Capacity-Raising Reversible Data Hiding Using Empirical Plus–Minus One in Dual Images
Abstract
:1. Introduction
2. Related Work
2.1. Reversible Data-Hiding Scheme Based on Dual Stegano Images Using Orientation Combinations
2.2. A Square-Lattice-Oriented Reversible Information-Hiding Scheme with Reversibility and Additivity for Dual Images
3. Proposed Method
3.1. Proposed Method Framework
3.2. Preprocessing
3.2.1. Secret Preprocessing
3.3. Rules Table Generation
3.4. Embedding Process
- In any entry, the difference in the modifications of the two stego images at neighboring pixels is at most 2 (low distortion ds), as seen in Table 1.
- The modification entry can be determined uniquely from stego image 1 and stego image 2 (hence extracting the hidden message). For example, during the extraction process, the rules used for embedding can be found, and these rules can be used to look up from the EQUATION to uniquely identify the entry. It should be noted that an additional value p (as defined in Equation (3)) is necessary to determine the actual entry used in the extended table. Figure 2 shows an example of how Table 1 can be extended to a table with 262,133 entries. Examples of the fourteenth and fifteenth entries (s = 13 and s = 14, respectively) are shown. From this figure, it is demonstrated that the table can be extended without altering the 13 rules. Each rule can be used for multiple table entries, and therefore the difference between neighboring pixels does not change when the table size increases. The maximum difference between neighboring pixels (distortion) is 2. Each next entry in the table references an entry that is already in Table 1; however, it should be noted that the table size increases to accommodate cases where the secret has more than 13 distinct characters. The table size depends on the size of the images used. For instance, the proposed method used 512 × 512 image sizes; therefore, the highest multiple of 13 (262,132) that was less than 262,144 (512 × 512) was found to be the maximum secret number, as can be seen in the last row (colored in blue) of the extended table in Figure 2. In Figure 2, the different colors represent entries that use different rules, while the same colors represent entries that use the same rule. Figure 3 demonstrates the embedding procedure. Figure 4 and Table 1 demonstrate the extraction procedure and how the entries alongside p are uniquely used during the extraction process.
3.5. Embedding Example
- Stego image 1 = (50 − 1, 48), stego image 2 = (50, 48).
- Stego image 1 = (49, 48), Stego image 2 = (50, 48).
Algorithm 1 Embedding Algorithm Pseudocode. |
Input: secret data s, cover image CI. |
Output: 2 stego images |
for i = 0 …H − 1 do |
For j = 0 …W − 2 do |
block = [(x, y), (x, y + 1) |
p = floor(s/12) |
if s < 12 then |
t = 1; |
else |
t = p × 12 + 1; |
end if |
if t = 1 then |
Esr = s; |
else |
Esr = s − t; |
end if |
stego1_block=[]; |
stego2_block=[]; |
for pixel in block do |
adjust stego1_pixel &stego2_pixel using rules; |
stego 1_block.append(stego1_pixel); |
stego2_block.append(stego2_pixel); |
end for |
end for |
end for |
3.6. Extraction and Recovery Process
Algorithm 2 Extraction and Recovery Algorithm Pseudocode. |
Input: p, stego_image_1, stego_image_2. |
Output: secret data s, cover image CI. |
lookup_table = {table used during embedding process}; |
recovered_image = …; |
rule = {}; |
for i in range(stego_image_1.width): |
for j in range(stego_image_1.height): |
x1, y1 = stego_image_1.get_pixel(i, j); |
x2, y2 = stego_image_2.get_pixel(i, j); |
x = (x1 + x2 + 1)//2; y = (y1 + y2 + 1)//2; |
recovered_image.set_pixel(i, j, (x, y)); |
rule[(i, j)] = (x1 − x, y1 − y, x2 − x, y2 − y); |
end for |
end for |
secret_message = ‘ ’; |
for i in range(stego_image_1.width): |
for j in range(stego_image_1.height): |
key = (rule[(i, j)][0], rule[(i, j)][1]); |
secret_number = lookup_table[key]; |
t = p × 12 + 1; |
secret_message + = t+secret_number; |
end for |
end for |
3.7. Extraction and Recovery Example
3.8. Additional Information
4. Experimental Results
4.1. Security Analysis
- (a)
- The regular groups with RM and R-M;
- (b)
- The singular group with SM and S-M;
- (c)
- The unusable group.
- Find out the total number of POVs of the image.
- Sort the pixel values and calculate the average of the j-th group of pixels to the total concurrencies using Equation (10).
- 3.
- Compute the representative number of pixel value pairs in the j group, where nj = numbers of index 2j.
- 4.
- Compute the chi-square statistics using Equation (11).
- 5.
- Use the chi-square distribution characteristics to compute the image-hiding probability p using (12).
4.2. Visual Analysis
4.3. Stego Analysis Using StegoExpose
5. Application
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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s | (x1, y1) | (x2, y2) | ds |
---|---|---|---|
0 | (x, y) | (x, y) | 0 |
1 | (x, y) | (x − 1, y) | 1 |
2 | (x − 1, y) | (x, y) | 1 |
3 | (x, y) | (x, y − 1) | 1 |
4 | (x, y − 1) | (x, y) | 1 |
5 | (x, y) | (x − 1, y − 1) | 2 |
6 | (x − 1, y − 1) | (x, y) | 2 |
7 | (x + 1, y) | (x − 1, y) | 2 |
8 | (x − 1, y) | (x + 1, y) | 2 |
9 | (x, y + 1) | (x, y − 1) | 2 |
10 | (x, y − 1) | (x, y + 1) | 2 |
11 | (x − 1, y) | (x, y − 1) | 2 |
12 | (x, y − 1) | (x − 1, y) | 2 |
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Huang, C.-T.; Weng, C.-Y.; Shongwe, N.S. Capacity-Raising Reversible Data Hiding Using Empirical Plus–Minus One in Dual Images. Mathematics 2023, 11, 1764. https://doi.org/10.3390/math11081764
Huang C-T, Weng C-Y, Shongwe NS. Capacity-Raising Reversible Data Hiding Using Empirical Plus–Minus One in Dual Images. Mathematics. 2023; 11(8):1764. https://doi.org/10.3390/math11081764
Chicago/Turabian StyleHuang, Cheng-Ta, Chi-Yao Weng, and Njabulo Sinethemba Shongwe. 2023. "Capacity-Raising Reversible Data Hiding Using Empirical Plus–Minus One in Dual Images" Mathematics 11, no. 8: 1764. https://doi.org/10.3390/math11081764