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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.
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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.
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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
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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.
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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
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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
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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.
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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
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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
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This work was supported in part by the Natural Science Foundation
[20] X. Wu, J. Li, A. Bhatti, W. Chen, Logistic map and Contourlet-based robust
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