a technique 𝐹𝑛 𝐹𝑛−1 that combines the multi-scale analysis capabilities of the discrete The determinants of the Q-matrix can be expressed as: wavelet transform with high computational efficiency. It utilizes a tree structure with low-pass and high-pass filter banks to decompose the Det(𝑄𝑛 ) = 𝐹𝑛+1 𝐹𝑛−1 − 𝐹𝑛2 = (−1)𝑛 (6) real and imaginary parts of the image into multiple scales. At each scale, the DTCWT generates a low-frequency component and six detail The corresponding inverse matrix 𝑄−𝑛 of the 𝑄𝑛 is given below: [ ] components with different orientations (±15◦ , ±45◦ , ±75◦ ). Recently, 𝐹 −𝐹𝑛 DTCWT has been widely adopted in image fusion [39]. The DTCWT 𝑄−𝑛 = 𝑛−1 (7) −𝐹𝑛 𝐹𝑛+1 efficiently extracts multi-scale image details, producing fused images with richer content and improved visual quality. 2.3. Image normalization (1) DTCWT of the image. Apply DTCWT to the original images for 1-level decomposition to obtain the low-frequency coefficients (𝐿𝐿1 , Normalization is a critical step in image processing and computer 𝐿𝐿2 , . . . , 𝐿𝐿𝑘 ) and high-frequency coefficients (𝐻𝐿1 , 𝐿𝐻 2 , . . . , 𝐻𝐻 𝑘 ,) vision [39]. Generally, the cover image after the normalization oper- with the following equations: ation is transformed into a standard form that can resist attacks from [ ] ( ) affine transformations, such as translation, rotation, and scaling. The 𝐿𝐿𝑘 , 𝐻𝐿𝑘 , 𝐿𝐻 𝑘 , 𝐻𝐻 𝑘 = DTCWT 𝐼𝑘 (11) two-dimensional (𝑝 + 𝑞)-order moment of the cover image 𝑓 (𝑥, 𝑦) is where 𝑘 = 1, 2, . . . , 𝑛. defined as: (2) Fusion of high-frequency coefficients. Calculate the energy of ∑∑ each coefficient and its neighboring region in the high-frequency sub- 𝑚𝑝𝑞 = 𝑥𝑝 𝑦𝑞 𝑓 (𝑥, 𝑦) (8) 𝑥 𝑦 bands of all images. The window size is set to 2r + 1, where 𝑟 is the window radius. Within this window, each coefficient is given a where 𝑝, 𝑞 = 0, 1, 2, 3..., and the image central moments are defined 1 weight of (2𝑟+1) 2 . The local energy 𝐸(𝑥, 𝑦) of image 𝑘 at position (𝑥, 𝑦) as ∑∑ is calculated as: 𝑝 𝑢𝑝𝑞 = (𝑥 − 𝑥) ̄ 𝑞 𝑓 (𝑥, 𝑦) ̄ (𝑦 − 𝑦) (9) ∑ 𝑥+𝑟 ∑ 𝑦+𝑟 𝑥 𝑦 𝐸𝑘 (𝑥, 𝑦) = (|𝑓𝑘 (𝑚, 𝑛)|2 ) (12) 𝑚 𝑚 𝑚=𝑥−𝑟 𝑛=𝑦−𝑟 where (𝑥, ̄ is the center of mass of the image with 𝑥̄ = 𝑚10 and 𝑦̄ = 𝑚01 . ̄ 𝑦) 00 [ 00] The fused high-frequency coefficients are selected from the image 𝑢20 𝑢11 The covariance matrix M of the cover image is defined as with maximum energy at each position. The relevant formula is shown 𝑢11 𝑢02 below. . The normalization operation for the image is based on the invariance of the matrix as follows: 𝐻𝐹 (𝑥, 𝑦) = arg max 𝐸𝑘 (𝑥, 𝑦) (13) 𝑘 [ 𝑚] [ ]⎡ 𝑐 0 ⎤[ 𝑒 ][ ] 𝑥 cos𝛼 sin𝜕 ⎢ √𝜆1 ⎥ 1𝑥 𝑒1𝑦 𝑥 − 𝑥̄ For positions where multiple images have equal maximum energy, the = 𝑐 (10) average of the coefficients of those images is taken. 𝑦𝑚 −sin𝜕 cos𝛼 ⎢ 0 √ ⎥ −𝑒1𝑦 𝑒1𝑥 𝑦 − 𝑦̄ ⎣ 𝜆2 ⎦ (3) Fusion of low-frequency coefficients. The maximum coefficients where 𝜆1 and 𝜆2 are the eigenvalues of M, and the corresponding across all images in the LSs are selected. [ ]𝑇 [ ]𝑇 eigenvectors are 𝑒1𝑥 , 𝑒1𝑦 and 𝑒2𝑥 , 𝑒2𝑦 , respectively. 𝐿𝐹 (𝑖, 𝑗) = max(𝐿𝐿1 (𝑖, 𝑗), 𝐿𝐿2 (𝑖, 𝑗), … , 𝐿𝐿𝑘 (𝑖, 𝑗)) (14) The images before and after the normalization process are shown in Fig. 2 for four standard medical images, where the original images (4) Image reconstruction. With the fused high-frequency details are shown in (a)–(d), and the corresponding normalized versions are and low-frequency coefficients, the reconstructed image is obtained by shown in (e)–(h). applying the inverse DTCWT to the fused data. 3 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 The functions 𝑃𝑛𝑚 (𝑟, 𝜃) are orthogonal within the unit circle, where (0 ≤ 𝑟 ≤ 1, 0 ≤ 𝜃 ≤ 2𝜋). 2𝜋 1 𝑃𝑛𝑚 (𝑟, 𝜃)𝑃𝑘𝑙 (𝑟, 𝜃)𝑟𝑑𝑟𝑑𝜃 = 𝛿𝑛𝑚𝑘𝑙 (18) ∫0 ∫0 where 𝛿𝑛𝑚𝑘𝑙 is the Kronecker delta, the image function 𝑓 (𝑟, 𝜃) can be decomposed orthogonally in the polar coordinate system by the func- tional system 𝑃𝑛𝑚 (𝑟, 𝜃). Reconstruction using the CHFMs is thus made possible, and the image reconstruction function 𝑓 (𝑟, 𝜃) can subsequently Fig. 3. Experimental results of image fusion using DTCWT. be written as: ∑ ∞ ∑ +∞ 𝑓 (𝑟, 𝜃) = 𝜙𝑛𝑚 𝑅𝑛 (𝑟) exp(𝑗𝑚𝜃) (19) 𝑛=0 𝑚=−∞ where 𝜙𝑛𝑚 is the CHFM for image 𝑓 (𝑟, 𝜃). 2𝜋 1 𝜙𝑛𝑚 = 𝑓 (𝑟, 𝜃)𝑅𝑛 (𝑟) exp(−𝑗𝑚𝜃)𝑟𝑑𝑟𝑑𝜃 (20) ∫0 ∫0 2.7. Fast Fourier transform The FFT is a fast algorithm based on the discrete Fourier transform (DFT) that leverages the inherent properties of the DFT, including sym- metry, periodicity, and the relationship between odd and even terms. It works by using its intrinsic periodicity and symmetry to decompose a long sequence of DFTs into the sum of many short sequences of Fig. 4. Schematic of the CT. DFTs [43]. The FFT can be represented mathematically as follows: ∑ 𝑁−1 𝑛 𝑥𝑘 = 𝑥𝑛 ∙ 𝑒−𝑖2𝜋𝑘 𝑁 (21) The normalized images from Fig. 2 were fused using the afore- 𝑛=0 mentioned image fusion technique. Fig. 3 presents the fusion results, where k = 0, 1, 2, . . . ..., N -1. where Fig. 3(a) shows the fusion of images Fig. 2(f) and (g); Fig. 3(b) Computing the DFT of a discrete signal using Eq. (21) requires displays the fusion incorporating images Fig. 2(e)–(g); and Fig. 3(c) 𝑁 × 𝑁 steps, whereas the FFT computes the DFT of a discrete signal by demonstrates the fusion combining images Fig. 2(e)–(h). dividing the DFT equation into two independent components, as shown in Eq. (22). 2.5. Contourlet transform (𝑁 )−1 (𝑁 )−1 ∑ 2 𝑚 −𝑖2𝜋𝑘 (𝑁∕2) 1 −𝑖2𝜋𝑘 𝑁 ∑ 2 𝑚 −𝑖2𝜋𝑘 (𝑁∕2) 𝑥𝑘 = 𝑥2𝑚 ∙ 𝑒 +𝑒 𝑥2𝑚+1 ∙ 𝑒 (22) The Contourlet transform (CT) [40] is a dual-filter structure that is 𝑚=0 𝑚=0 effective in obtaining sparse extensions of typical images with smooth ∑𝑁∕2−1 𝑚 −𝑖2𝜋𝑘 𝑁∕2 contours due to its unique multi-resolution and multidirectional capa- where 𝑚=0 𝑥2𝑚 ⋅ 𝑒 represents the even-indexed DFT and 1 ∑𝑁∕2−1 𝑚 −𝑖2𝜋𝑘 𝑁∕2 bility. The Laplace Pyramid is utilized to capture point discontinuities 𝑒 −𝑖2𝜋𝑘 𝑁 𝑥2𝑚+1 ⋅ 𝑒 means the odd-indexed DFT. 𝑚=0 in the image, while a bank of directional filters connects these discon- tinuities into a linear structure. Basic elements such as contour lines 3. Effect of the attacks on the stability of extracted feature vectors are used for image expansion, which facilitates the reconstruction of complex image features. Fig. 4 shows a schematic of the decomposition The performance of zero-watermarking methods against attacks of a 512 × 512 image using CT. mainly depends on whether the essential features extracted when con- structing a zero-watermarking signal exhibit strong robustness against 2.6. Chebyshev-Fourier moments attacks. In this study, we first normalized and fused multiple images. Then, we extracted the effective regions of the fused images and The Chebyshev–Fourier moments (CHFMs) were proposed by Ping performed CT and CHFMs to generate the magnitude sequence. Finally, et al. [41] in 2002 and entail the following key steps: an FFT was performed on the generated magnitude sequence to obtain In polar coordinates (𝑟, 𝜃), the Chebyshev–Fourier function 𝑃𝑛𝑚 (𝑟, 𝜃) 64-bit feature vectors. To validate the ability of the proposed method consists of two components: the radial function 𝑅𝑛 (𝑟) and the angular to resist attacks, the following two experiments were conducted: function exp(𝑗𝑚𝜃). (1) The stability of the extracted feature vectors of the cover image against various attacks was verified on the Chest X-ray image shown 𝑃𝑛𝑚 (𝑟, 𝜃) = 𝑅𝑛 (𝑟) exp(𝑗𝑚𝜃) (15) in Fig. 5. Table 1 shows the corresponding results. As observed, the extracted feature vectors (64 bits) under different attacks are almost where unchanged, and the correlation coefficients are all higher than 0.984, √ 𝑛+2 indicating that the extracted feature vectors exhibit strong robustness 8 1−𝑟 4 ∑ 1 2 (𝑛 − 𝑘)! 𝑅𝑛 (𝑟) = ( ) (−1)𝑘 [2(2𝑟 − 1)]𝑛−2𝑘 (16) in the face of various attacks. 𝜋 𝑟 𝑘!(𝑛 − 2𝑘)! 𝑘=0 (2) The uniqueness of the feature vectors generated from the fused In 2007, Ping et al. [42] showed that CHFMs are deformations of the images was verified on the feature vectors extracted from the images Jacobi–Fourier moments (𝑝 = 2, q = 3/2), and thus the radial function in Fig. 5 after fusion. The experimental results are shown in Tables 2 𝑅𝑛 (𝑟) of CHFMs can be expressed as and 3, where 𝑃1 , 𝑃2 , 𝑃3 , and 𝑃4 denote the Heart, Chest X-ray, Brain, √ and Knee images, respectively. The results show that the extracted 8 1−𝑟 4 ∑ 1 𝑛 (𝑛 + 𝑘 + 1)!22𝑘 𝑠 feature vectors from different fused images differ, with a similarity 𝑅𝑛 (𝑟) = ( ) (−1)𝑘 𝑟 (17) 𝜋 𝑟 𝑘=0 (𝑛 − 𝑘)!(2𝑘 + 1)! of approximately 0.5. In contrast, the feature vectors from the same 4 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Fig. 5. Four original medical images and their fusion. Table 1 Feature vectors generated under different attacks (64-bit). Type of attack Generated feature vectors NC No attacks 1111111100111111000000000000001100000000000000010111111111111111 – JPEG compression (QF = 15) 1111111100111111000000000000001100000000000000011111111111111111 0.998 Median filtering (3 × 3) 1111111100111111000000000000001100000000000000000111111111111111 0.999 Wiener filtering (3 × 3) 1111111100111111000000000000001100000000000000010111111111111111 1.000 Gaussian noise (0.1) 1111111101111111000000000000001100000000000000011111111111111111 0.991 Salt & pepper noise (0.1) 1111111101111111000000000000001100000000000000011111111111111111 0.993 Rotation attack (10◦ ) 1111111101101111000000000000001100000000000000010111111111111111 0.984 Scaling attack (Shrink 0.25x) 1111111100111111000000000000001100000000000000010111111111111111 1.000 Cropping attack (Upper left 1/16) 1111111110011111000000000000001100000000000000010111111111111111 0.992 Table 2 Feature vectors generated by different images fusion (64-bit). Fusion of different images Generated feature vectors 𝑃1 , 𝑃 2 , 𝑃 3 00000011100001111111011111101111111001111100011111000000000001111 𝑃1 , 𝑃 2 , 𝑃 4 01100011111111111111111111111111111111111111111111000000000000011 𝑃2 , 𝑃 3 , 𝑃 4 00001110111100111000011111100011100011111100000000000000000000000 𝑃1 , 𝑃 3 , 𝑃 4 00011111111111111111011111111111111111111111111111000000000000111 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 00010000000000111011111000111111111111111111000111111111111000111 Table 3 protection of multiple images, a robust zero-watermarking method Similarity of feature vectors generated from different images fusion. combining image moments and multi-scale transformation is proposed. Fusion of 𝑃1 , 𝑃2 , 𝑃3 𝑃1 , 𝑃2 , 𝑃4 𝑃2 , 𝑃 3 , 𝑃 4 𝑃1 , 𝑃 3 , 𝑃 4 𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 Figs. 6–8 show the flowcharts for the copyrighted image encryption and different images decryption, zero-watermarking construction, and detection algorithms, 𝑃1 , 𝑃 2 , 𝑃 3 1.000 0.574 0.554 0.546 0.528 respectively. 𝑃1 , 𝑃 2 , 𝑃 4 0.576 1.000 0.501 0.512 0.563 𝑃2 , 𝑃 3 , 𝑃 4 0.552 0.501 1.000 0.581 0.515 𝑃1 , 𝑃 3 , 𝑃 4 0.546 0.512 0.591 1.000 0.530 4.1. Copyrighted image encryption 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 0.528 0.563 0.515 0.530 1.000 To enhance the security of the method, a copyrighted image CI of size 𝑚 × 𝑛 was encrypted using the Lorenz chaotic system and Fibonacci Q-matrix. Fig. 6 shows the experimental results after encrypting the fused images are identical, with a similarity of 1.000. This indicates copyrighted image using the following key steps: that the extracted feature vectors can effectively distinguish the fusion Step 1: Using the original copyrighted image CI of size 𝑚 × 𝑛, the of different images. initial key 𝑥1 of the Lorenz chaotic system is computed. The experimental results demonstrate that the constructed feature ∑𝑚 ∑𝑛 signal exhibits robust performance, providing a theoretical basis for 𝑖=1 𝑗=1 𝐶𝐼(𝑖, 𝑗) + (𝑚 × 𝑛) 𝑥1 = (23) utilizing the feature signal to generate a robust zero-watermarking 2000 + (𝑚 × 𝑛) signal. Two new values, 𝑥2 and 𝑥3 , are then obtained by iterating twice. Finally, 𝑥1 , 𝑥2 , and 𝑥3 are chosen as the initial values of the state 4. Proposed method variables x, y, and z, respectively. Step 2: Based on the selected initial values, three vectors, X, Y and To address the poor performance of most methods in resisting Z are generated using Eq. (1), from which three sub-vectors of length diversity attacks and the high storage space required for centralized 𝑚 × 3𝑛 are chosen to construct a vector V of length 𝑚 × 𝑛. 5 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Fig. 6. An example of simple copyrighted image encryption and decryption. Step 3: The copyrighted image CI is first reshaped into a one- method is used to construct the binary feature image 𝐹 = {𝑓 (𝑖, 𝑗), 1 ≤ dimensional vector G, and then, the sequence V is sorted in ascending 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛}. order to obtain index IX. Finally, G is permuted using IX to generate a { 1, 𝐶(𝑖, 𝑗) ≥ 𝑀 scrambled one-dimensional vector R. 𝐹 (𝑖, 𝑗) = (25) 0, 𝐶(𝑖, 𝑗) < 𝑀 Step 4: The R vector is reshaped into a matrix of size 𝑚 × 𝑛, and the matrix is partitioned into blocks of size 2 × 2. Step 9: Perform an XOR operation between the feature matrix Step 5: Set the parameter 𝑛 = 20 in Eq. (5) to compute 𝑄𝑛 . Then, F obtained in Step 8 and the encrypted copyrighted image ECI in perform a modulo-2 operation on each term in 𝑄𝑛 to obtain a binary Section 4.1 to get a robust zero-watermarking image, which is then matrix. authenticated and registered with a third-party intellectual property Step 6: Using the Fibonacci Q-matrix construction method intro- rights (IPR). The unique ID number is then saved as the basis for duced in Section 2.2, an exclusive-OR operation is performed between copyright extraction. The zero-watermarking image construction and each block of size 2 × 2 and the Fibonacci Q-matrix to obtain an registration processes are thus completed. encrypted copyrighted image (ECI ). 𝑍 = XOR (𝐸𝐶𝐼, 𝐹 ) (26) The image decryption step is simply the reverse of the encryption step and is not described here. 4.3. Zero-watermarking detection 4.2. Zero-watermarking construction The zero-watermarking detection process is the reverse of the zero- watermarking construction method. Below is a description of the key Assuming that the sizes of the four cover images I and the copy- steps. righted image CI are 𝑀 × 𝑁 and 𝑚 × 𝑛, respectively. A robust feature Step 1: Same as Step 1 of the zero-watermarking signal generation image is constructed by combining image moments and multi-scale process, four gray-scale images of size 𝑀 × 𝑁 are normalized using transforms, and a robust zero-watermarking signal is generated by the method described in Section 2.3, followed by scaling and rotation performing an exclusive-OR (XOR) operation with the encrypted copy- normalizations to produce standard normalized images. righted image. The key steps of the proposed method are outlined as Step 2: The corresponding feature image is obtained by performing follows. the normalized standard images following Steps 2–8 in Section 4.2. Step 1: Using the moment-based image normalization technique in Step 3: A zero-watermarking image saved by a third-party au- Section 2.3, four gray-scale images of size 𝑀 × 𝑁 are subjected to thentication center can be obtained using the ID number. Then, an the corresponding normalization process. Then, scaling and rotation XOR operation is performed on the zero-watermarking image and the normalizations are applied to obtain four standard normalized images. generated feature image, resulting in an undecrypted copyright image Step 2: A new fused image (FI ) is generated by fusing the informa- (UCI ). tion of the four normalized images using the image fusion method in ( ) Section 2.4. 𝑈 𝐶𝐼 = XOR 𝑍, 𝐹 ′ (27) Step 3: For a fused image FI of size 𝑀 × 𝑁, the geometric center of Step 4: The original copyrighted image CI can be recovered by FI is defined as 𝑥 = 𝑀2 , 𝑦 = 𝑁2 . The effective region (ER) of size 𝑃 × 𝑄 decrypting the undecrypted copyrighted image UCI using the Lorenz is extracted from the fused image FI using Eq. (24). chaotic system and the Fibonacci Q-matrix. Because the original CI [ ] 𝑃 𝑃 𝑄 𝑄 is a meaningful and recognizable image, the human eye can directly 𝐸𝑅 = FI (𝑥 − ) ∶ (𝑥 + − 1), (𝑦 − ) ∶ (𝑦 + − 1) (24) authenticate the recovered copyrighted image. 2 2 2 2 Step 4: Using the Contourlet transform, the LSs are obtained from 5. Experimental results and analysis the extracted ER. A square region (SR) of size ((𝑀 +𝑁)∕2)×((𝑀 +𝑁)∕2) is then selected from LSs. 5.1. Experimental parameters Step 5: The maximum-order 𝑛max = 25 is selected, and the region SR is computed using Eq. (15) to obtain (𝑛max + 1)(2𝑛max − 1) CHFMs. To verify the effectiveness of our method, a simulation experiment Step 6: To make the number of CHFMs the same size as the was conducted in two software environments: one configured with copyrighted image, 𝑚 × 𝑛 moment values are obtained by expanding MATLAB R2023a and the other with Microsoft Windows 11. Four the amplitude sequence of the (𝑛max + 1)(2𝑛max − 1) moments, converting 512 × 512 standard medical images: Heart, Chest X-ray, Brain, and them into an 𝑚 × 𝑛 one-dimensional vector 𝐴 = {𝑎(𝑖), 1 ≤ 𝑖 ≤ 𝑚 × 𝑛}. Knee were chosen as experimental images, as shown in Fig. 9(a)∼(d). Step 7: FFT is performed on one-dimensional vector A to generate Fig. 9(e) shows the original binary copyrighted image, which is a one-dimensional vector 𝐵 = {𝑏(𝑖), 1 ≤ 𝑖 ≤ 𝑚 × 𝑛}. 64 × 64 pixel binary image composed of a binary sequence of length Step 8: Reshape the vector B into a two-dimensional matrix C. 4096. Fig. 9(f) displays a zero-watermarking image generated by this Calculate the mean value M of the matrix C and binarize it using M as proposed method. As can be seen, the resulting zero-watermarking a threshold. Specifically, if the value of an element of C is greater than image looks cluttered and, if not recovered, unrecognizable to the or equal to M, the feature bit is 1; otherwise, the feature bit is 0. This human eye. 6 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Fig. 7. Flowchart of zero-watermarking construction method. Fig. 8. Flowchart of zero-watermarking detection method. Fig. 9. Original medical image, original copyrighted image, and generated zero-watermarking image. 5.2. Evaluation indicators reconstruction error (MSRE) to objectively assess the methods’ perfor- mance. (1) Normalized correlation The attack resistance of the methods is measured using a gen- The NC value is commonly used to measure the similarity between eralized normalized correlation coefficient (NC), and the quality of a copyrighted image extracted from an attacked cover image and the the reconstructed image is evaluated using a generalized mean-square original copyrighted image. The NC value typically falls between 0 and 7 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Fig. 10. (a) Variations in the value of 𝑅𝑛 (𝑟) with 𝑟, in the interval 0 < 𝑟 ≤ 1, 𝑛max = 1, 2, 9, 10. (b) MSRE values corresponding to CHFMs with different orders for the grayscale image Heart. 1, where 0 indicates that the two images are not similar, and 1 indicates Table 4 that they are identical. In other words, the higher the NC value, the Results of resisting JPEG compression. more similar the two images are, suggesting that the method is more Fusion of Quality factors (QF) Average resistant to attacks. different images 5 10 15 20 25 ∑𝑚 ∑𝑛 𝑖=1 𝑗=1 [𝑂𝐶𝐼(𝑖, 𝑗)𝐸𝐶𝐼 (𝑖, 𝑗)] 𝑃1 0.985 0.990 0.986 0.989 0.996 0.989 𝑁𝐶 (𝑂𝐶𝐼, 𝐸𝐶𝐼) = √ √ (28) 𝑃2 0.996 0.999 0.998 1.000 0.999 0.998 ∑𝑚 ∑𝑛 2 ∑𝑚 ∑𝑛 2 𝑗=1 𝑂𝐶𝐼(𝑖, 𝑗) 𝑗=1 𝐸𝐶𝐼(𝑖, 𝑗) 𝑃3 0.995 0.994 0.997 0.998 0.998 0.996 𝑖=1 𝑖=1 𝑃4 0.987 0.996 0.998 1.000 0.999 0.996 where both OCI and ECI are of size 𝑚 × 𝑛; OCI refers to the original 𝑃1 , 𝑃 2 0.992 0.994 0.998 0.998 0.999 0.996 copyrighted image, while ECI is the copyrighted image extracted after 𝑃1 , 𝑃 3 0.991 0.991 0.993 0.993 0.993 0.992 𝑃1 , 𝑃 4 0.989 0.995 0.994 0.994 0.995 0.993 the cover image has undergone an attack. 𝑃2 , 𝑃 3 0.994 0.996 0.996 0.998 0.998 0.996 (2) Mean-squared reconstruction error 𝑃2 , 𝑃 4 0.994 0.997 0.998 0.999 0.999 0.997 As a generalized tool, the quality of the reconstructed images can be 𝑃3 , 𝑃 4 0.989 0.996 0.998 1.000 0.999 0.996 objectively assessed using the MSRE in Eq. (29). In general, the smaller 𝑃1 , 𝑃 2 , 𝑃 3 0.991 0.997 0.998 0.998 0.999 0.996 the MSRE value, the lower the error between the reconstructed and 𝑃1 , 𝑃 2 , 𝑃 4 0.994 0.996 0.998 0.998 0.999 0.997 𝑃2 , 𝑃 3 , 𝑃 4 0.994 0.995 0.998 0.997 0.999 0.996 original images, indicating better image quality; conversely, a higher 𝑃1 , 𝑃 3 , 𝑃 4 0.987 0.991 0.989 0.990 0.989 0.989 MSRE value suggests poorer reconstruction quality. 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 0.994 0.999 0.999 0.998 0.998 0.997 +∞ +∞ [ ′ ]2 ∫ ∫ 𝐼 (𝑥, 𝑦) − 𝐼(𝑥, 𝑦) 𝑑𝑥𝑑𝑦 𝑀𝑆𝑅𝐸 = −∞ −∞ +∞ +∞ (29) ∫−∞ ∫−∞ [𝐼 ′ (𝑥, 𝑦)]2 𝑑𝑥𝑑𝑦 images is 1.000. In addition, the proposed method exhibited strong where I and 𝐼 ′ denote the original and reconstructed images, respec- robustness when the cover image was attacked. To perform a systematic tively. and robust assessment of the proposed method, images, as well as two-, three-, and four-fused images, were tested for their resistance to attacks. 5.3. Image reconstruction experiments The detailed experiments are described below. Fig. 10(a) shows the variation in the values of the radial polynomial 5.4.1. JPEG compression attack function 𝑅𝑛 (𝑟) (Eq. (16)) in the interval [0, 1]. It can be seen that 𝑅𝑛 (𝑟) The ability to resist JPEG compression attacks is summarized in has n zeros, which satisfy a uniform distribution in the interval [0, 1], Table 4. It can be seen that the proposed method is more resistant to and the values of the function located near the zeros of different orders JPEG compression, with an average NC value of 0.995. This may be are almost the same. because the proposed method chooses to compute the CHFMs in the To verify the reconstruction ability of the CHFMs, experiments were LSs of the CT transform, where the information is more concentrated, conducted by setting the parameters 𝑝 = 2 and q = 1.5 and selecting thereby enhancing its resistance to JPEG compression. a standard medical heart image of size 512 × 512. Figs. 10(b) and 11 show the corresponding MSRE values and reconstructed images for n 5.4.2. Noise attack = 0, 5, . . . , 25. As shown in Fig. 10(b), the MSRE is the lowest when Table 5 summarizes the experimental results against Gaussian white 𝑛𝑚𝑎𝑥 = 25. The best-quality reconstructed image is observed in Fig. 11 noise and salt & pepper noise attacks, and Table 6 lists the results for for 𝑛𝑚𝑎𝑥 = 25. Gaussian noise and speckle noise attacks. Note that for the Gaussian white noise attack, the values of the parameter intensity are 0, 0.5, and 5.4. Resistance to regular attack experiments 1. It is observed that our method has high resistance to noise attacks with NC values of 0.968, 0.963, 0.958, and 0.989 against Gaussian In this section, the NC value is used to quantitatively assess the white noise, salt & pepper noise, Gaussian noise, and speckle noise quality of the extracted copyrighted image, which reflects the method’s attacks, respectively. This may be because the technique used in the resistance to attacks. The results show that when the cover image is not proposed method has a suppression effect on noise in the transform attacked, the NC value between the extracted and original copyrighted domain, which enhances its ability to resist noise attacks. 8 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Fig. 11. Samples of CHFMs reconstructed with different orders. Table 5 Results of resisting Gaussian white noise and salt & pepper noise attacks. Fusion of Gaussian white noise Average Salt & pepper noise Average different images (0.05, (0.1, (0.2, (0.1, (0.5, 0.1 0.2 0.3 0.4 0.5 0.025,0) 0.05,0) 0.1,0) 0.1,0.1) 0.25,0) 𝑃1 0.936 0.914 0.904 0.915 0.900 0.914 0.951 0.922 0.903 0.894 0.896 0.913 𝑃2 0.996 0.993 0.990 0.990 0.979 0.990 0.993 0.990 0.985 0.979 0.974 0.984 𝑃3 0.964 0.956 0.944 0.958 0.931 0.950 0.978 0.958 0.951 0.941 0.937 0.953 𝑃4 0.976 0.959 0.943 0.960 0.930 0.954 0.982 0.966 0.949 0.938 0.932 0.954 𝑃1 , 𝑃 2 0.992 0.988 0.979 0.981 0.962 0.981 0.984 0.980 0.971 0.959 0.950 0.969 𝑃1 , 𝑃 3 0.978 0.963 0.956 0.963 0.937 0.959 0.978 0.968 0.950 0.940 0.925 0.952 𝑃1 , 𝑃 4 0.979 0.973 0.959 0.970 0.948 0.966 0.980 0.971 0.953 0.950 0.935 0.958 𝑃2 , 𝑃 3 0.994 0.991 0.985 0.986 0.976 0.986 0.990 0.984 0.980 0.979 0.973 0.981 𝑃2 , 𝑃 4 0.994 0.988 0.979 0.985 0.966 0.982 0.991 0.983 0.976 0.971 0.958 0.976 𝑃3 , 𝑃 4 0.984 0.970 0.963 0.974 0.948 0.968 0.986 0.974 0.964 0.956 0.952 0.966 𝑃1 , 𝑃 2 , 𝑃 3 0.991 0.980 0.974 0.976 0.957 0.976 0.987 0.971 0.966 0.959 0.953 0.967 𝑃1 , 𝑃 2 , 𝑃 4 0.988 0.979 0.971 0.977 0.963 0.976 0.986 0.978 0.968 0.963 0.956 0.970 𝑃2 , 𝑃 3 , 𝑃 4 0.987 0.982 0.974 0.980 0.961 0.977 0.986 0.981 0.973 0.961 0.953 0.971 𝑃1 , 𝑃 3 , 𝑃 4 0.978 0.971 0.961 0.967 0.947 0.965 0.978 0.969 0.957 0.949 0.936 0.958 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 0.990 0.984 0.976 0.972 0.965 0.977 0.997 0.994 0.989 0.990 0.987 0.991 Table 6 Results of resisting Gaussian noise and speckle noise attacks. Fusion of Gaussian noise Average Speckle noise Average different images 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 𝑃1 0.926 0.903 0.894 0.898 0.894 0.903 0.993 0.987 0.983 0.983 0.977 0.985 𝑃2 0.991 0.986 0.977 0.972 0.960 0.977 0.996 0.995 0.993 0.992 0.992 0.993 𝑃3 0.965 0.954 0.947 0.939 0.932 0.947 0.997 0.996 0.994 0.994 0.992 0.995 𝑃4 0.971 0.952 0.939 0.934 0.930 0.945 0.997 0.997 0.993 0.991 0.990 0.993 𝑃1 , 𝑃 2 0.984 0.971 0.959 0.950 0.944 0.962 0.991 0.988 0.989 0.979 0.978 0.985 𝑃1 , 𝑃 3 0.970 0.957 0.941 0.931 0.919 0.944 0.992 0.986 0.987 0.980 0.981 0.986 𝑃1 , 𝑃 4 0.977 0.963 0.949 0.938 0.933 0.952 0.988 0.988 0.987 0.980 0.980 0.985 𝑃2 , 𝑃 3 0.989 0.984 0.980 0.977 0.974 0.981 0.996 0.993 0.992 0.991 0.989 0.992 𝑃2 , 𝑃 4 0.987 0.979 0.970 0.962 0.957 0.971 0.995 0.994 0.986 0.987 0.987 0.990 𝑃3 , 𝑃 4 0.976 0.970 0.960 0.956 0.950 0.963 0.997 0.995 0.989 0.989 0.990 0.992 𝑃1 , 𝑃 2 , 𝑃 3 0.983 0.967 0.961 0.950 0.942 0.961 0.992 0.988 0.985 0.984 0.979 0.986 𝑃1 , 𝑃 2 , 𝑃 4 0.979 0.970 0.964 0.957 0.951 0.964 0.989 0.992 0.989 0.983 0.984 0.988 𝑃2 , 𝑃 3 , 𝑃 4 0.982 0.975 0.966 0.959 0.956 0.968 0.994 0.996 0.985 0.990 0.986 0.990 𝑃1 , 𝑃 2 , 𝑃 4 0.973 0.960 0.952 0.940 0.935 0.952 0.990 0.991 0.985 0.983 0.986 0.987 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 0.990 0.984 0.976 0.972 0.965 0.977 0.997 0.994 0.989 0.990 0.987 0.991 5.4.3. Filtering attack offset rank attacks and cropping attacks, with an average NC of 0.989 Table 7 lists the experimental results for Median and Wiener fil- against offset rank attacks and 0.965 against cropping attacks. This tering attacks, and Table 8 gives the experimental results for Gaussian finding can be attributed to two key reasons. First, the orthogonality low-pass and mean filtering attacks. As observed, our method has high of the Chebyshev polynomials is independent of each other within resistance to filtering attacks with NC values of 0.994, 0.997, 1.000, a specific interval, which helps reduce interference between different and 0.994 against median filtering, Wiener filtering, Gaussian low-pass frequency components, thereby improving the stability and robustness filtering, and mean filtering attacks, respectively. This may be because of the signal. Second, the FFT converts the amplitude signal from the the CT transform used in the proposed method provides a nuanced and time domain to the frequency domain, resulting in the loss of key compelling characterization of the local and global features of the cover information when the original signal is affected by an offset or crop- image in the transform domain, which makes the cover image highly ping attack in the time domain. In contrast, the information remains stable when subjected to a filtering attack, effectively enhancing its relatively intact in the frequency domain. Consequently, the proposed ability to resist the filtering attack. method can effectively resist offset-rank and cropping attacks. 5.5. Resistance to geometric attack experiments 5.5.2. Scaling attack The results of the scaling attack are summarized in Table 10. Note 5.5.1. Offset rows, columns, and cropping attacks that after scaling the image by a factor of 𝑥, it needs to be scaled Table 9 provides experimental results for offset rows, columns, and again by a factor of 𝑥1 before constructing the feature image. As seen, cropping attacks. The proposed method exhibits robustness against the proposed method demonstrated outstanding resistance to scaling 9 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Table 7 Results of resisting median and Wiener filtering attacks. Fusion of Median filtering Average Wiener filtering Average different images 3 × 3 5 × 5 7 × 7 9 × 9 11 × 11 3 × 3 5 × 5 7 × 7 9 × 9 11 × 11 𝑃1 0.990 0.973 0.968 0.966 0.965 0.972 0.998 0.998 0.998 0.998 0.998 0.998 𝑃2 0.999 0.999 0.999 0.999 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 𝑃3 1.000 0.997 0.995 0.994 0.992 0.995 1.000 0.999 0.998 0.997 0.997 0.998 𝑃4 1.000 0.999 0.997 0.996 0.995 0.997 1.000 1.000 1.000 1.000 0.999 1.000 𝑃1 , 𝑃 2 0.998 0.996 0.996 0.995 0.993 0.996 0.999 0.998 0.997 0.997 0.995 0.997 𝑃1 , 𝑃 3 1.000 0.997 0.994 0.992 0.988 0.994 1.000 0.999 0.998 0.994 0.992 0.996 𝑃1 , 𝑃 4 0.999 0.996 0.994 0.990 0.986 0.993 1.000 0.998 0.995 0.991 0.989 0.994 𝑃2 , 𝑃 3 0.999 0.999 0.998 0.995 0.993 0.997 1.000 0.999 0.998 0.997 0.995 0.998 𝑃2 , 𝑃 4 0.998 0.997 0.994 0.994 0.993 0.995 0.999 0.998 0.994 0.993 0.992 0.995 𝑃3 , 𝑃 4 0.999 0.997 0.994 0.991 0.986 0.993 0.998 0.998 0.997 0.996 0.995 0.997 𝑃1 , 𝑃 2 , 𝑃 3 0.999 0.997 0.995 0.994 0.990 0.995 0.999 0.998 0.995 0.995 0.995 0.996 𝑃1 , 𝑃 2 , 𝑃 4 0.998 0.994 0.990 0.989 0.987 0.992 0.998 0.995 0.992 0.989 0.987 0.992 𝑃2 , 𝑃 3 , 𝑃 4 0.999 0.996 0.993 0.990 0.985 0.993 0.999 0.997 0.993 0.991 0.989 0.994 𝑃1 , 𝑃 3 , 𝑃 4 0.998 0.998 0.996 0.992 0.989 0.995 0.999 0.998 0.998 0.996 0.995 0.997 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 1.000 0.999 0.998 0.998 0.997 0.998 1.000 1.000 0.999 0.998 0.997 0.999 Table 8 Results of resisting Gaussian low-pass and mean filtering attacks. Fusion of Gaussian low-pass filtering Average Mean filtering Average different images 3 × 3 5 × 5 7 × 7 9 × 9 11 × 11 3 × 3 5 × 5 7 × 7 9 × 9 11 × 11 𝑃1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.997 0.996 0.996 0.997 𝑃2 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.998 0.998 0.998 0.997 0.998 𝑃3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.996 0.996 0.995 0.997 𝑃4 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.998 0.997 0.998 𝑃1 , 𝑃 2 1.000 1.000 1.000 1.000 1.000 1.000 0.996 0.995 0.994 0.992 0.990 0.993 𝑃1 , 𝑃 3 1.000 1.000 1.000 1.000 1.000 1.000 0.997 0.996 0.994 0.990 0.990 0.993 𝑃1 , 𝑃 4 1.000 1.000 1.000 1.000 1.000 1.000 0.995 0.994 0.991 0.990 0.984 0.991 𝑃2 , 𝑃 3 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.996 0.995 0.993 0.991 0.995 𝑃2 , 𝑃 4 0.999 0.999 0.999 0.999 0.999 0.999 0.995 0.993 0.991 0.991 0.989 0.992 𝑃3 , 𝑃 4 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.995 0.993 0.992 0.988 0.993 𝑃1 , 𝑃 2 , 𝑃 3 1.000 1.000 1.000 1.000 1.000 1.000 0.997 0.994 0.993 0.991 0.988 0.992 𝑃1 , 𝑃 2 , 𝑃 4 0.999 0.999 0.999 0.999 0.999 0.999 0.996 0.993 0.988 0.986 0.986 0.990 𝑃2 , 𝑃 3 , 𝑃 4 1.000 1.000 1.000 1.000 1.000 1.000 0.997 0.994 0.989 0.986 0.985 0.990 𝑃1 , 𝑃 3 , 𝑃 4 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.996 0.995 0.991 0.989 0.994 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.997 0.995 0.994 0.992 0.995 Table 9 Results of resisting offset and cropping attacks. Fusion of Offset direction Average Cropping position Average different images Shift right Shift left Shift up Shift down Upper left Upper left Upper left Center 2 columns 2 columns 2 rows 2 rows 1/16 1/8 1/4 1/4 𝑃1 0.995 0.993 0.992 0.994 0.993 0.976 0.963 0.923 0.933 0.949 𝑃2 0.997 0.995 0.995 0.997 0.996 0.992 0.987 0.962 0.979 0.980 𝑃3 0.996 0.992 0.993 0.993 0.993 0.983 0.973 0.922 0.955 0.958 𝑃4 0.993 0.995 0.998 0.997 0.996 0.979 0.963 0.932 0.952 0.957 𝑃1 , 𝑃 2 0.991 0.992 0.985 0.986 0.989 0.993 0.981 0.958 0.982 0.978 𝑃1 , 𝑃 3 0.985 0.984 0.988 0.984 0.985 0.980 0.970 0.941 0.968 0.965 𝑃1 , 𝑃 4 0.981 0.981 0.984 0.979 0.981 0.987 0.967 0.928 0.948 0.958 𝑃2 , 𝑃 3 0.989 0.993 0.992 0.986 0.990 0.989 0.979 0.960 0.971 0.975 𝑃2 , 𝑃 4 0.988 0.988 0.988 0.987 0.988 0.992 0.981 0.959 0.956 0.972 𝑃3 , 𝑃 4 0.986 0.989 0.988 0.982 0.986 0.987 0.962 0.925 0.930 0.951 𝑃1 , 𝑃 2 , 𝑃 3 0.989 0.989 0.988 0.985 0.988 0.991 0.981 0.952 0.975 0.975 𝑃1 , 𝑃 2 , 𝑃 4 0.986 0.987 0.985 0.982 0.985 0.991 0.974 0.945 0.964 0.969 𝑃2 , 𝑃 3 , 𝑃 4 0.987 0.987 0.987 0.981 0.986 0.989 0.974 0.948 0.951 0.966 𝑃1 , 𝑃 3 , 𝑃 4 0.986 0.982 0.986 0.979 0.983 0.979 0.960 0.924 0.953 0.954 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 0.992 0.995 0.992 0.989 0.992 0.993 0.983 0.958 0.963 0.974 attacks, as evidenced by its average NC value of 0.998. The main 5.5.3. Rotation attack reasons for this are as follows: The CT transform can effectively capture The results of rotation attacks are summarized in Table 11. It can the local features of the cover image, and the method’s resistance to be seen that the proposed method achieves strong resistance to rotation scaling attacks is improved by the normalizing process based on image attack, with an average NC value of 0.964. It is mainly attributed to the moments. These properties enable the proposed method to maintain fact that CHFMs possess rotational invariance when computing the LSs. the stability of the extracted feature vectors when an image undergoes This property ensures that even if the LSs are rotated, the feature data a scaling attack. Consequently, the proposed method has enhanced its can still be effectively extracted in the low-frequency part. Additionally, ability to resist scaling attacks. the FFT transforms the amplitude sequence, further enhancing the 10 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Table 10 Results of resisting scaling attack. Fusion of Scaling Average different images Shrink Shrink Shrink Shrink Magnify Magnify Magnify Magnify 0.25x 0.4x 0.5x 0.7x 4x 2x 2.4x 1.3x 𝑃1 0.999 0.999 1.000 0.998 1.000 1.000 1.000 0.999 0.999 𝑃2 1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 1.000 𝑃3 0.998 0.998 1.000 0.997 1.000 1.000 1.000 0.999 0.999 𝑃4 0.999 1.000 1.000 0.999 1.000 1.000 1.000 1.000 1.000 𝑃1 , 𝑃 2 0.998 0.993 1.000 0.994 1.000 1.000 0.998 0.999 0.998 𝑃1 , 𝑃 3 0.997 0.994 1.000 0.995 1.000 1.000 0.998 0.999 0.998 𝑃1 , 𝑃 4 0.996 0.988 0.999 0.992 1.000 1.000 0.999 0.999 0.997 𝑃2 , 𝑃 3 0.999 0.996 0.999 0.996 1.000 1.000 1.000 1.000 0.998 𝑃2 , 𝑃 4 0.997 0.992 0.999 0.994 1.000 1.000 0.999 0.999 0.997 𝑃3 , 𝑃 4 0.998 0.993 1.000 0.994 1.000 1.000 1.000 1.000 0.998 𝑃1 , 𝑃 2 , 𝑃 3 0.998 0.992 1.000 0.995 1.000 1.000 0.999 0.999 0.998 𝑃1 , 𝑃 2 , 𝑃 4 0.994 0.989 0.999 0.992 1.000 1.000 0.997 0.999 0.996 𝑃2 , 𝑃 3 , 𝑃 4 0.996 0.989 1.000 0.990 1.000 1.000 0.998 0.999 0.996 𝑃1 , 𝑃 3 , 𝑃 4 0.996 0.991 0.999 0.992 1.000 1.000 0.998 0.999 0.997 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 1.000 0.999 1.000 0.999 1.000 1.000 1.000 1.000 1.000 Table 11 Results of resisting rotation attack. Fusion of Rotation angle Average different images 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 90◦ 𝑃1 0.973 0.958 0.956 0.955 0.956 0.960 0.965 0.980 0.996 0.967 𝑃2 0.984 0.970 0.958 0.940 0.941 0.958 0.972 0.984 0.999 0.967 𝑃3 0.986 0.981 0.982 0.979 0.981 0.982 0.987 0.988 0.995 0.985 𝑃4 0.975 0.951 0.937 0.924 0.924 0.941 0.961 0.979 0.997 0.954 𝑃1 , 𝑃2 0.998 1.000 1.000 1.000 0.993 0.994 0.998 0.999 0.993 0.997 𝑃1 , 𝑃3 0.981 0.969 0.967 0.962 0.958 0.963 0.961 0.971 0.992 0.969 𝑃1 , 𝑃4 0.973 0.965 0.962 0.950 0.938 0.947 0.954 0.973 0.994 0.962 𝑃2 , 𝑃3 0.980 0.959 0.950 0.934 0.935 0.949 0.958 0.975 0.998 0.960 𝑃2 , 𝑃4 0.969 0.948 0.925 0.912 0.899 0.925 0.954 0.969 0.995 0.944 𝑃3 , 𝑃4 0.978 0.957 0.937 0.922 0.916 0.939 0.952 0.974 0.996 0.952 𝑃1 , 𝑃2 , 𝑃3 0.978 0.962 0.957 0.947 0.942 0.944 0.956 0.977 0.994 0.962 𝑃1 , 𝑃2 , 𝑃4 0.969 0.968 0.954 0.944 0.933 0.937 0.949 0.971 0.991 0.957 𝑃2 , 𝑃3 , 𝑃4 0.972 0.948 0.933 0.920 0.918 0.936 0.949 0.973 0.994 0.949 𝑃1 , 𝑃3 , 𝑃4 0.978 0.964 0.961 0.955 0.950 0.951 0.958 0.979 0.988 0.965 𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 0.978 0.969 0.956 0.944 0.938 0.950 0.959 0.979 0.999 0.963 rotational invariance in the frequency domain and thereby increasing staying above 0.970, and the performance difference between various the robustness of the method against rotation attacks. Consequently, groups is not apparent. These results demonstrate that even when the the proposed method improves the resistance to rotation attacks. number of fused images increases dramatically, the proposed method can still effectively resist multiple types of attacks and can be applied 5.6. Combined attack to the fusion needs of different numbers of images. To further measure the anti-attack capability of the method, the 5.8. Experiments on image datasets cover image was subjected to combined attacks, and the corresponding results are listed in Table 12. As shown in the table, the average NC To verify the generalizability of the proposed method, we conduct value of the suggested method against the combined attacks is still as experiments on four benchmark image datasets: BossBase [44], BOWS- high as 0.980. According to the results above, the proposed method 2 [45], COVID [46], and SIPI [47]. For the experiments, 100 images is capable of resisting a range of combined attacks, in addition to were randomly selected from each image dataset for evaluation. The conventional and geometric attacks, indicating that it can withstand proposed method is first used to construct a zero-watermarking image various types of attacks and exhibits strong, robust performance. for each test image. Then, an anti-attack test is performed to quantify the performance by calculating the NC value between the extracted 5.7. Impact of multiple images fusion on the performance of the proposed image and the original copyrighted image. The average test results method and the standard deviation STD of the 100 images are shown in Table 15. It can be seen that the average NC values of the proposed To objectively evaluate the impact of multiple image fusion on method are always higher than 0.95, and the STDs are less than the performance of the proposed method, nine sets of images were 0.032 in all datasets, indicating that the proposed method not only first randomly selected from the image dataset BossBase [44], with exhibits excellent robustness on different datasets but also has excellent the numbers of 5, 10, 15, 20, 30, 40, 60, 80, and 100, respectively. generalization ability. Although the experiments are conducted on Then, for each set of images, a fused image is generated using the standard datasets, the possible attacks on real-world natural images fusion technique described in Section 2.4. The proposed method is and medical images are simulated, which validates the ability of our then utilized to construct a zero-watermarking image for the fused method to resist attacks and generalization. COVID [46] is a publicly image and to perform experiments on various attacks. The experimental open dataset of chest X-rays and CT images of patients, containing 930 results are shown in Tables 13 and 14, from which it can be seen images. The proposed method demonstrates superior attack resistance that the proposed method exhibits stable robustness under different on this dataset, indicating its potential application in real-world image numbers of image fusion conditions, with the average NC value always copyright protection scenarios. 11 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Table 12 Results of resisting combination attacks. Fusion of Type of combination attack Average different images Rotation (10◦ ) JPEG (QF = 5) JPEG (QF = 10) Gaussian noise Gaussian noise Median filtering Median filtering + Scaling + Scaling + Wiener (0.2) + Rotation (0.2) + Wiener (3 × 3) + Salt & (3 × 3) + Gaussian (Shrink 0.25x) (Shrink 0.25x) filtering (3 × 3) (10◦ ) filtering (3 × 3) pepper noise (0.2) noise (0.2) 𝑃1 0.979 0.987 0.989 0.903 0.902 0.920 0.903 0.943 𝑃2 0.967 0.997 0.998 0.986 0.985 0.990 0.985 0.991 𝑃3 0.957 0.995 0.995 0.954 0.955 0.958 0.953 0.972 𝑃4 0.979 0.989 0.995 0.953 0.953 0.967 0.953 0.973 𝑃1 , 𝑃 2 0.973 0.992 0.996 0.970 0.970 0.978 0.970 0.981 𝑃1 , 𝑃 3 0.967 0.991 0.992 0.958 0.956 0.968 0.957 0.974 𝑃1 , 𝑃 4 0.956 0.987 0.994 0.960 0.964 0.973 0.962 0.977 𝑃2 , 𝑃 3 0.978 0.996 0.998 0.984 0.985 0.987 0.986 0.990 𝑃2 , 𝑃 4 0.975 0.993 0.999 0.982 0.980 0.983 0.978 0.987 𝑃3 , 𝑃 4 0.967 0.988 0.998 0.969 0.970 0.973 0.971 0.981 𝑃1 , 𝑃 2 , 𝑃 3 0.977 0.992 0.997 0.968 0.968 0.977 0.970 0.981 𝑃1 , 𝑃 2 , 𝑃 4 0.963 0.994 0.994 0.970 0.969 0.975 0.970 0.981 𝑃2 , 𝑃 3 , 𝑃 4 0.964 0.994 0.995 0.978 0.976 0.980 0.975 0.985 𝑃1 , 𝑃 3 , 𝑃 4 0.976 0.985 0.991 0.959 0.960 0.963 0.960 0.974 𝑃1 , 𝑃 2 , 𝑃 3 , 𝑃 4 0.978 0.995 0.999 0.982 0.983 0.983 0.983 0.989 Table 13 Experimental results of multi-image fusion against common attacks. Number of JPEG compression Median filtering Wiener filtering Gaussian low-pass Mean filtering Gaussian fusion images (QF = 15) (3 × 3) (3 × 3) filtering (3 × 3) (3 × 3) noise (0.1) 5 images 0.998 1.000 1.000 1.000 0.994 0.994 10 images 0.998 1.000 1.000 1.000 0.998 0.989 15 images 1.000 1.000 1.000 1.000 0.999 0.991 20 images 0.997 0.999 0.999 1.000 0.996 0.989 30 images 0.996 1.000 1.000 1.000 0.996 0.990 40 images 0.999 1.000 1.000 1.000 0.996 0.990 60 images 0.997 0.999 0.999 1.000 0.996 0.980 80 images 0.998 1.000 1.000 1.000 0.990 0.979 100 images 0.999 1.000 1.000 1.000 0.998 0.992 Table 14 Experimental results of multi-image fusion against geometric attack. Number of Salt & pepper Speckle Gaussian white Rotation Scaling attack Cropping attack fusion images noise (0.1) noise (0.1) noise (0.1,0.05,0) attack (10◦ ) (Shrink 0.25x) (Upper left 1/16) 5 images 0.994 0.997 0.994 0.970 0.998 0.992 10 images 0.992 0.994 0.994 0.974 0.999 0.996 15 images 0.993 0.994 0.993 0.983 1.000 0.998 20 images 0.982 0.994 0.988 0.974 0.997 0.994 30 images 0.992 0.991 1.000 0.992 1.000 0.999 40 images 0.978 0.996 0.994 0.976 0.998 0.994 60 images 0.990 0.994 0.991 0.971 0.997 0.990 80 images 0.980 0.993 0.991 0.974 0.998 0.996 100 images 0.979 0.995 0.993 0.990 1.000 0.996 5.9. Comparison with similar methods in [22] employs FQGPCET, a nonlinear transformation method based on quaternions and polar coordinates, which is more sensitive to noise To highlight the superiority of the proposed method, six representa- and shifts due to its nonlinear nature, potentially leading to distortion tive similar methods were selected for comparison experiments under of the extracted features. The method in [25] is less robust to cropping the same conditions, and the results are shown in Table 16, where the and offset attacks due to the sensitivity of the polar harmonic invariant proposed method is generally superior to the six similar methods in moments to cropping and offset. Specifically, the NC value obtained terms of robustness. The reasons for this can mainly be attributed to by the method is only 0.872 for the center 1/16 cropping attack the following four aspects: First, the methods in [16–19] all use block since the cropping part is not used in the computation. However, the processing, and the ‘‘block effect’’ introduced by these methods can proposed method constructs binary eigenvectors using CHFMs and FFTs lead to discontinuities or blurring of the boundaries between neigh- based on frequency-domain feature extraction. This makes the proposed boring image blocks, which reduces the stability and accuracy of the method robust to this type of attack. Third, compared with the NSST feature vectors, whereas the proposed method generates the amplitude used by the method in [16], the CT transform has sparse properties sequences by calculating the CHFMs of the effective regions of the and better detail characterization capabilities. Thus, it can filter or LSs and performs the FFT transform. The proposed method not only perform specific processing to reduce the noise in the cover image to avoids the ‘‘block effect’’ inherent in these methods but also leverages fewer coefficients, allowing for effective noise suppression and thereby the rotational invariance and scaling invariance of CHFMs to construct improving the ability to resist noise attacks. Fourth, unlike the DTCWT feature vectors by computing CHFMs in the effective regions of the LSs. used in [17], the proposed method constructs features by introducing Performing the FFT transform further enhances the method’s resistance the CT transform, which enables the extraction of more stable principal to geometric attacks. Second, the methods in [22,25] both construct component information. When subjected to noise, filtering, and JPEG a zero-watermarking image based on image moments. The method compression, the CT transform can effectively remove high-frequency 12 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Table 15 Comparative experimental results for four different datasets. Type of attack BossBase BOWS-2 COVID SIPI Average NC STD Average NC STD Average NC STD Average NC STD JPEG compression (QF = 5) 0.9899 0.0185 0.9962 0.0038 0.9971 0.0036 0.9963 0.0036 JPEG compression (QF = 15) 0.9973 0.0052 0.9989 0.0012 0.9991 0.0014 0.9990 0.0013 Median filtering (3 × 3) 0.9985 0.0020 0.9992 0.0008 0.9996 0.0008 0.9991 0.0005 Median filtering (11 × 11) 0.9933 0.0069 0.9961 0.0026 0.9985 0.0020 0.9965 0.0030 Wiener filtering (3 × 3) 0.9997 0.0005 0.9998 0.0003 0.9999 0.0002 0.9999 0.0004 Wiener filtering (11 × 11) 0.9979 0.0013 0.9987 0.0009 0.9995 0.0009 0.9994 0.0006 Gaussian low-pass filtering (3 × 3) 0.9999 0.0003 0.9999 0.0002 0.9999 0.0001 0.9999 0.0002 Gaussian low-pass filtering (11 × 11) 0.9998 0.0004 0.9996 0.0003 0.9995 0.0006 0.9996 0.0003 Mean filtering (3 × 3) 0.9985 0.0009 0.9987 0.0008 0.9993 0.0010 0.9982 0.0010 Mean filtering (11 × 11) 0.9946 0.0026 0.9956 0.0021 0.9977 0.0028 0.9961 0.0026 Gaussian noise (0.1) 0.9665 0.0309 0.9846 0.0114 0.9881 0.0116 0.9812 0.0212 Gaussian noise (0.5) 0.9528 0.0244 0.9510 0.0302 0.9609 0.0318 0.9542 0.0245 Salt & pepper noise (0.1) 0.9786 0.0240 0.9912 0.0065 0.9932 0.0070 0.9802 0.0390 Salt & pepper noise (0.5) 0.9537 0.0230 0.9617 0.0267 0.9710 0.0269 0.9622 0.0253 Speckle noise (0.1) 0.9948 0.0041 0.9946 0.0035 0.9962 0.0036 0.9900 0.0017 Speckle noise (0.5) 0.9868 0.0084 0.9845 0.0081 0.9889 0.0088 0.9859 0.0106 Gaussian white noise (0.1,0.05,0) 0.9750 0.0329 0.9936 0.0088 0.9927 0.0095 0.9815 0.0291 Gaussian white noise (0.5,0.25,0) 0.9558 0.0079 0.9714 0.0242 0.9765 0.0252 0.9744 0.0214 Rotation attack (10◦ ) 0.9695 0.0086 0.9703 0.0071 0.9824 0.0120 0.9844 0.0140 Rotation attack (80◦ ) 0.9694 0.0083 0.9716 0.0076 0.9644 0.0304 0.9714 0.0242 Scaling attack (Shrink 0.25x) 0.9990 0.0012 0.9994 0.0010 0.9991 0.0014 0.9991 0.0010 Scaling attack (Magnify 4x) 0.9991 0.0009 0.9990 0.0012 0.9999 0.0002 0.9998 0.0003 Cropping attack (Upper left 1/16) 0.9963 0.0053 0.9988 0.0012 0.9971 0.0032 0.9969 0.0028 Cropping attack (Upper left 1/8) 0.9766 0.0072 0.9859 0.0106 0.9872 0.0186 0.9845 0.0081 Table 16 Experimental results of the proposed method and six similar methods. Type of attack Method Method Method Method Method Method Proposed [16] [17] [18] [19] [22] [25] method JPEG compression (QF = 15) 0.983 0.985 0.995 0.989 0.997 0.996 0.998 Median filtering (3 × 3) 0.996 0.989 0.998 0.980 0.972 1.000 0.999 Wiener filtering (3 × 3) 0.998 0.995 1.000 0.999 0.996 1.000 1.000 Gaussian low-pass filtering (3 × 3) 0.999 0.999 1.000 0.996 0.998 1.000 1.000 Mean filtering (3 × 3) 0.997 0.995 0.995 0.989 0.992 0.979 0.998 Gaussian noise (0.1) 0.987 0.979 0.985 0.981 0.966 0.944 0.991 Salt & pepper noise (0.1) 0.962 0.939 0.964 0.997 0.959 0.954 0.993 Speckle noise (0.1) 0.969 0.966 0.974 0.987 0.971 0.980 0.997 Gaussian white noise (0.1, 0.05, 0) 0.988 0.925 0.984 0.960 0.958 0.940 0.993 Rotation attack (10◦ ) 0.899 0.939 0.890 0.896 0.985 0.985 0.984 Scaling attack (Shrink 0.25x) 0.997 0.995 1.000 0.981 0.992 0.997 1.000 Cropping attack (Upper left 1/16) 0.998 0.996 0.976 0.998 1.000 1.000 0.992 Offset attack (Shift up 2 rows) 0.98 0.969 0.977 0.981 0.973 0.950 0.995 Table 17 Summary of improvement rates from Table 16. Type of attack Method [16] Method [17] Method [18] Method [19] Method [22] Method [25] Average JPEG compression (QF = 15) 0.910% 1.114% 0.910% 1.012% 0.706% 0.706% 0.893% Median filtering (3 × 3) 0.909% 1.835% 0.706% 2.567% 2.884% −0.100% 1.467% Wiener filtering (3 × 3) 0.908% 0.806% 0.000% 0.806% 0.806% 0.000% 0.555% Gaussian low-pass filtering (3 × 3) 0.100% 0.100% 0.000% 0.402% 0.200% 0.000% 0.134% Mean filtering (3 × 3) 0.504% 0.302% 0.302% 0.910% 0.605% 2.675% 0.883% Gaussian noise (0.1) 0.405% 1.226% 0.814% 0.916% 2.588% 4.757% 1.784% Salt & pepper noise (0.1) 3.115% 5.751% 3.008% 3.762% 3.545% 4.088% 3.878% Speckle noise (0.1) 2.890% 3.209% 3.746% 4.180% 2.678% 1.735% 3.073% Gaussian white noise (0.1,0.05,0) 1.120% 7.701% 1.223% 3.438% 4.088% 5.638% 3.868% Rotation attack (10◦ ) 9.821% 9.333% 10.562% 10.438% 0.306% 0.204% 6.777% Scaling attack (Shrink 0.25x) 0.705% 0.908% 0.000% 1.833% 0.908% 0.705% 0.843% Cropping attack (Upper left 1/16) −0.601% 0.405% 0.303% 0.609% −0.800% −0.800% −0.147% Offset attack (Shift up 2 rows) 3.323% 2.683% 1.842% 2.577% 3.323% 4.737% 3.081% signals while retaining the low-frequency signals that represent the 5.10. Ablation experiment cover image, resulting in a more stable extracted feature vector. In summary, our method is robust against most attacks compared to In this study, a zero-watermarking method that combines CT, similar methods. CHFMs, and FFT is proposed. The experimental results show that it Based on the data in Table 16, the improvement rate of the proposed provides excellent performance. In general, CT is a multi-scale trans- method compared to the other methods is given in Table 17. It can be form that can resist noise and filtering attacks. However, it is difficult seen that, for most attacks, the proposed method outperforms various to adaptively adjust due to the fixed orientation of its basis functions, techniques with an average improvement rate of approximately 2%, resulting in limited adaptive capability against geometric attacks such indicating that the proposed method is effective. as rotation. CHFMs utilize the rotational and scaling invariance of 13 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Table 18 Results of ablation experiments. Type of attack Our method Without CT Without CHFMs Without FFT JPEG compression (QF = 15) 0.998 0.973 0.988 0.981 Median filtering (3 × 3) 0.999 0.964 0.998 0.991 Wiener filtering (3 × 3) 1.000 0.968 0.999 0.997 Gaussian low-pass filtering (3 × 3) 1.000 0.989 0.999 0.990 Mean filtering (3 × 3) 0.998 0.950 0.998 0.989 Gaussian noise (0.1) 0.991 0.884 0.968 0.988 Salt & pepper noise (0.1) 0.993 0.798 0.975 0.977 Speckle noise (0.1) 0.997 0.827 0.986 0.967 Gaussian white noise (0.1,0.05,0) 0.993 0.827 0.977 0.974 Rotation attack (10◦ ) 0.984 0.982 0.893 0.964 Scaling attack (Shrink 0.25x) 1.000 0.997 0.902 0.970 Cropping attack (Upper left 1/16) 0.992 0.981 0.881 0.906 ( ) moments to resist geometric distortion attacks; however, their global The space complexities in [16–19,22], and [25] are 𝑂 73 𝑁 2 + ( ) ( ) ( ) 16 integration property makes them highly sensitive to local distortions, 𝑂(5𝑛2 ), 𝑂 513 𝑁 2 + 𝑂(5𝑛2 ), 𝑂 593 𝑁 2 + 𝑂(5𝑛2 ), 𝑂 657 𝑁 2 + 𝑂(6𝑛2 ), such as compression and noise. FFT-based global spectral analysis ( ) 64 ( 64) 64 enhances robustness to geometric attacks and resists interference in the 𝑂 19364 𝑁 2 + 𝑂(5𝑛2 ), and 𝑂 69 𝑁 2 + 𝑂(5𝑛2 ), respectively. 64 frequency domain, but it is weak against localized cropping attacks. In summary, the computational complexity of the proposed method To verify how CT, CHFMs, and FFT enhance robustness in our is approximately 𝑂(𝑁 3 ), and the space complexity is 𝑂(𝑁 2 ). The in- method, we performed ablation experiments. The experimental results crease in computational complexity of the proposed method com- are shown in Table 18. It can be seen that these three transforms are pared to similar methods is primarily due to the introduction of im- complementary in their ability to resist attacks. CT provides resistance age moments, which enhance resistance against geometric attacks. In to noise and filtering attacks through multi-scale frequency domain fea- terms of space complexity, the proposed method is comparable to its tures; CHFM provides resistance to geometric attacks, such as rotation counterparts, indicating that the fusion technique effectively mitigates and scaling, through geometric invariant features; and FFT enhances the problem of increasing storage overhead as the number of images resistance to conventional and geometric attacks through frequency increases. domain stability. These three transformations can provide resilience against different types of attacks separately, and their synergistic effect 5.12. Key space and sensitivity analysis together enhances the overall robustness of our method. A simple image encryption method based on the Lorenz chaotic 5.11. Complexity comparison system and the Fibonacci Q-matrix is proposed to improve the security of the original binary copyrighted images. Next, the security of the Table 19 summarizes the average running time of the seven methods proposed image encryption scheme is analyzed in terms of key space for processing 100 images under the same experimental conditions. It can be seen that methods [16,17,25] have the shortest running and sensitivity. time because they are zero-watermarking methods for a single image; methods [18,19] have an increased running time due to the need 5.12.1. Key space to perform operations such as fusion and normalization on multiple In general, the security of an encryption scheme depends critically images. The method [22] has a relatively long running time due to the on the quantity of its key space. A sufficiently large key space is essen- need to compute image moments, even though it only processes a single tial to provide resistance against exhaustive attack. The security of the image. The proposed method has the longest running time among the proposed encryption scheme primarily relies on the initial conditions of seven methods because it combines operations such as multiple images the Lorenz chaotic system, as described by Eq. (1). In a 64-bit operating fusion, CT, CHFMs, and FFT. In practice, the runtime of the proposed system environment, each parameter is represented as a 64-bit double- method is feasible within 60 s on an ordinary personal computer. precision floating-point number. Consequently, the total key space Taking this into account, the running time of the proposed method is amounts to (264 )3 = 2192 . A key space of this magnitude is considered approximately 31.5 s, which is within the acceptable level. adequate to ensure the cryptographic strength of the encryption scheme In the experiments, the sizes of the original cover image and the against exhaustive attack, thereby enhancing its robustness in practical copyrighted image are assumed to be 𝑁 × 𝑁 and 𝑛 × 𝑛, respectively. applications. The proposed method mainly consists of the following steps: image fusion, CT, CHFMs, FFT, copyrighted image encryption, and zero- 5.12.2. Sensitivity analysis watermarking generation. ( )The (computational ) ( complexities ) of these Key sensitivity is regarded as one of the fundamental metrics for steps are 𝑂(4𝑁 2 ), 𝑂 14 𝑁 2 , 𝑂 18 𝑁 3 , 𝑂 641 𝑁 2 log 𝑁 , 𝑂(2𝑛2 ), and evaluating the security of cryptographic schemes. A cryptosystem with 𝑂(𝑛2 ), respectively. If some details of the method implementation high security strength should exhibit significant sensitivity to even are ignored, the overall computational ( complexity of the proposed ) minor perturbations in the key. That is, a slight modification in the key method can be approximated as 𝑂 18 𝑁 3 + 64 1 𝑁 2 log 𝑁 + 17 4 𝑁 2 + 3𝑛2 . should prevent the decryption algorithm from successfully recovering Accordingly, the computational complexities in [16–19,22], and [25] the original plaintext image. The experimental results, depicted in Fig. ( ) ( ) ( ) 12, demonstrate that when the decryption key matches the encryption 192 2 205 2 624 2 are 𝑂 𝑁 + 𝑂(2𝑛2 ), 𝑂 𝑁 + 𝑂(2𝑛2 ), 𝑂 𝑁 + 𝑂(2𝑛2 ), 64) 64 64 key precisely, the decrypted image is perfectly consistent with the ( ( ) ( ) 624 2 1 2 192 2 original. However, when a subtle perturbation is introduced to the 𝑂 𝑁 + 𝑂(2𝑛2 ), 𝑂 𝑁 + 2𝑁 2 log 𝑁 + 𝑂(2𝑛2 ), and 𝑂 𝑁 64 32 64 decryption key parameter 𝑥, i.e., 𝑥′1 = 𝑥1 +10−16 , the resulting decrypted + 𝑂(2𝑛2 ), respectively. Similarly, the space complexities ( of) the (six steps ) image becomes severely distorted and entirely unrecognizable to the of the proposed method are 𝑂(4𝑁 2 ), 𝑂(2𝑁 2 ), 𝑂 45 64 𝑁 2 , 𝑂 16 1 𝑁2 , human eye. This result indicates that the proposed image encryption 𝑂(3𝑛2 ), and 𝑂(4𝑛2 ), respectively. The overall space complexity ( of the ) scheme possesses a high level of key sensitivity, thereby enhancing its proposed method can be approximately expressed as 433 64 𝑁 2 + 7𝑛2 . resistance against key-related attacks. 14 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Table 19 Comparison of the running times of seven similar methods. Type of attack Method [16] Method [17] Method [18] Method [19] Method [22] Method [25] Proposed method Running time (s) 0.846 1.066 4.326 4.612 5.004 0.907 31.522 Fig. 12. Experimental results of key sensitivity analysis. 5.13. Discussions 6. Conclusion A robust zero-watermarking method is proposed considering the Aiming to address the limitations of existing zero-watermarking advantages of CT, CHFMs, and FFT. Experimental results show the methods, which often exhibit poor performance against specific at- superior attack resistance of the proposed method against conventional tacks and can only process a single image, a multi-image robust zero- image processing, geometric attacks, and combinatorial attacks. The watermarking method based on CT, CHFMs, and FFT is proposed. ablation experimental results show that without CT, the ability to resist First, a high-dimensional chaotic system and a Fibonacci Q-matrix noise attacks is weaker; without CHFMs, the ability to resist geometric are employed to encrypt a copyrighted image, thereby enhancing the attacks decreases significantly; and without FFT, the ability to resist security of the proposed method. Second, multiple images are fused noise and cropping attacks decreases slightly. In addition, compared into a single image, and the advantages of the CT, CHFMs, and FFT are with the methods in [16–19,22,25], our proposed method achieves combined to construct a feature vector. Numerous experimental results superior robustness against most attacks. Although these results demon- demonstrate that the NC values remain above 0.95 for conventional strate the effectiveness of the proposed method, its limitations remain image processing attacks, geometric attacks, and combined attacks, in the following three aspects. indicating the proposed method is effective against various types of attacks. Compared to the latest representative methods, it achieves superior performance with an average improvement of approximately 5.13.1. Ability to resist Gaussian noise 2%. The ablation experiments also confirmed the effectiveness of the From the experimental results in Tables 5 and 6, it can be concluded combined approach, which utilized CT, CHFMs, and FFT. Although that the proposed zero-watermarking scheme exhibits strong robustness the proposed method can withstand most attacks, its performance still against Speckle noise and Salt & pepper noise. However, its perfor- needs improvement. Overall, the limitations of the proposed method mance under Gaussian noise is not satisfactory, indicating a limited are primarily reflected in three aspects. First, the extracted feature resistance to such interference. Consequently, the method’s capability vectors are sensitive to noise, resulting in insufficient resilience against to withstand Gaussian noise attacks requires further improvement to attacks such as Gaussian noise. Second, the computational load associ- enhance its overall robustness. ated with using CHFMs is high, making it less suitable for real-time applications. Third, the current design is optimized for images and 5.13.2. Low efficiency in calculating CHFMs does not directly support videos. To address the limitations above, From the experimental results in Table 19, it can be concluded future work may be focused on the following three perspectives. First, that the proposed zero-watermarking method requires approximately explore the construction of noise-robust feature vectors using advanced 30 s to run on a general-purpose personal computer, indicating that feature extraction methods to enhance resistance against noise attacks. it is not directly applicable to real-time multimedia streaming envi- Second, improve the computational approach for CHFMs to enhance ronments or large datasets. Experiments revealed that the computation efficiency, enabling the proposed method to be applied in scenarios of CHFMs constitutes the most time-consuming component in the pro- with high time-sensitivity requirements. Third, attempt to adapt the posed method, accounting for the majority of the overall execution proposed method for video by considering its unique spatial and tem- time. Efficiently computing CHFMs to reduce runtime further is a key poral characteristics. Additionally, we plan to integrate blockchain and issue to be addressed by our method, enabling it to meet real-time smart contract technology to create a more comprehensive copyright requirements. protection model. 5.13.3. Scalability of the proposed method CRediT authorship contribution statement The proposed zero-watermarking generation framework is primarily designed for cover image; therefore, it cannot be directly extended to Xinhui Lu: Writing – original draft, Software, Methodology. video watermarking. Evidently, video covers are not only composed Guangyun Yang: Visualization, Methodology. Yu Lu: Visualization, of individual frames but also possess inherent relationships between Methodology. Xiangguang Xiong: Writing – review & editing, adjacent frames. Applying the proposed technique directly to video Supervision, Methodology. scenes often yields unsatisfactory performance. In addition, the zero- watermarking signal generated by the proposed method is stored Declaration of competing interest in a third-party trusted IPR, without considering integration with blockchain technology. The extension of the proposed method to video The authors declare that they have no known competing financial applications and its integration with blockchain technology would be interests or personal relationships that could have appeared to one of the future research perspectives worthy of in-depth exploration. influence the work reported in this paper. 15 X. Lu et al. Computer Standards & Interfaces 97 (2026) 104115 Acknowledgments [19] B. Wang, W. Wang, P. Zhao, A zero-watermark algorithm for multiple im- ages based on visual cryptography and image fusion, J. Vis. Commun. Image Represent. 87 (2022) 103569, http://dx.doi.org/10.1016/j.jvcir.2022.103569. This work was supported in part by the Natural Science Foundation [20] X. Wu, J. Li, A. Bhatti, W. 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