Dual Partial Reversible Data Hiding Using Enhanced Hamming Code

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This work proposes a new reversible data hiding (RDH) method based on two images, using enhanced Hamming code (HC(7,4)). This method aims to improve data embedding capacity while maintaining high image quality. Experimental results show a significant increase in data embedding rate (1.5 bpp) compared to existing techniques, with superior image quality.

I emphasize the following aspects:

- The methodology is described in detail and is appropriate for the study's objective. The description of the development, implementation, experiments, and results analysis stages is clear and understandable.

- The results are presented coherently and organized. The evaluation metrics are relevant and well-used to compare the proposed method with existing methods.

- The discussion of the results is comprehensive and well-founded. The authors explain the reasons why the proposed method surpasses previous methods, addressing aspects such as image quality, insertion capacity, and computational efficiency.

- The conclusions effectively summarize the main findings of the research. The authors highlight significant improvements in insertion capacity, image quality, robustness against steganalysis, reversibility, and computational efficiency.

- The study is relevant to the RDH field, especially in applications that require high insertion capacity and preservation of image quality. The innovative approach using enhanced Hamming code contributes to the advancement of the field.

- The work offers significant contributions to the RDH field. The insertion capacity of up to 1.5 bpp and the maintenance of high image quality are important advances. Robustness against steganalysis and computational efficiency are also valuable contributions.

To further improve the quality of the work, the authors should consider:

- Updating the Literature Review: Include references from the last 3 years to ensure the review is up-to-date.

- Citing Tables and Figures: Ensure all Tables and Figures are cited in the text before they appear.

- Referencing Used Software: Include references to the software used in the methodology description.

Author Response

We would like to thank the reviewers for their valuable feedback and comments on our manuscript. We have carefully considered each point raised and have made the necessary revisions to address the concerns. Below, we provide detailed responses to each of the reviewers' comments.

 

Reviewer

Comment 1 :“Updating the Literature Review: Include references from the last 3 years to ensure the review is up-to-date.

Author Response:
We appreciate the reviewer’s valuable suggestion regarding the inclusion of more recent literature to strengthen the contextual relevance of our study. In response, we have updated Section 1 to incorporate and discuss several newly published works from the past three years.

Specifically, we have included:

• Lee and Chan (2024), who introduced a dual-image RDH method based on vector coordinate embedding with Triangular Order Coding (TOC), offering enhanced embedding flexibility and image quality without auxiliary data.
• Wan et al. (2025), who proposed a texture-guided hierarchical quantization-based multi-party RDH scheme, demonstrating practical application in medical image sharing while preserving reversibility and security.
• Zhan et al. (2024), who presented a fragile watermarking system using dual tampering detection and adaptive recovery to ensure image integrity and high PSNR quality in critical tampered regions.

The citations have been incorporated into the manuscript (Section 1) and added to the reference list accordingly.

 

Comment 2: Citing Tables and Figures: Ensure all Tables and Figures are cited in the text before they appear.

Author Response:
We thank the reviewer for the helpful observation. In response, we carefully reviewed the entire manuscript to ensure that all tables and figures are properly cited in the main text before their first appearance. Specifically, we verified that:

• Figure 2 (Standard array of HC(7,4)) is now referenced in the explanation of the codeword selection process in Section 3.1.
• Figures 3 to 5, which illustrate numerical examples and visual comparisons, are all introduced in the text before being displayed.
• Tables 1 to 3, which summarize embedding capacity, PSNR, and comparative performance, are each cited appropriately in the Results section prior to presentation.

 

Comment 3: Referencing Used Software: Include references to the software used in the methodology description.

Author Response:
We thank the reviewer for this valuable suggestion. In response, we have revised the “Experimental results” section to clearly specify the computational environment used in our experiments. Specifically, we have added the following sentence:

“All experiments were conducted on a system equipped with a Core i5-8250U processor (1.60 GHz) and 8GB of RAM, utilizing MATLAB R2019b as the simulation platform.”

 

We sincerely appreciate the reviewers’ detailed and thoughtful feedback, which helped us refine the manuscript considerably. We have addressed all comments point by point, and we believe the revised version reflects these improvements both in clarity and in scientific rigor.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This article is devoted to the development of a method for Partially Reversible Data Hiding (PRDH) using an enhanced Hamming Code (HC(7,4)) across image pairs, enabling high embedding capacity while maintaining image quality. The topic of the paper is timely and relevant, as reversible data hiding has extensive applications in secure data transmission, medical imaging, and copyright protection. The article’s structure conforms to the standard MDPI format for research articles (introduction, problem statement and related work, method description, experimental results, conclusion). The level of English is acceptable; the article reads relatively easily, although some syntactic inaccuracies are present throughout. The figures are of adequate quality and illustrate the algorithms and comparative results, although the diagrams could be rendered more clearly. The article cites 24 sources, all of which are relevant and cover both classical approaches and the latest advancements in RDH.

The following comments and recommendations can be made regarding the manuscript:

1. Despite the claimed novelty, the article lacks a rigorous formalisation of the problem. In Section 1 and subsequently in Sections 3.1–3.3, no clearly defined mathematical functional to be optimised is presented. The method is described primarily in algorithmic terms (see Algorithms 1–3), but is not formalised as a problem of optimal encoding or distortion minimisation. This omission hinders thorough theoretical analysis of the method’s efficiency and precludes comparisons with known lower bounds for classes of RDH schemes. Moreover, the authors do not consider the method’s robustness to errors beyond the Hamming space, which undermines its theoretical soundness.
2. The concept of partial reversibility is introduced somewhat arbitrarily. In Section 2.3, it is stated that “recovery is performed not to the original image, but to a partially modified cover” (p. 4). However, there is no explanation of how this modified cover deviates from the original, how this deviation is measured (e.g., via PSNR), or whether the transformation could be reversed to the original image if supplementary information were available. The lack of a quantitative criterion blurs the distinction between reversible and irreversible data hiding and raises doubts regarding the method’s applicability in sensitive domains, such as legal or medical contexts.
3. The complexity of the proposed method is not analysed. Algorithms 1–3 describe procedures for selecting the optimal code from the Hamming code table (see formulas 5 and 6), but no assessment of execution time or computational overhead is provided. Although the conclusion (p. 17) asserts that the method is “suitable for real-time applications”, this claim is not supported by quantitative experiments or comparative analysis with existing techniques. The absence of a performance evaluation is particularly critical for implementations in resource-constrained systems.
4. The experimental results are limited to the analysis of only three metrics: PSNR, embedding capacity, and embedding rate. There is no statistical analysis of the significance of differences between methods. Tables 2 and 3 present only average values, without confidence intervals, standard deviations, or significance tests. This approach weakens the credibility of the empirical conclusions. For instance, the reported 1–2 dB improvement in PSNR may be statistically insignificant in the absence of proper analysis.
5. The comparison with existing methods is superficial. Although results from other techniques are included (Tables 2 and 3), the authors do not account for differences in usage scenarios, permissible distortion levels, data density, or resilience to adversarial attacks. Notably, there is no evaluation of robustness against common modern attacks, including neural network-based attacks or JPEG compression. The only robustness test performed is RS analysis (Section 4, p. 15), which is insufficient as a standalone indicator of imperceptibility in contemporary conditions.
6. Figure 1, which presents the block diagram of the method, is uninformative. It merely repeats descriptive text blocks and does not reveal the architecture of module interactions. Elements are not annotated with numerical parameters; the stages of virtual pixel generation are not distinguished; data flows are not indicated. Consequently, the diagram provides little value for engineering applications and could be significantly improved.
7. The justification of the method’s practical value is declarative in nature. The conclusion (pp. 17–18) states that the method can be applied in medical imaging and secure communication systems. However, the text lacks specific examples or scenarios for such applications. There is no discussion of how integration into existing protocols or software would be achieved, what constraints would apply to image types, or whether the level of distortion would be acceptable in the mentioned use cases.

Author Response

Author Response to Reviewer Comments

Comment 1:

Despite the claimed novelty, the article lacks a rigorous formalisation of the problem. In Section 1 and subsequently in Sections 3.1--3.3, no clearly defined mathematical functional to be optimised is presented. The method is described primarily in algorithmic terms (see Algorithms 1--3), but is not formalised as a problem of optimal encoding or distortion minimisation. This omission hinders thorough theoretical analysis of the method’s efficiency and precludes comparisons with known lower bounds for classes of RDH schemes. Moreover, the authors do not consider the method’s robustness to errors beyond the Hamming space, which undermines its theoretical soundness.

Author Response:

We thank the reviewer for this constructive suggestion. In response, we revised Section 3.1 to introduce a formal representation of our method as a distortion minimization problem under the constraint of syndrome validity. Specifically, for a given pixel pair, we define a virtual 7-bit pixel y derived from selected bits of x_j^1 and x_j^2. When y does not satisfy the condition yH^T=0, the objective becomes:

min_{h ∈ C_0} ε(h) = (α - α̃)^2 + (β - β̃)^2                           (5)

where h is a codeword in the zero-syndrome coset C_0, and α, β are decimal values from the bit segments. This corresponds to Equation (5) and gives a precise distortion measure for codeword selection. The limitation regarding robustness beyond Hamming space is acknowledged and now explicitly noted as future work in the revised Conclusion.

Comment 2:

The concept of partial reversibility is introduced somewhat arbitrarily. In Section 2.3, it is stated that ‘recovery is performed not to the original image, but to a partially modified cover’ (p. 4). However, there is no explanation of how this modified cover deviates from the original, how this deviation is measured (e.g., via PSNR), or whether the transformation could be reversed to the original image if supplementary information were available. The lack of a quantitative criterion blurs the distinction between reversible and irreversible data hiding and raises doubts regarding the method’s applicability in sensitive domains, such as legal or medical contexts.

Author Response:

We appreciate the reviewer’s insight on this conceptual issue. We have revised Section 2.3 to clarify that partial reversibility refers to the ability to reconstruct the two deterministically modified cover images from the stego pair, not the original image. This design avoids extra side information, but prevents exact recovery of the original. We now clearly state this in both Section 2.3 and the Conclusion. To quantify deviation, we now report that the PSNR between the original and cover images exceeds 48 dB in all cases, supporting minimal perceptual distortion. We also clarify that this approach is not intended for strict-lossless applications like medical archiving or forensics, and this limitation is now mentioned in the revised Conclusion.

Comment 3:

The complexity of the proposed method is not analysed. Algorithms 1–3 describe procedures for selecting the optimal code from the Hamming code table (see formulas 5 and 6), but no assessment of execution time or computational overhead is provided. Although the conclusion (p. 17) asserts that the method is "suitable for real-time applications", this claim is not supported by quantitative experiments or comparative analysis.

Author Response:

Thank you for this important observation. In Section 4, we now include a discussion of computational complexity, noting that the method searches over a fixed-size (16-codeword) HC(7,4) table per pixel, yielding O(N) complexity for N-pixel images. Empirically, the full embedding and extraction of a 512×512 grayscale image takes approximately 0.97 seconds on a standard Intel i5-8250U processor using MATLAB. Accordingly, we have revised our claim to state that the method is "lightweight and suitable for low-latency scenarios" rather than asserting real-time applicability.

Comment 4:

Tables 2 and 3 present only average values, without confidence intervals, standard deviations, or significance tests. This approach weakens the credibility of the empirical conclusions.

Author Response:

We thank the reviewer for highlighting the importance of statistical reliability. For Table 2, we clarified that the embedding ratio is exactly 1.50 for all images, with standard deviation 0.00. This is a deterministic outcome of our method, not a rounding artifact. A footnote was added for clarification. For Table 3, individual PSNR and bpp values were already shown. We now note that bpp is constant at 1.50 and that PSNR standard deviation across test images is less than 0.06 dB. This information has been added beneath each table.

Comment 5:

The comparison with existing methods is superficial. Although results from other techniques are included (Tables 2 and 3), the authors do not account for differences in usage scenarios, permissible distortion levels, data density, or resilience to adversarial attacks. Notably, there is no evaluation of robustness against common modern attacks, including neural network-based attacks or JPEG compression. The only robustness test performed is RS analysis (Section 4, p. 15), which is insufficient as a standalone indicator of imperceptibility in contemporary conditions.

Author Response:

We appreciate this valuable point. In the revised Section 4, we clarify that our method targets uncompressed domains like BMP or TIFF where perfect reversibility is essential, and is inherently incompatible with lossy formats like JPEG due to its LSB-based design. RS analysis was used in alignment with common RDH literature to check imperceptibility. We now explicitly note that robustness against adversarial and transform-domain attacks is a limitation, and propose transform-domain RDH extensions as future work.

Comment 6:

Figure 1, which presents the block diagram of the method, is uninformative. It merely repeats descriptive text blocks and does not reveal the architecture of module interactions. Elements are not annotated with numerical parameters; the stages of virtual pixel generation are not distinguished; data flows are not indicated. Consequently, the diagram provides little value for engineering applications and could be significantly improved.

Author Response:

We sincerely thank the reviewer for this constructive suggestion. Originally, Figure 1 served as a conceptual sketch, but we agree that it lacked clarity for engineering use. Accordingly, we introduced a new modular diagram (Figure 2), illustrating three core modules: Cover Image Generator (Algorithm 1), Data Embedder (Algorithm 2), and Data Extractor (Algorithm 3). Each is depicted as an independent block with directional data flows and high-level inputs/outputs. Although we omit parameter annotations in the figure due to space constraints, Sections 3.1--3.3 now detail the bit-level formulations (e.g., Equations (4)--(5)) referenced by each block.

Comment 7:

The justification of the method’s practical value is declarative in nature. The conclusion (pp. 17–18) states that the method can be applied in medical imaging and secure communication systems. However, the text lacks specific examples or scenarios for such applications. There is no discussion of how integration into existing protocols or software would be achieved, what constraints would apply to image types, or whether the level of distortion would be acceptable in the mentioned use cases.

Author Response:

We thank the reviewer for pointing this out. While our work primarily focuses on algorithmic contributions and theoretical analysis, we agree that clearer framing of application scenarios is important. The revised Conclusion now explains that our method is suitable for domains such as secure image communication and metadata-preserving archiving, but may not meet the needs of lossless medical or forensic systems. This clarification reflects a more realistic scope of applicability and outlines future directions for domain-level integration studies.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

The conducted analysis of the revisions confirms a thorough and responsible approach to improving the manuscript, resulting in a substantial enhancement of its scientific and practical quality.

The formalisation of the problem in terms of distortion minimisation under a zero-syndrome constraint is particularly commendable. The introduction of Equation (5) adds methodological rigour and aligns the proposed algorithm with a class of optimisation problems within the codeword space. The explicit acknowledgement of limited robustness beyond the Hamming space, and the identification of this issue as a direction for future research, also reflects a mature and academically sound perspective.

The concept of partial reversibility is now presented with greater clarity and coherence. The clarification that the method reconstructs deterministically modified cover images, rather than the original image, and that the PSNR consistently exceeds 48 dB, makes the method’s assumptions more transparent for potential users. The clear statement in the Conclusion regarding its inapplicability to strictly lossless domains such as medicine and forensics adds to the credibility of the work.

It is also worth noting the added explanations accompanying the tables, which highlight the statistical determinism of the embedding ratio and the stability of PSNR values. This contributes to improved interpretability of the results. While an in-depth comparative analysis with alternative methods remains limited, the authors have appropriately delineated the scope of applicability (i.e., uncompressed formats) and acknowledged current limitations in terms of robustness to distortions and attacks.

The submitted revisions are substantial and sufficient. The article may be recommended for publication in its current form.