Dual-prediction and adaptive complexity for reversible watermarking of color images

1 Introduction

The rapid development of mobile Internet and cloud computing technologies has led to a substantial proliferation of user-generated multimedia content on social media platforms. When such data is transmitted over public networks or stored in cloud environments, the inherent privacy-sensitive elements (including personally identifiable information and biometric identifiers) become vulnerable to unauthorized access and malicious exploitation, thereby posing significant security risks. Traditional cryptographic encryption methods for protecting such data tend to increase data saliency, thereby making encrypted content more susceptible to detection by attackers. In contrast, digital watermarking techniques leverage the masking effects of human perceptual systems, rendering embedded data essentially imperceptible to direct observation (Shi, 2004). This fundamental distinction establishes digital watermarking as a critical approach for safeguarding the security of digital content transmitted over public networks. Among current watermarking techniques, reversible watermarking (RW) stands out for its capability to completely restore the original carrier image after extracting the embedded information (Zhang et al., 2024). This unique capability has thus driven its widespread adoption in content-sensitive fields requiring stringent integrity preservation, including medical imaging, remote sensing, and public security applications (Kaur et al., 2021).

In recent years, significant research efforts have been dedicated to enhancing the maximum embedding capacity (EC) and post-embedding image quality of RW methods, with notable progress made by techniques including lossless compression (Celik et al., 2005; Celik et al., 2006), difference expansion (DE) (Tian, 2003; Hu et al., 2008; Ding et al., 2022), prediction error expansion (PEE) (Thodi and Rodríguez, 2007; Li J. et al., 2013; Yu et al., 2022), and histogram shifting (HS) (Ni et al., 2006; Jia et al., 2019). The lossless compression approach primarily embeds payload data into the pixel redundancy space of host images through lossless encoding. However, to ensure reversibility, this method is inherently constrained by a fundamental trade-off between achievable embedding capacity and visual fidelity preservation. To further enhance embedding performance, researchers have increasingly focused on exploiting the intrinsic pixel correlation features for data embedding. Tian proposed a DE-based RW algorithm (Tian, 2003), which embeds data by expanding the differences between pixel pairs and achieves superior embedding performance through multi-pixel vector expansion while maintaining the essential reversibility constraints. The other RW method of PEE was proposed by Thodi and Rodríguez (2007), which utilizes the predicted values based on the neighboring context pixels and then generates prediction errors through the differences between the predicted and original pixel values for data embedding. Optimizations were achieved in terms of embedding capacity and quality. Ni et al. proposed the HS method (Ni et al., 2006), which strategically manipulates global pixel statistics by relocating extreme pixels (maximum/minimum frequency values in histogram) to create embedding space for ciphertext insertion. This approach achieves favorable visual fidelity under low computational complexity while preserving image structural integrity. Over time, the pixel value ordering (PVO) framework proposed by Li et al. has demonstrated significant advancements (Li X. et al., 2013). This technique partitions cover images into non-overlapping blocks of uniform size, sorts intra-block pixels in ordering sequence, and embeds data using the extreme values (max/min) of the sorted sets. Building on this foundation, Peng et al. established an improved PVO (IPVO) methodology that optimizes embedding by targeting zero-difference bins, thus enhancing stego-image quality (Peng et al., 2014). Ou et al. devised a pixel-based PVO (PPVO) approach which replaces block-based embedding with neighboring pixel contexts to predict pixel values, consequently improving the capacity-distortion trade-off (Qu and Kim, 2015). Wu et al. subsequently enhanced PPVO by proposing the IPPVO scheme (Wu et al., 2020), which extends the neighborhood scope from a single bottom-right region to encompass bottom-left, bottom-right, and right-side regions. Concurrently, they adaptively adopt prediction contexts of varying sizes based on local image complexity, generating multiple prediction-error histograms (PEHs). Data embedding was then implemented through multi-histogram modification. He et al. proposed a multi-pass IPVO scheme (He and Cai, 2020), where non-extremal pixels are predicted using the maximum/minimum pixels as predictors (instead of the second largest/smallest pixels), and data embedding is achieved through shrinking/magnifying these non-extremal pixels, thereby further optimizing embedding performance. Recent years have also witnessed the development of attention-guided methods (Huang et al., 2023; Chen et al., 2025; Zhang et al., 2025) and learning-based RW methods (Ahmadi et al., 2020; Hu et al., 2024; Wang et al., 2023; Huang et al., 2025). Attention-guided schemes integrate attention mechanisms to locate robust and imperceptible image regions for watermark embedding (Huang et al., 2023), enhancing the balance between invisibility and robustness (Zhang et al., 2025). Learning-based methods, such as those based on deep learning (Wang et al., 2023), neural networks (Huang et al., 2025), and residual networks (Ahmadi et al., 2020), leverage powerful feature extraction capabilities to optimize embedding performance in complex scenarios. However, these methods often rely on complex network structures and large-scale training datasets, resulting in high computational complexity.

The aforementioned classic RW algorithms primarily focus on grayscale cover images, whereas the majority of images transmitted on social media and mobile networks today are colorful images (Li J. et al., 2013; Ou et al., 2015; Yao et al., 2017; Chang et al., 2022; Mao et al., 2023). Unlike traditional grayscale-based RW methods, color images have multiple channels (e.g., RGB) that contain richer information. However, effectively leveraging these characteristics for RW poses specific challenges: (i) how to optimally exploit inter-channel correlations for prediction and correction to minimize embedding distortion, (ii) how to accurately evaluate the embedding capacity of complex color textures to balance embedding payload and visual fidelity, and (iii) how to adaptively manage the embedding process across multiple channels to prevent error propagation and pixel overflow (Li et al., 2024). Existing PVO-based and cross-channel methods often address these aspects in isolation or with fixed strategies, leaving room for a more integrated and adaptive solution. Therefore, leveraging inter-channel correlations for prediction or dynamic capacity allocation represents a unique approach to implementing RW for color images. Li et al. enhanced pixel prediction accuracy in color images by extracting edge information from adjacent channels and leveraging inter-channel correlations to select optimal contextual pixels for pixel value prediction (Li J. et al., 2013). Ou et al. implemented a channel correlation-based exhaustive search to adaptively integrate payload partitioning into the optimization objective function, minimizing invalid displacements while preserving visual quality and reducing embedded-data disturbances in color images (Ou et al., 2015). Yao et al. leverages inter-channel correlation via a guided filtering predictor to establish linear models between channels and prediction error to optimize embedding parameters (Yao et al., 2017). Chang et al. proposed constructing a three-dimensional (3D) prediction error histogram using the three image channels of color image and designed an adaptive 3D histogram generation strategy tailored for color images (Chang et al., 2022). Mao et al. enhanced the fidelity of color images by leveraging channel correlations throughout the entire data embedding process, including prediction stages, block selection, and capacity allocation (Mao et al., 2023). However, existing reversible watermarking methods for color image still have room for improvement:

• While the multi-pass IPVO enhances intra-block prediction, it operates within a single channel and lacks mechanisms to exploit or correct errors using cross-channel information, limiting its efficiency for color images.
• Although IPPVO adapts the size of prediction context based on local complexity, it relies on fixed thresholds and a unidirectional embedding flow, which cannot dynamically optimize the trade-off across interdependent color channels.
• Existing cross-channel correlation methods often utilize inter-channel relationships for prediction or allocation but typically in a one-way, open-loop manner, without a backward correction mechanism to refine the initial reference channel based on embedded results. These issues motivate the development of the proposed adaptive scheme for color images.

Nevertheless, further improvements remain necessary. Compared with existing PVO-based and PPVO-based methods that rely on unidirectional prediction and single-dimensional complexity evaluation, as well as cross-channel correlation methods that insufficiently exploit inter-channel synergy. Our work proposes a novel optimized scheme to address the limitations of traditional PVO-based methods in color images while maintaining the advantages of simplicity and low complexity, this work introduces a novel scheme combining bidirectional correction, dual-complexity evaluation and multi-threshold control to optimize embedding parameters, primarily the embedding window size and regional priority, in an adaptive manner. The remainder of this paper is organized as follows: Section 2 reviews the related work in this field. Section 3 details the proposed methodology. Section 4 describes the framework for embedding and extraction. Section 5 presents the comparative experiments and analysis results. Section 6 concludes the paper and outlines future work.

2 Related work

Reversible watermarking techniques have evolved along a clear technical trajectory, from foundational frameworks like DE-based, PEE-based, and HS-based methods to more sophisticated prediction-based methods. Among these schemes, PVO and its variants have become dominant due to their superior capacity-distortion trade-off performance. This section details two advanced PVO-based schemes that are most pertinent to our work: the multi-pass IPVO scheme (He and Cai, 2020) and the Improved Pixel-based PVO (IPPVO) scheme (Wu et al., 2020). Their respective strategies for enhancing prediction and managing complexity form a direct technical foundation for the methodology proposed in this paper.

2.1 Multi-pass IPVO scheme (18)

The multi-pass IPVO algorithm proposed by He et al. overcomes the limitation of conventional PVO methods—which predict only extreme value pixels—through a multi-round prediction mechanism (He and Cai, 2020). By sequentially processing sub-maximal and non-extremum pixels, this approach fully exploits intra-block spatial correlations, thereby enhancing embedding performance. First, the cover image is divided into non-overlapping blocks of size n ₁ × n ₂, for each block, the pixels (x ₁, x ₂, … , x _n) within this set will subsequently be sorted in ascending order to form a new set (x _*σ* (Shi, 2004), x _*σ* (Zhang et al., 2024), …, x _*σ*(*n*)), where σ(n) denotes a unique one-to-one mapping, satisfying that σ(i) < σ(j) if x _*σ*(*i*) = x _*σ*(*j*) and i < j. Then, the maximum prediction error P E max is calculated as follows Equations 1, 2:

P E max i = x u − x v (1)

u = min i , σ n v = max i , σ n i ∈ σ 1 , σ 2 , … , σ n − 1 (2)

Based on the obtained prediction error, the ciphertext can be embedded into x at this time and the pixel values x ′ can be adjusted by Equation 3:

x ′ = x i − b , i f P E max i ∈ 0 , 1 x i − 1 , i f P E max i ∉ 0 , 1 (3)

Where b ∈ {0, 1} is the to be embedded secret bit. It is evident that the pixel embedding operation does not alter the ordering within the block of pixels, so the embedding can continue using the minimum value. At this point, the modified block is reordered, and data is embedded into non-minimum pixels as follows Equations 4, 5:

P E min i = x s − x t (4)

s = min σ 1 , i t = max σ 1 , i i ∈ σ 1 , σ 2 , … , σ n − 1 (5)

And the modification process is as following Equation 6:

x ′ = x i + b , i f P E min i ∈ 0 , 1 x i + 1 , i f P E min i ∉ 0 , 1 (6)

At the extraction end, pixels within each block are first sorted using the identical strategy as in embedding. Given the multi-stage nature of the embedding process, extraction proceeds in corresponding multi-rounds, exhibiting reverse-order symmetry with the embedding workflow. First, the predicted error P E min for the current minimum pixel is calculated following a process analogous to the embedding phase. Using P E min the secret data is extracted, and the pixel values modified in the first embedding stage are restored, which can be obtained as follows Equations 7, 8:

b = P E min i − 1 , i f P E min i ∈ 1 , 2 − P E min i , i f P E min i ∉ − 1 , 0 (7)

x i = x ′ − b , i f P E min i ∈ − 1 , 0 , 1 , 2 x ′ − 1 , i f P E min i < − 1 o r P E min i > 2 (8)

For these resorted pixels, the predicted error P E min (with the maximum pixel as the reference) is recalculated. Based on P E max, the original block data is fully recovered as follows Equation 9:

x i = x ′ + b , i f P E max i ∈ − 1 , 0 , 1 , 2 x ′ + 1 , i f P E max i < − 1 o r P E max i > 2 (9)

2.2 Improved pixel-based PVO scheme (17)

Building on the PPVO approach where Wu et al. used the nearest n pixels as the prediction context for pixel prediction, the neighborhood region is extended from the single bottom-right area to encompass the bottom-left, bottom, and right regions. The size of the prediction context and the distribution of neighboring pixels for the pixel to be predicted are selected, as shown in Figure 1.

Here, x represents the pixel to be predicted, and different colors in Figure 1 correspond to prediction context windows of different sizes, denoted as C n. Assuming the set of n neighboring pixels determined based on the threshold at this point is (x ₁, x ₂, … x _*n*), as with the previously introduced multi-pass IPVO algorithm, the IPPVO method adopts the same sorting approach to form the sorted set (x _σ(1), x _σ(2), …, x _σ(n)), thus, the prediction context can be partitioned into four sets {S1, S2, S3, S4}, as shown in following Equation 10.

S 1 = x | ⁡ max C n ≠ min C n a n d x ≥ max C n S 2 = x | ⁡ max C n ≠ min C n a n d x ≤ min C n S 3 = x | ⁡ max C n = min C n = U v a n d x = U v ≠ 254 S 4 = x | ⁡ max C n = min C n = U v a n d x = U v = 254 (10)

Where, min C n and max C n stand for the minimum and maximum values in the selected subset, respectively, U v denote the uniform value of all pixels in the context when max C n = min C n. Subsequently, the predicted value x ∼ and prediction error for the target pixel are derived through the following Equations 11, 12.

x ∼ = max C n , i f x ∈ S 1 min C n , i f x ∈ S 2 ⋃ S 3 254 , i f x ∈ S 4 (11)

P E = x − x ∼ (12)

Thus, the modified pixel value x ′ after embedding can be obtained as shown in Equation 13:

x ′ = max C n + b , i f x ∈ S 1 , P E = 0 a n d max C n + b ≤ 255 min C n − b , i f x ∈ S 2 , P E = 0 a n d min C n − b ≥ 0 x + 1 , i f x ∈ S 1 , P E > 0 a n d x + 1 ≤ 255 x − 1 , i f x ∈ S 2 , P E < 0 a n d x − 1 ≥ 0 U v − b , i f x ∈ S 3 , U v ≠ 254 254 + b , i f x ∈ S 4 , U v = 254 (13)

At the extraction end, the prediction context of the marked pixel is also partitioned into four sets (consistent with the embedding phase) to derive the predicted value. The embedded information bit b is extracted according to the following criteria: b = 0, when x ′ − x ∼ = 0 and b = 1 when x ′ − x ∼ = ± 1. Subsequently, the original pixels can be perfectly reconstructed through the inverse operation of the embedding process, as demonstrated in Equation 14.

x ′ = max C n , i f x ′ ∈ S 1 , x ′ − x ∼ = 0 min C n , i f x ′ ∈ S 2 , x ′ − x ∼ = 0 x ′ − 1 , i f x ′ ∈ S 1 , x ′ − x ∼ = 1 x ′ + 1 , i f x ′ ∈ S 2 , x ′ − x ∼ = 1 U v , i f x ∈ S 3 254 , i f x ∈ S 4 (14)

In summary, while Multi-pass IPVO and IPPVO have achieved improvements in grayscale image reversible watermarking, their direct extension to color images faces notable limitations. Their context selection relies on fixed thresholds and a predefined, unidirectional embedding flow, which restricts dynamic optimization and cross-channel synergy. Moreover, their expanded neighborhood range is not fully optimized for proximity to the target pixel. These limitations motivate our proposed framework integrating bidirectional correction and an adaptively refined context selection mechanism. Table 1 summarizes their key aspects in terms of underlying assumptions, primary prediction models, and overflow/underflow handling strategies.

TABLE 1. Comparison of representative reversible watermarking methods.

Following this comparative perspective, Section 3 details our proposed scheme which is designed to address the limitations observed in prior art.

3 Proposed methods

This paper introduces a novel reversible watermarking scheme for color images, which is based on complexity evaluation of inter-channel correlations and multi-threshold joint constraints. First, a two-dimensional complexity evaluation model integrating local variance features and visual saliency features is constructed. It accurately quantifies the variation patterns of spatial texture features and achieves more precise image feature localization by incorporating the perceptual characteristics of the human visual system. Subsequently, a bidirectional prediction correction mechanism and a window-adaptive embedding strategy are designed based on the three channel reference components, which selects the channel with the lowest pixel overflow rate as the main reference component for spatial correlation prediction. Dynamically adjusting the embedding window size maintains the balance between visual quality and embedding capacity can be maintained in regions exhibiting high prediction correlation. Finally, a multi-step embedding strategy with controllable thresholds is adopted. By leveraging multi-level regional classification and adaptive adjustment of embedding positions, flexible adjustment is achieved.

3.1 Bidirectional prediction correction module

This section details the core algorithmic mechanisms of the proposed bidirectional prediction-correction module. Owing to the high inter-channel correlation and inherent synergistic properties in color images, this work proposes a closed-loop embedding architecture incorporating cross-channel primary prediction and secondary corrective prediction mechanisms–the latter optimizes the initial prediction results using cross-channel information to reduce embedding distortion. In the forward direction, the framework utilizes predictive correlations from reference channels to non-reference channels; conversely, it harnesses reconstructive correlations from previously embedded channels to reference channels. Through this bidirectional interaction, the architecture maximizes inter-channel correlation potential, thereby enabling joint nonlinear optimization of both predictive and reconstructive processes, which adaptively adjusts modification amplitudes based on local complexity to avoid distortion caused by linear scaling. The specific implementation is as follows:

First, dynamic unidirectional prediction is applied to the R, G, and B channels of a color image based on the overflow rate O R C. By quantifying the distribution characteristics of overflow pixels (i.e., pixels with values of 0 or 255) in each channel, the color channel with the lowest overflow rate is selected as the optimal reference channel, defined as I r e f, and O R C is calculated by Equation 15: O R C represents the embedding overflow metric for channel C where C ∈ {R, G, B}, I c i denotes the grayscale value of the ith pixel in channel C, and (x, y) is the coordinate position of the current pixel to be embedded, δ (x, y) denotes the indicator function with δ (x, y) = 1 when x = y and 0 otherwise. At this stage, owing to a more concentrated distribution of initial prediction errors, it typically demonstrates higher image information density and lower cost in shifting operations. The choice of O R C as the selection criterion is driven by its direct correlation with embedding distortion risk. Specifically, overflow pixels (0 or 255) cannot be shifted during embedding without causing data loss or irreversible distortion, making them the primary source of invalid shifts and auxiliary information overhead. Minimizing the number of such pixels in the reference channel directly reduces the initial set of problematic pixels, thereby lowering the potential for invalid shifts during subsequent bidirectional prediction and preserving more capacity for secret data. To further justify this choice, we compare O R C with other plausible selection criteria commonly discussed in the literature:

O R C = 1 N ∑ i = 1 N δ I c i , 0 + δ I c i , 255 C l u C = arg ⁡ max j d e n s i t y j (15)

Local variance: While local variance reflects texture complexity, it does not directly quantify overflow risk. A region with high variance may still contain few overflow pixels, whereas a smooth region with near-0 or near-255 values can be highly prone to overflow.

Inter-channel correlation strength: Although high correlation between channels can improve prediction accuracy, it does not account for the pixel value limits. Two highly correlated channels may both suffer from overflow issues, propagating rather than mitigating distortion.

Prediction error magnitude: This is a post-prediction metric and is not available prior to embedding; using it for reference selection would introduce circular dependency.

In contrast, O R C directly measures the proportion of pixels that are at the boundary of the representable range, i.e., those most vulnerable to underflow/overflow during expansion-based embedding. Moreover, if multiple channels have the same minimum overflow count, the one with the highest average correlation with the other two channels can also be chosen as the reference channel, which still maintains good prediction accuracy and embedding stability. To validate the effectiveness of the proposed reference channel selection strategy based on O R C, we conducted a comparative experiment on three representative images (Lena, Lake, Airplane). For each image, we separately designated the R, G, and B channels as the reference and recorded the resulting PSNR at a payload of 10,000 bits. Table 2 summarizes the overflow counts, inter-channel correlation coefficients, and the corresponding PSNR values.

TABLE 2. Performance comparison with different reference channels and the corresponding overflow counts.

Ref. denotes the selected reference channel (R, G, or B) for data embedding; Overflow represents the number of overflow pixels when the corresponding channel is used as the reference; Corr. indicates the inter-channel correlation coefficient; PSNR, is the peak signal-to-noise ratio achieved under the corresponding reference channel setting. As can be observed from Table 2, the channel with the lowest overflow rate consistently achieves the highest PSNR, performance. For example, in the Lena image, the G channel yields the best PSNR, of 65.67 dB. in the lake image, the R channel outperforms the other channels. For the Airplane image, both the R and B channels deliver the same optimal performance. These results confirm that selecting the reference channel based on the overflow rate is a reasonable and effective strategy, which can effectively reduce embedding distortion and improve the overall performance of the proposed method.

Bold values indicate the best performance among all compared methods.

Therefore, by quantifying overflow potential, O R C directly identifies the channel with the lowest embedding risk as the most adaptable prediction reference. This selection minimizes the initial set of problematic pixels, thereby reducing the potential for invalid shifts during subsequent bidirectional prediction. When serving as a prediction reference, the reference channel can enhance the prediction accuracy for the other two channels (designated as I r e s 1 and I r e s 2), thus preserving more complete image features, as illustrated in Figure 2:

As shown in Figure 2, a bidirectional constraint based phased embedding strategy is adopted to enable flexible transitions between high fidelity and large capacity while preserving color correlation. Here, correlation is employed for predictive compensation, and difference is utilized for hierarchical embedding optimization—this refers to prioritizing embedding in low-complexity regions, then adjusting parameters for complex regions to balance capacity and fidelity. Specifically, during embedding, complexity information from the first reference component is utilized to perform window prediction and embedding on the other two components. After embedding data into the non-reference components, the two embedded non-reference components are treated as secondary reference components to extract their complexity map, which is then used to guide the embedding of the original reference component, thus completing the embedding of the entire color image. To quantify the bidirectional interaction, key formulas are defined as follows:

First, for the forward prediction from reference channel to associated channels. Based on the fused complexity map F C M a p of the primary reference component I r e f and complexity threshold Ts—which will be detailed in the next section, Data embedding is performed on the other two components to obtain the embedded components I r e s 1 and I r e s 2. The embedded value of associated channel pixels I r e s 1 ′ and I r e s 2 ′ is given by Equation 16 where D denotes the data to be embedded, E b · represents the embedding function, and Ω (F C M a p f, Ts) stands for the neighborhood pixel set filtered by F C M a p and Ts.

I r e s 1 , 2 ′ = E b I r e s 1 , 2 , Ω F C M a p f , T s , D (16)

Second, the embedded components I r e s 1 ′ and I r e s 2 ′ are used as secondary reference components. Their joint fused complexity map F C M a p 1 , 2 is generated by fusing the F C M a p of I r e s 1 ′ and I r e s 2 ′. The neighborhood pixel set Ω is determined with the complexity threshold T, and data embedding is implemented on the original primary reference component I r e f to generate the final embedded component I r e f, expressed as:

I r e f ′ = E b I r e f , Ω F C M a p 1 , 2 , T s , D (17)

Overall, the three embedded components are integrated to obtain the complete embedded image I ′ by Equation 18

I ′ = I r e f ′ , I r e s 1 ′ , I r e s 2 ′ (18)

In essence, the proposed bidirectional prediction-correction mechanism operates as a closed-loop, two-stage process. First, in the forward stage, the optimally selected reference channel I r e f (with the lowest overflow rate) guides the embedding into the two associated channels I r e s 1 and I r e s 2. This process can be mathematically characterized as a nonlinear mapping from the reference channel to the associated channels. Specifically, let f forward : I r e f → I r e s 1 ′ , I r e s 2 ′ denote the forward prediction function that embeds data into the associated channels, guided by the reference channel’s complexity map. This is achieved by transferring the fused complexity map F C M a p r e f from I r e f to the associated channels, thereby determining their adaptive embedding windows and reducing prediction errors. Subsequently, in the backward stage, the already-embedded associated channels I res 1 ′ and I res 2 ′ jointly provide a refined complexity map F C M a p joint, which is then applied to refine the embedding of the original reference channel I ref. This backward correction can be expressed as g backward : I res 1 ′ , I res 2 ′ → I ref ′. The overall bidirectional process can be expressed as the composite mapping I ′ = h I = g backward f forward I ref , I res 1 , I res 2, where the nonlinearity arises from the adaptive threshold-based window selection and the conditional pixel modification rules. This bidirectional interaction ensures that inter-channel correlations are fully exploited—the reference channel initially benefits the others, and the embedded results, in turn, feedback to optimize the reference channel itself. The entire procedure is tightly integrated with the multi-threshold joint constraints (detailed in Section 3.3; Algorithm 1), which dynamically adjust prediction windows and exclude unsuitable pixels at each stage.

During the embedding process for correlated images, we leverage the pixel-by-pixel concept from IPPVO to perform a traversal scan of the pixels. At this stage, the weight distribution within the neighborhood surrounding the pixel to be embedded is illustrated in Figure 3 below, where different colors represent window sizes corresponding to specific thresholds yellow corresponds to C_n = 4 green to C_n = 10 blue to C_n = 18 and white to C_n = 24. When selecting neighboring pixels for prediction, pixels with complexity differences exceeding T _*cd* (a threshold from the multi-dimensional threshold set Ts to be detailed later) are excluded. For C_n = 4 to C_n = 24 windows, the number of excluded pixels typically ranges from 1 to 5 with slightly more excluded in complex regions to ensure prediction accuracy. The selectable neighborhood content within the predicted pixel set can subsequently be dynamically adjusted based on the quality of the embedded image, and these pixels excluded due to excessive complexity differences do not participate in embedding avoiding invalid shifts that consume capacity.

To enhance the performance of the IPPVO algorithm for color images, this paper introduces two embedding constraint mechanisms based on reference component pixels. First, during the embedding process for I r e s 1 and I r e s 2, the IPPVO algorithm utilizes the maximum and minimum values of neighboring pixels as prediction pixels. When a to-be-embedded pixel exceeds the range defined by its neighbors and requires shifting, I r e f is introduced as a reference. If the co-located I r e f pixel also exceeds its neighboring pixel range, the original pixel value remains unchanged, thereby reducing the number of invalid shifted pixels. Second, to address pixel overflow caused by ciphertext embedding, traditional RW schemes (e.g., IPPVO) typically pre-count the coordinates of potential overflow pixels (pixels with values 0 or 255). As shown in Figure 4, the traditional overflow rule records 7 overflow points in the LM for the illustrated region. This data is used to construct the LM, which undergoes lossless compression into a constraint bitstream using arithmetic coding. Each overflow point occupies approximately 19 bits (derived from the 512 × 512 image size) and 7 such points consume about 133 bits of embedding capacity. In contrast, the proposed scheme eliminates pre-embedding overflow counting. Instead, it dynamically constrains the participation of overflow-prone pixels (all 7 potential overflow points in the example) in the embedding process through conditional rules, thereby reducing the size of the compressed LM data by approximately 133 bits without compress, and this released capacity is allocated to secret data embedding directly improving the effective embedding capacity.

3.2 Dual-dimensional complexity detection module

Through analyzing traditional RW methods, we observed that the accuracy of perceptual features extraction for texture regions and statistical distributions in color images significantly affect both the flexible allocation of embedding capacity and the accurate selection of optimal embedding regions. To address the limitations inherent in traditional single-dimensional complexity evaluation methods, this paper proposes a dual-dimensional complexity detection framework that integrates local variance and visual saliency features. By combining texture structural features and visual perception characteristics, we construct a multi-scale discriminative complexity characterization scheme. This framework significantly enhances the recognition capability of high-frequency details and regional structures in color images, while improving embedding performance. First, the variation trend of pixel intensity centered on luminance transitions in the cover image is computed to obtain the local features contrast map (C M a p), as expressed by the following Equation 19:

C M a p = 1 M N ∑ x = 1 M ∑ y = 1 N I x , y − μ 2 μ = 1 M N ∑ x = 1 M ∑ y = 1 N I x , y (19)

Here, M and N denote the dimensions of partitioned block sizes, respectively. Leveraging these formulas, a distribution map can be generated for any color channel of the cover image. In subsequent calculations, this map enables the fine-grained characterization of the information distribution features within the image. While the C M a p effectively characterizes edge regions, its performance degrades in weakly textured areas where uniform pixel distributions yield minimal deviation from local means. Consequently, it generates low feature response amplitude, making it challenging to effectively capture salient image features. To address this limitation, this paper also proposes an active visual attention mechanism. This mechanism quantitatively detects and prioritizes visually salient regions, thereby establishing a unified feature representation framework. By fusing hierarchical contrast features with this framework, we enhance multi-scale texture stratification capabilities, achieving comprehensive feature characterization spanning edge-level (macroscopic) to texture-level (microscopic) granularity, as shown in Equations 20–23.

F u , v = ∑ x = 0 M − 1 ∑ y = 0 N − 1 I x , y e − j 2 π u x M + v y N (20)

I x , y = 1 M N ∑ u = 0 M − 1 ∑ v = 0 N − 1 F u , v e j 2 π u x M + v y N (21)

The phase spectrum A u , v is extracted from F u , v to reconstruct the dominant structural components of the image:

A u , v = tan − 1 I m F u , v R e F u , v (22)

Thus, the phase angle is subsequently employed to define the saliency map (S M a p) according to the following Equation 23:

S M a p = F − 1 e j · A u , v α (23)

Here, α > 1 represents the significance enhancement coefficient, which adjusts the contrast intensity of salient regions, and α = 2 is generally adopted. To avoid interference from extreme peaks, a Gaussian filter can be further used to smooth the S M a p. Thus, in the dual-dimensional complexity evaluation model, we employ the C M a p to capture statistical properties of macroscopic structures, and utilize the S M a p to reinforce attentional allocation toward fine-texture features. The complementary fusion of these two components yields a fused complexity feature map (F C M a p), enabling mutual compensation between dual feature sources in complex scenarios to improve resistance against interference factors such as illumination variations and object occlusion in color images using Equation 24:

F C M a p = β · C M a p + 1 − β · S M a p (24)

Here, C M a p and S M a p are the local contrast map and saliency map defined by Equations 19, 23, respectively, and β denotes the fusion coefficient. In actual embedding, adaptive adjustment of β enables selection of neighborhood prediction window sizes: small-sized windows ensure higher embedding capacity in low-complexity regions, while large-sized windows preserve image fidelity in high-complexity areas. The implementation is illustrated in Figure 5 below:

This figure shows an example for the F C M a p model extracts and comparatively analyzes local variance features and saliency features in reference images, subsequently constructing a fused composite feature map by fusing them. At the spatial domain level, the local variance operator accurately quantifies the statistical distribution characteristics of the image microstructure, effectively capturing full-spectrum information from fine textures to macro contours. At the frequency domain level, coverage extends the analysis across the entire spectrum from extremely high-frequency details to region-level structures, thereby enhancing the assessment granularity of image content. To ensure the rationality of feature fusion, the fusion coefficient β is adaptively determined based on the local complexity of the image with a value range of [0.4, 0.6]. When the local complexity is low, β takes a larger value close to 0.6 to emphasize the contribution of C M a p in capturing macro-structural features. When the local complexity is high, β takes a smaller value close to 0.4 to enhance the weight of S M a p in highlighting fine-grained salient features. This adaptive adjustment mechanism of β ensures that the fused F C M a p can match the texture characteristics of different image regions. Notably, the performance metrics (image quality and embedding capacity) remain near-optimal and exhibit minimal variation when β is set around its mid-range value of 0.5. To further justify the selection of the fusion coefficient β and demonstrate its impact on embedding performance, we conducted an ablation study on the Lake and Lena images at an embedding payload of 10,000 bits. The results, shown in the accompanying Figure 6, illustrate the number of distorted pixels corresponding to different values of β.

Figure 6 illustrates the impact of different fusion coefficients β on the number of distorted pixels for the Lake and Lena images at a payload of 10,000 bits. The results demonstrate that fusing Cmap and Smap (β = 0.5) significantly reduces distortion compared to using either map individually. For Lake, the number of distorted pixels decreases from 6,457 (β = 0, Smap only) and 8,562 (β = 1.0, Cmap only) to 5,210 (β = 0.5), representing reductions of 19.3% and 39.1%, respectively. For Lena, distortion drops from 7,211 (β = 0) and 9,735 (β = 1.0) to 6,459 (β = 0.5), with reductions of 10.4% and 33.7%. These improvements confirm that the complementary nature of local contrast (capturing macro-structure) and visual saliency (highlighting fine details) enables more accurate adaptive window selection, and that their fusion yields a substantial performance gain beyond either feature used independently. Therefore, considering the balance between embedding complexity and computational overhead, we adopt this value in our subsequent experiments.

3.3 Multi-threshold joint constraints module

To achieve an adaptive hierarchical embedding strategy for color images, this paper proposes a dynamically adaptive embedding framework under multi-threshold coordinated control. This framework aims to mitigate visual quality degradation induced by embedding capacity during multi-channel watermarking. Its core innovation lies in an adaptive multi-channel hierarchical embedding strategy regulated via multivariate thresholds. Specifically, embedding priorities and invalid displacement losses are controlled by dynamically allocated prediction window thresholds, while multi-threshold coordination further minimizes such losses.

We construct a multi-dimensional threshold set T s = T c , T w , T l w , T v d , T c d, which integrates various threshold constraint parameters including complexity threshold, prediction window threshold, local variance window threshold, variance fluctuation constraint threshold, and complexity difference threshold. The specific value ranges and determination methods of each threshold are as follows: T c (complexity threshold) ∈ [0, 255], T w (prediction window threshold) ∈ [1, 24] corresponding to the four window sizes in Figure 3, T l w (local variance window threshold) ∈ [1, 24] (fixed based on image block division experience), T v d (variance fluctuation constraint threshold) ∈ [0, 50], T c d (complexity difference threshold) ∈ [0, 255]. All thresholds except T l w are determined through grid search-based tuning on the test dataset, with the objective of maximizing the average PSNR while ensuring embedding capacity meets practical requirements. This five-element threshold set enables cascaded discrimination of texture features in color images, facilitating the establishment of hierarchical adaptive window selection mechanism with self-adaptive threshold. Quantitative analysis as shows in Figure 6.

Figure 7 illustrates the trade-off between embedding capacity and image fidelity (PSNR) under different threshold set Ts configurations. Overall, PSNR exhibits a decreasing trend with the increase of embedding capacity (5,000–25,000 bits), which conforms to the fundamental characteristics of reversible watermarking technology. Comparative results show that the multi-threshold combination including T c , T w , T l w , T v d and T c d consistently maintains a higher PSNR at the same embedding capacity and demonstrates superior performance stability. In contrast, single-window configurations (e.g., win 4, win 18) not only have a lower initial PSNR but also exhibit a more pronounced distortion trend as the capacity increases. These results verify the effectiveness of multi-threshold joint constraints in optimizing the “capacity-fidelity” balance.

To elucidate the individual contribution of each threshold, we conduct a detailed analysis of their effects on the capacity–fidelity trade-off.

• Impact of T c and T w on embedding performance. Figure 7 compares fixed windows (4 vs. 18 neighbors) with the proposed adaptive window selection. At 10,000 bits, a larger window (win 18) improves PSNR by ∼2 dB over a small window (win 4). However, at the maximum capacity, the smaller window permits a higher payload. The adaptive strategy balances this trade-off: smooth regions use small windows to preserve capacity, while textured regions use larger windows to maintain fidelity.
• Role of T l w and T v d. These thresholds refine the prediction window based on local texture differences. Adding T l w and T v d boosts PSNR by 0.99 dB, 0.68 dB, and 0.47 dB at 5,000, 10,000, and 20,000 bits, respectively. The gain diminishes at higher payloads because the room for adaptive adjustment shrinks.
• Effect of T c d. T c d excludes neighboring pixels with excessive complexity deviation, improving the effective embedding ratio R = N e m b e d / N s h i f t. For Lena, a suitable T c d (restriction 1: [10,20,30,40]) increases R from 1.983 (unrestricted) to 1.990, while a looser restriction ([15,20,25,45]) reduces R to 1.938. Similar behavior is observed for Lake, confirming that moderate T c d enhances efficiency, whereas overly large values discard useful neighbors and degrade performance.

Consequently, it achieves dynamic capacity-distortion coupling through complexity-driven optimization, forming a progressive pathway to optimal window selection. Among these, the dual-channel variance extrema serve as the texture segmentation criterion, explicitly discriminating between smooth and complex regions. Quantified analysis facilitates differentiated processing of image content variations, thereby improving prediction accuracy through adaptive feature weighting. The variance fluctuation constraint threshold quantifies disparities among color components, facilitating adaptive window selection. This enables dynamic optimization of embedding regions while improving identification accuracy for complex textures. Complexity difference filtering is employed to remove discrete extreme points while preserving detail gradients and variations in flat regions. Such a process optimizes neighborhood consistency, thereby further enhancing prediction accuracy and embedding capacity. Some threshold settings and optimization procedures are shown in Figure 8 below:

Figure 8 summarizes the construction of the multi-dimensional threshold set Ts, highlighting its role as the central control parameter suite for our adaptive strategy. The flowchart illustrates a sequential decision process: first selecting the reference channel, then determining prediction windows based on local complexity and inter-channel variance disparity, and finally excluding pixels with excessive complexity differences. As can be observed from Figure 8 above, the multi-dimensional thresholds Ts within the proposed threshold set participate as control information throughout the entire embedding process. To ensure the rationality and stability of the threshold set, these thresholds are determined according to the statistical characteristics of all test images. A validation-based strategy is further applied to reduce the risk of overfitting to specific image content. Building upon the bidirectional correction and multi-dimensional evaluation mentioned in the previous section, we can further provide an illustrative workflow of bidirectional correction involving these thresholds in following Figure 9.

Figure 9 then details how these thresholds govern the bidirectional prediction-correction flow, explicitly linking the control parameters to the adaptive embedding logic. Specifically, by computing the fusion complexity map to assess local texture characteristics and adaptively identifying smooth and complex channels through comparing the variances of residual channels, the multi-threshold control mechanism can dynamically select corresponding neighborhood sizes based on complexity levels and variance differences. Finally, under the constraint of complexity tolerance, two customized embedding neighborhoods are reconstructed. This described procedure is concisely presented in Algorithm 1 below.

Algorithm 1 Threshold-Controlled embedding neighborhood construction algorithm.

By introducing the multi-dimensional threshold set to control and fine-tune the overall embedding process, we can adjust the abnormal regions with significant image quality degradation during large-capacity embedding, thereby reducing the loss caused by image distortion. Additionally, the joint action of the complexity model and the multi-threshold controller modify the criteria for pixel selection. By dynamically adjusting classification thresholds (e.g., T c, T c d), pixels that would be excluded by a conventional fixed-threshold scheme can be re-evaluated and deemed suitable for embedding under the adaptive framework. This potential for increased embeddable content is conceptually illustrated in Figure 10, which contrasts the embeddable regions under fixed thresholds versus our adaptive scheme, showing a clear expansion of usable pixels.

4 Embedding and extraction workflow

This section outlines the complete operational workflow for embedding and extraction, building upon the algorithmic foundations presented in Section 3. Notably, since reversible information hiding requires ensuring the complete recoverability of the cover image before and after embedding and extraction, it is necessary to record additional auxiliary information for potential overflow points that may arise. Consistent with established methodology, we utilize a LM to efficiently process this critical information.

The length of the embedded data occupies ⌈log₂ (height × weight) ⌉ bits; The length of the multi-threshold subset T s used during embedding occupies 2 + 40 + 96 + 16 = 154 bits.; The reference channel marker occupies 2 bits, and the coefficient β in the fused image occupies 7 bits. The bit-lengths for these auxiliary parameters are determined as follows: the ciphertext length must be able to represent any possible payload size up to the total number of pixels (height × weight), hence the logarithmic representation; the 154 bits for T s are the sum of fixed-length encodings assigned to each of the five thresholds based on their empirically observed dynamic ranges; the reference channel is one of three (R, G, B), requiring 2 bits; and β is quantized to 7 bits to balance the precision of representing its adaptive value. For channels with large overflow volume (overflow pixels >5% of total pixels), the LM includes 1-bit flags for 16 × 16 sub-blocks to avoid excessive data volume. These parameters are combined into a binary stream, which is embedded into the starting position of the image via lossless compression using the Least Significant Bit (LSB) method. The LM is similarly compressed using arithmetic coding before being embedded as part of the auxiliary information. After this, the information hiding process for the entire image can be initiated. To present the embedding and extraction processes of the proposed scheme more clearly and intuitively, Figures 11, 12 below provide a flow chart illustrating the entire embedding and extraction procedures.

The extraction process reverses the embedding steps. First, the auxiliary information is retrieved from the LSBs to obtain the parameters and the compressed LM. After decompression and restoration of overflow pixels, the main extraction proceeds. Crucially, error propagation is prevented because all prediction processes during extraction are strictly deterministic and symmetrical to those in embedding. The saliency-based prediction relies on the F C M a p, which is derived from the stego-image in both embedding and extraction phases. Since the same dual-dimensional complexity detection module (Section 3.2) is applied to the same input image at both ends, it generates identical prediction contexts. Furthermore, the order of extraction is the exact inverse of embedding, and the reference channel is restored first using the corrected data from associated channels, ensuring a stable foundation for subsequent steps. This reverses the bidirectional prediction and threshold-adaptive embedding steps to recover the original image and secret data.

5 Experiment results

This section evaluates the performance of the proposed scheme across diverse color images. We focus on the core evaluation metrics in RW, specifically EC and image quality measured by peak signal-to-noise ratio (PSNR), to clarify the advantages and limitations of our method when compared with state-of-the-art approaches in the same category. Those comparative methods include Yao et al. (2017), Chang et al. (2022), Mao et al. (2023), He and Cai (2024). The evaluation was performed on a typical personal computer with the Windows 10 platform, an Intel i9 CPU, 16G Memory, utilizing MATLAB R2021a. To ensure consistency, we verified the effectiveness of the proposed method on 7 color images from the classic USC-SIPI image database (http://sipi.usc.edu/database/database.php?volume=misc (accessed on 20 December 2023)), including benchmark 24-bit color images such as Lena, Splash, Tiffany, Peppers, House, Lake and Airplane, with an image size of 512 × 512 in the uncompressed TIFF format. Subsequently, we further expanded the performance tests across varying embedding payloads (e.g., 10,000 bits to 50,000 bits in increments of 5,000 bits). The corresponding thresholds, overflow markers, standard deviations, and other relevant information used for selected images under different embedding capacities in the experiment are shown in Table 3 below.

TABLE 3. Detailed experimental parameter settings.

As the algorithm is deterministic, the reported PSNR and capacity values for each (image, payload) pair are unique and repeatable, with its operations, which include complexity map generation, threshold selection, and pixel modification, all uniquely defined by the input cover image and the secret payload. Consequently, under fixed experimental conditions, the corresponding experimental results are provided in Tables 4–6 below.

TABLE 4. Performance evaluation of maximum PSNR (dB) for a payload of 10,000 bits.

TABLE 5. Performance evaluation of maximum PSNR (dB) for a payload of 20,000 bits.

TABLE 6. Performance evaluation of maximum PSNR (dB) for a payload of 50,000 bits.

As shown in Table 4, the proposed scheme exhibits good performance at an embedding capacity of 10,000 bits, outperforming other comparable algorithms overall. The average image quality is improved by 3.30 dB, 1.89 dB, 0.65 dB, 0.55 dB, respectively. When the capacity is further increased to 20,000 bits, as indicated in Table 5, the proposed scheme still maintains stable performance, achieving better image quality in the vast majority of images. Moreover, it yields average improvements of 2.73 dB, 1.69 dB, 0.36 dB and 0.76 dB over the comparative schemes. This indicates that the proposed scheme can better utilize pixel correlation to provide more accurate quality control at low embedding capacity. Subsequently, as the embedding payload further increases, image quality begins to decrease rapidly, with significant loss. Table 6 presents the PSNR results when the embedding capacity is continuously increased to 50,000 bits. It can be observed that for the vast majority of images, particularly those with smoother edge regions and stronger color contrast (e.g., “Lena”, “Tiffany”, “Splash”), the scheme maintains relatively high performance. However, in images exhibiting high color consistency across color channels, such as “Peppers” and “Airplane”, the image quality experiences a slight decline. This is presumably because the dual-dimensional complexity reference map places greater emphasis on delineating edge regions and visually salient areas during the fusion process, consequently reducing recognition accuracy for uniformly distributed content. Specifically, this performance decline is directly linked to F C M a p behavior in smooth regions: smooth images have uniformly distributed pixels, leading to extremely low local variancevalues and weak feature response. The visual saliency map also fails to capture distinct salient features in these regions, resulting in low S M a p values. Since the fusion coefficient β adapts to local complexity and prioritizes C M a p in low-complexity regions, the weak C M a p response reduces F C M a p's ability to distinguish fine-grained texture differences. This makes F C M a p unable to accurately guide adaptive embedding window selection, ultimately leading to suboptimal prediction accuracy and increased distortion. We further analyze the performance degradation observed on highly uniform images such as Peppers and Airplane. In such regions, low local variance and weak visual saliency cause the fused F C M a p to lack sufficient discriminative power, which hinders the multi-threshold control (T c, T w) from reliably assigning appropriate prediction windows. This leads to suboptimal window selection, increased prediction errors, and ultimately higher distortion. The effect explains the reduced performance margin on uniform images, while the method retains its advantage on textured images with stronger feature responses. This explains why the proposed method slightly underperforms certain competitors on specific test images. For instance, on Peppers at 10,000 bits, the PSNR (64.50 dB) is 1.51 dB lower than Mao et al. (2023) (66.01 dB); on Airplane at 10,000 bits, it is marginally lower than Chang et al. (2022) (69.03 dB vs. 69.28 dB). Both images contain large uniform regions, where the limited discriminative power of F C M a p leads to suboptimal window selection. Meanwhile, methods like Mao et al. employ specialized strategies for smooth blocks, giving them an advantage in such cases. This observation reinforces the trade-off discussed above and motivates further refinement of the complexity model for uniform areas.

Among the comparative algorithms, such as Mao et al. (2023) method, prioritize the generation of marker bitmaps and predictive modification in smooth blocks. As a result, they exhibit slightly better quality than other algorithms on images like Pepper, which offers feasible insights for our subsequent research on enhancing related performance. However, in terms of overall effectiveness, the proposed method in this paper still achieves better image quality on most images with the same payload and demonstrates favorable average performance in terms of overall performance.

To further evaluate the generalization ability of the proposed method, additional experiments were conducted on the Kodak image dataset, which comprises 24 color images covering diverse scenes and content types. The Kodak images are displayed in Figure 13.

Based on this dataset, comprehensive comparisons with state-of-the-art color image reversible data hiding methods are conducted (EC = 10,000 bits), and the corresponding experimental results are shown in Figure 14.

As shown in Figure 14, the proposed method achieves better overall performance than several state-of-the-art methods under the same embedding payload, with average PSNR improvements of 4.10 dB, 1.48 dB, 3.97 dB and 0.16 dB, and obtains the optimal visual quality in most test images. The proposed method presents a smoother and more stable performance trend with smaller fluctuations than the comparison methods, and still maintains favorable balance even on images with drastic content changes such as kodim08, kodim13, kodim18 and kodim20. Although the peak PSNR is slightly lower than that of Mao et al.’s method on some texture-rich images (e.g., kodim03 and kodim08), our method still achieves a higher average PSNR, which benefits from the fused complexity perception and auxiliary information constraints for better distortion suppression, as well as the multi-threshold joint optimization that effectively alleviates the performance degradation caused by structural jumps.

To further illustrate the trend of performance variation of the proposed algorithm, we present the comparison curves of embedding capacity versus average PSNR in Figure 15 below, where the embedding capacity is gradually increased from 10,000 bits to 50,000 bits in increments of 5,000 bits. The specific results are as follows Figure 15. Note that the curves depict unique, deterministic results for each payload point; no performance variance exists across runs for these algorithms under fixed conditions.

As shown in Figure 15, the proposed method demonstrates an efficient balance in overall embedding performance. Except for the case of “Peppers”, where the concentrated color distribution and smooth textural characteristics result in slightly lower image quality compared to Mao et al. (2023) method, the proposed scheme outperforms other comparative schemes on most test images including “Lena”, “Tiffany”, and “Splash”. Moreover, the degradation of image quality is more gradual as the embedding payload capacity increases. This characteristic benefits from the effective restriction on the increase of image distortion by the proposed multi-threshold joint control mechanism, which has better practical application value in some real-world scenarios, such as the security protection of social multimedia information in mobile Internet.

In addition to capacity and fidelity, the computational efficiency of a reversible watermarking scheme is crucial for practical applications. The running time of the proposed algorithm is primarily influenced by the dual-dimensional complexity detection (involving local variance calculation and frequency-domain transformation for saliency) and the multi-threshold joint control mechanism that guides adaptive embedding. The overall computational complexity is dominated by the saliency map generation, resulting in an embedding complexity of O (MN log MN) for an M × N image. The extraction process, being the symmetric inverse, has a comparable or slightly lower complexity of O (MN) if the saliency map can be reused or efficiently regenerated.

Local variance calculation and frequency-domain transformation for saliency are both lightweight per-pixel operations, and the multi-threshold control avoids redundant computations, making the additional overhead from the adaptive components manageable. Compared to single-channel or fixed-window methods (Li X. et al., 2013; Peng et al., 2014; Yao et al., 2017; Chang et al., 2022) with O (MN) complexity, our scheme’s efficiency is comparable for standard 512 × 512 color images. Conversely, our threshold-based mechanism is more efficient than methods relying on exhaustive search or complex optimization (Qu and Kim, 2015; Wu et al., 2020; Ou et al., 2015; Mao et al., 2023), which often incur higher complexity. It is acknowledged that the incorporation of dual-dimensional complexity detection and multi-threshold control introduces additional computational overhead compared to simpler single-channel methods. However, this increase is a direct consequence of achieving superior rate-distortion performance and remains within acceptable limits for standard image processing tasks. Moreover, several avenues exist for further efficiency optimization within the current framework. For instance, future studies could adopt lightweight feature extraction strategies, accelerate complexity map calculation, or simplify the adaptive decision-making process without compromising performance. These optimization strategies will be explored in future work to enhance the method’s practicality while preserving its performance advantages. It is worth noting, however, that even in its current form, the framework achieves a favorable balance between computational cost and embedding performance. The observed time complexity aligns with the sophistication of the operations: our method aims to strike a balance between achieving superior performance and maintaining acceptable computational cost for standard color image processing. It is also instructive to consider the method’s performance under varying inter-channel correlation. The test images exhibit diverse color characteristics: images like Tiffany and Splash, with smooth color regions, are likely to possess strong correlations, where the proposed method achieves substantial gains (e.g., 0.93 dB at 20,000 bits for Tiffany). In contrast, images such as House and Lake, with more varied color distributions, probably have weaker correlations; here the improvements are modest (0.08–0.22 dB at 10,000 bits). This suggests that the bidirectional correction is most beneficial when strong inter-channel dependencies exist. Importantly, even on weakly correlated images, the method remains competitive with state-of-the-art approaches, thanks to the intra-channel features of the dual-dimensional complexity map and the adaptive multi-threshold control. A systematic quantitative analysis linking correlation coefficients to performance is left for future work.

While the experimental results demonstrate the overall superiority of the proposed integrated scheme, it is insightful to discuss the expected contribution of each core module to the final performance. The three modules are designed to address distinct challenges in color image RW. The Bidirectional Prediction Correction Module is primarily responsible for harnessing inter-channel correlations more completely than unidirectional approaches, which likely forms the foundation for improved prediction accuracy and capacity allocation, especially in images with strong channel dependencies (e.g., Tiffany, Splash). The Dual-dimensional Complexity Detection Module introduces a more nuanced content characterization by combining local variance and visual saliency. This enhancement in feature discrimination is expected to be the key driver for the adaptive and precise localization of embedding regions, leading to the observed graceful degradation in PSNR as payload increases (as shown in Figure 11). Finally, the Multi-threshold Joint Constraints Module operationalizes the complexity information by providing a coordinated control mechanism. Its role is crucial in translating the fine-grained complexity map into concrete, adaptive embedding decisions (window size, embedding order), thereby mitigating invalid pixel shifts and distortion, which is reflected in the maintained high fidelity at moderate payloads. The synergistic interplay among these modules is essential; for instance, the accuracy of the complexity map directly affects the efficacy of the threshold controls, and the bidirectional correction benefits from the adaptively selected embedding regions. The presented holistic performance gain is a result of this coordinated design. A detailed quantitative ablation study to isolate each module’s exact contribution will be a valuable focus of our future work.

6 Conclusion

In this paper, we introduce a novel reversible watermarking algorithm tailored for color digital images. By identifying the reference channel and correlation channels among different color channels, a bidirectionally corrected predicted pixel embedding mechanism based on the PVO algorithm is established. Furthermore, we leverage image content through local complexity and visual saliency, constructing a dual-dimensional complexity evaluation method. Through joint control of multi-element thresholds, this method enables the adaptive optimal selection of embedding parameters. Finally, through comparative experiments, we evaluate the advantages and limitations of the proposed algorithm compared to several state-of-the-art algorithms in the same category. The results verify the application value of the proposed method in reducing image distortion, especially in scenarios with low capacity and high image quality, as well as in scenarios with smoothly varying content. To further expand the application scope and generalization ability of the proposed method, future work will focus on three key directions: First, conducting comprehensive performance evaluation on more diverse datasets to verify the adaptability of the method to varying resolutions and texture characteristics of modern digital images; Second, optimizing the dual-dimensional complexity detection module to enhance its sensitivity to fine-grained details of modern images, thereby improving prediction accuracy and embedding stability for smooth and uniform content; Third, investigating the robustness of the proposed method against common image attacks such as JPEG compression and noise interference to enhance its reliability in complex application environments, thereby enhancing its practicality and scalability.

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Footnotes

Edited by: Hicham Karmouni, Sidi Mohamed Ben Abdellah University, Morocco

Reviewed by: Gurpreet Kaur, Amity University, India

Mohamed Amine Tahiri, Universite Sidi Mohamed Ben Abdellah Laboratoire d’Ingenierie Systemes et Applications, Morocco