# High-Capacity Embedding Method Based on Double-Layer Octagon-Shaped Shell Matrix

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^{2}

^{3}

^{*}

## Abstract

**:**

**PSNR**) within an acceptable range with the embedding time less than 2 s.

## 1. Introduction

^{78}solutions) for resisting attacks. In 2019, Chang and Liu proposed two enhanced real time turtle shell-based data hiding schemes [16]. Both the schemes mapped each cover pixel pair onto the original or altered the turtle shell matrix to find out its associate set for embedding secret data. Then, the cover pixel pair was modified with minimum distortion according to the associate set. In 2020, Nguyen et al. proposed a new data hiding approach to embed secret data based on an x-cross-shaped reference-affected matrix [17]. The reference matrix consisted of three parts: petal matrix, calyx matrix, and stamen matrix, which were combined for executing the embedding procedure. Using this method, it was found that the smooth regions were more suitable for embedding secret data due to the smaller difference between pixel values. Unlike the traditional EMD method, the above-mentioned methods used a reference matrix instead of the extraction function in the embedding and extraction procedures.

- (1)
- The regular octagon-shaped shell method proposed in [12] carried only 5-bit of secret data for each pair of cover pixels. However, in our proposed scheme, additional 2-bits data was embedded in each cover pixel pair, leading to a higher embedding capacity of 3.5 bpp.
- (2)
- Peak signal to noise ratio (
**PSNR**) and structural index similarity (**SSIM**) are two measuring tools that are widely used in image quality assessment. Especially in the steganography image, these two measuring instruments are used to measure the quality of imperceptibility. Based on the experimental analysis, results show that the average value of the stego-image quality had an acceptable value of 37dB on an average while**SSIM**values were between [0.9, 0.95]. We contend that our proposed method is more suitable for complex images to obtain a higher**SSIM**value. Also, as seen previously, the**PSNR**values remained stable for all the test images irrespective of their image texture. - (3)
- Lastly, the computational cost in terms of embedding time is 1.82 s on an average.

## 2. Related Works

#### 2.1. The EMD Embedding Method and the EMD Extensions

**N**pieces with

**L**bits, and each secret piece is presented as a decimal value by

**D**digital numbers in a (2

**n**+ 1)-ary notational system, where

**n**is a parameter to determine how many pixels of cover image are used to hide one secret digit.

**P**= (

_{n}**p**

_{1},

**p**

_{2}, …,

**p**), which consists of

_{n}**n**cover pixels. A weight vector

**W**= (

_{n}**w**

_{1},

**w**

_{2}, …,

**w**) = (1, 2, …,

_{n}**n**) is defined. Therefore, the EMD method defines an embedding function

**f**as weighted sum function modulo (2

**n**+ 1) for each group, a secret digit

**d**can be carried by the

**n**cover pixels, and at most one pixel is increased or decreased by one.

**f**can be expressed as Equation (2):

**d**, the group

**P**is modified into

_{n}**Q**= (

_{n}**q**

_{1},

**q**

_{2}, …,

**q**), which is defined according to following conditions:

_{n}**Q**=_{n}**P**= (_{n}**p**_{1},**p**_{2}, …,**p**), if_{n}**d = f**.- When
**d**≠**f**, computes**s**=**d**-**f**mod (2**n**+ 1) and

**Q**= (

_{n}**q**

_{1},

**q**

_{2}, …,

**q**) by the following extraction function shown in Equation (3).

_{n}**n**= 2. For $\mathit{n}=2$, the

**PSNR**values of all test images averaged to 52.11 dB with embedding capacity of 1 bit per pixel (bpp).

**PSNR**was 51.157 dB when the embedding rate was 1 bit per grayscale pixel and the average

**PSNR**was 31.847 dB when the embedding rate was 4 bpp with good security and absence of overflow/underflow. In 2017, Lee et al. proposed an REMD [8] method using image interpolation and edge detection that achieved an embedding capacity of 3.01 bpp with an average image quality of 33 dB.

#### 2.2. Data Hiding Scheme Based on Turtle-Shaped Shells

**PSNR**of 49.4 dB and an average embedding capacity of 1.5 bpp. The limitation of the turtle shell scheme [10] was that the embedding rate was limited.

#### 2.3. The Regular Octagon-Shaped Shell Embedding Method

**PSNR**value obtained was 51.7 dB with an average embedding capacity of 2 bpp. In 2017, Leng and Tseng introduced the reference matrix based on regular octagon-shape shells [12] which achieved a payload of 2.5 bpp. The further improvement in [13] achieved the possible payload of 3.5 bpp.

## 3. Proposed Scheme

#### 3.1. Construction of the Double-Layer Octagon-Shaped Shell Reference Matrix

#### 3.2. Data Embedding and Data Extraction Procedures

Algorithm 1. Data Embedding Algorithm. |

Input: A cover image $I$ sized $W\times H$, the binary secret stream S with length L. |

Output: A stego-image ${I}^{\prime}$. |

Step 1: Construct a one-layer reference matrix $M={\left[m\left(i,j\right)\right]}_{256\times 256}$ according to the rules described in Section 2. Thereafter, generate a double-layer reference matrix, assigning the type value ${t}_{m}\left(i,j\right)$ corresponding to each matrix element $m\left(i,j\right)$ according to the rules described in Section 3.1. |

Step 2: Divide the cover image $I$ into non-overlapping pixel pairs (${p}_{k},{p}_{k+1}$) where $k=\left\{1,3,\dots ,\left(W\times H-1\right)\right\}$. Consider $\left({p}_{k},{p}_{k+1}\right)$ as the coordinates of the matrix M to specify the value $m\left({p}_{k},{p}_{k+1}\right)$ with the corresponding type ${t}_{m}$$({p}_{k},{p}_{k+1}$). |

Step 3: Divide the secret message S into sub-streams ${s}_{c}$ of 7 bits, where $c\in \left\{1,2,\dots ,\u2308\frac{L}{7}\u2309\right\}$. For each sub-stream ${s}_{c}$, convert the first 2 bits into a 4-ary digit ${s}_{c1}$ and the last 5 bits into a 32-ary digit ${s}_{c2}$, viz., ${s}_{c}={s}_{c1}\left|\right|{s}_{c2}$ where “||” denotes the string concatenation operator. |

Step 4: Embed each sub-stream ${s}_{c}$ into each pixel pair (${p}_{k},{p}_{k+1}$) according to the following rules: Find the closest element m(u,v) by searching in a square of $25\times 25$ centered on $m\left({p}_{k},{p}_{k+1}\right)$, where $m\left(u,v\right)={s}_{c2}$ and ${t}_{m}\left(u,v\right)={s}_{c1}$. Replace $\left({p}_{k},{p}_{k+1}\right)$ with $\left(u,v\right)$ such that the stego-pixel pairs $\left({q}_{k},{q}_{k+1}\right)=\left(u,v\right)$, to embed the sub-stream ${s}_{c}$ consisting of ${s}_{c1}$ and ${s}_{c2}$. |

Step 5: Repeat Step 4 until all the sub-streams are embedded. Finally, obtain the stego-image I’. |

Algorithm 2. Data Extraction Algorithm. |

Input: A stego-image I^{′} sized $W\times H$. |

Output: The binary secret stream S. |

Step 1: Reconstruct the double-layer reference matrix $M={\left[m\left(i,j\right)\right]}_{256\times 256}$ where each matrix element $m\left(i,j\right)$ corresponds to a type value ${t}_{m}\left(i,j\right)$, which is calculated according to the reules described in Section 3.1. |

Step 2: Divide the stego-image I’ into non-overlapping pixel pairs, where $k=\left\{1,3,\dots ,\left(W\times H-1\right)\right\}$. |

Step 3: For each stego-pixel pair $\left({q}_{k},{q}_{k+1}\right),$ find two digits m$\left({q}_{k},{q}_{k+1}\right)$ in 32-ary format and ${t}_{m}\left({q}_{k},{q}_{k+1}\right)$ in 4-ary format, respectively. The hidden secret data is ${s}_{c}={s}_{c1}\left|\right|{s}_{c2}$, where ${s}_{c2}=m\left({q}_{k},{q}_{k+1}\right)$ and ${s}_{c1}={t}_{m}\left({q}_{k},{q}_{k+1}\right)$, respectively. |

Step 4: Convert ${s}_{c}$ into binary bits of secret data. |

Step 5: Repeat Steps 2–4 to extract all the sub-streams. Combine all the sub-streams to form the secret binary stream S. |

#### 3.3. Example of Data Embedding and Data Extraction Procedures

## 4. Experimental Results

**EC**), Peak Signal to Noise Ratio (

**PSNR**), and Structural Similarity Index (

**SSIM**) were used as evaluation parameters to validate the performance of the proposed method.

**EC**is the amount of secret message per pixel that can be embedded in the image. If the number of secret messages embedded in the image is more and the image quality can be maintained to a certain standard, then the data hiding method is supposed to have high imperceptibility and high embedding capacity.

**PSNR**is used to measure the quality of the image. If the value of this parameter is high, it means that the image quality is good, and the secret message hidden in stego-image cannot be perceived easily by the human eye. Refer to Equations (6) and (7) to calculate

**PSNR**. Refer to Equation (8) to calculate the

**EC**.

^{′}represents the stego-image.

**MSE**represents mean-square error,

**EC**represents embedding capacity, and $\left|S\right|$ are the total number of secret bits that can be embedded).

**SSIM**measures similarity between the original image and the stego image. This is in line with the human eye’s judgment of image quality. The higher the

**SSIM**value, the higher will be the similarity between the original image and the stego image.

**SSIM**value is calculated using Equation (9) as shown below, where ${\mu}_{x}$ and ${\mu}_{y}$ are the average values of the original image and the stego image respectively; ${\sigma}_{xy}$ is the co-variation between the original and the stego images respectively; and ${\sigma}_{x}$ and ${\sigma}_{y}$ are the variation of the original image and the stego image respectively. ${C}_{1}$ and ${C}_{2}$ are constants.

**PSNR**with an average value of 36.91 dB compared to the average

**PSNR**of 30.62 dB obtained in [13]. While [14,15,16,20] have higher average

**PSNR**values of 44.12 dB, 46.37 dB, 46.84 dB, and 41.27 dB respectively, the embedding capacity of our proposed scheme is higher by 1.5 bpp.

**PSNR**for each image. For example, when the embedding capacity is $70\times {10}^{4}$ bits, the average

**PSNR**of each image remains at 38.09 dB. At the embedding capacity of $50\times {10}^{4}$ bits, the

**PSNR**remains at an average quality of 39.56 dB. Therefore, the

**PSNR**value remains stable for each test image. The magnitude of the pixels change during the data embedding process, which depends on the constructed double-layer reference matrix and the embedding procedure. In other words, the image quality of the stego-image is independent of the cover image and it is possible to maintain the image quality within an acceptable range using the proposed method.

**SSIM**values of six standard test images at a fixed

**EC**(3.5 bpp, which is 917,504 bits). Interestingly, the

**SSIM**value for complex images such as Baboon is higher compared to the

**SSIM**value for smooth images such as Airplane. This is a unique finding in our proposed method. Therefore, we contend that our proposed method is more suitable for complex images to obtain a higher

**SSIM**value. Also, as seen previously, the

**PSNR**values remains stable for all the test images irrespective of their image texture. To confirm this, we tested our proposed method on 50 additional test images using USC-SIPI database [18] and Kodak Image database [21] as shown in Figure 9.

**SSIM**values can be categorized into three groups as shown in the Figure 9. It is evident from the figure that the images which obtain

**SSIM**values between [0.90, 0.95] are the most suitable for the proposed method as these images show the highest

**PSNR**value. The

**PSNR**range obtained for

**SSIM**values between [0.90, 0.95] is [34.085597, 36.941688] dB, which falls in the acceptable image quality range.

**PSNR**values for six standard test images at different

**EC**(bpp) as shown in Figure 10 below. The figure clearly shows that as the embedding rate increases, the

**PSNR**value decreases similar to the property of methods based on the magic matrix hiding methods of [12,13,14,15,16,17,19,20,22]. However, interestingly, the

**PSNR**values for the test images do not have much difference from each other, which again shows the point that the

**PSNR**values remains stable using our proposed method irrespective of the image texture.

**EC**(bpp),

**PSNR**(dB), and embedding time (seconds) among Chang et al.’s turtle shell in a single-layer embedding, Leng and Tseng’s octagon-shaped shells scheme in a single-layer embedding, Xie et al.’s two-layer turtle shell matrix embedding, Shen et al.’s double-layer square magic matrix scheme, and the proposed method in a two-layer embedding using octagon-shaped shells. For simplicity, the representations [TS, 1,

**EC**1.5], [Octagon, 1,

**EC**2.5], [TS, 2,

**EC**2.5], [Square, 2,

**EC**2.5], [Ours, 2,

**EC**3.5] stand for the methods exploiting Chang et al.’s turtle shell in a single-layer embedding [10] with

**EC**= 1.5 bpp, Leng and Tseng’s octagon-shaped shells scheme in a single-layer embedding [12] with

**EC**= 2.5 bpp, Xie et al.’s two-layer turtle shell matrix embedding [19] with

**EC**= 2.5 bpp, Shen et al.’s double-layer square magic matrix scheme with

**EC**= 2.5 bpp, and the proposed method in a two-layer embedding using octagon-shaped shells with

**EC**= 3.5 bpp. As mentioned earlier, with regards to the information hiding method based on the magic matrix, under the same embedding capacity regardless of the image texture, the

**PSNR**and

**SSIM**values of all test images remain stable. Similarly, the execution time of each method also has this stable characteristic, which is presented in Table 4 and Figure 11.

**EC**(bpp),

**PSNR**(dB), and embedding time (seconds) among Chang et al.’s turtle shell in a single-layer embedding, Leng’s octagon-shaped shells scheme in a single-layer embedding, Xie et al.’s two-layer turtle shell matrix embedding, Shen et al.’s double-layer square magic matrix scheme, and the proposed method in a two-layer embedding using octagon-shaped shells. For simplicity, the representations [TS, 1,

**EC**1.5], [Octagon, 1,

**EC**2.5], [TS, 2,

**EC**2.5], [Square, 2,

**EC**2.5], [Ours, 2,

**EC**3.5] stand for the methods exploiting Chang et al.’s turtle shell in a single-layer embedding [10] with

**EC**= 1.5 bpp, Leng and Tseng’s octagon-shaped shells scheme in a single-layer embedding [12] with

**EC**= 2.5 bpp, Xie et al.’s two-layer turtle shell matrix embedding [19] with

**EC**= 2.5 bpp, Shen et al.’s double-layer square magic matrix scheme with

**EC**= 2.5 bpp, and the proposed method in a two-layer embedding using octagon-shaped shells with

**EC**= 3.5 bpp. As mentioned earlier, the

**PSNR**and

**SSIM**values of all test images remain stable for the information hiding method based on the magic matrix, under the same embedding capacity regardless of the image texture. Similarly, the execution time of each method also shows stability, which is presented in Table 4 and Figure 11.

## 5. Conclusions

**PSNR**37 dB on an average while

**SSIM**values were between 0.9–0.95. We contend that our proposed method is more suitable for complex images to obtain a higher

**SSIM**value as compared to other methods. Also, as seen previously, the

**PSNR**values were stable for all the test images irrespective of their image texture. Moreover, the computation cost in terms of embedding time was 1.82 s on average.

- (1)
- The first point is the results in Table 3 are showing the proposed algorithm is insecure because of the obtained
**SSIM**values with high**EC**. When**SSIM**= 0.95, most people will be satisfied with the visual image of the image. While**SSIM**is lower than 0.90, it means that the defect may be twice as much as 0.95, and the naked eyes may perceive the picture deterioration and cause insecurity. According to our experimental results, when the**EC**value reaches 3.5 bpp, approximately 10% of pictures will have an**SSIM**slightly below 0.9. The results under 0.95 should be the limit for increasing the**EC**. - (2)
- From the experimental results in Figure 11, although the proposed method has a larger maximum embedding capability, it takes more time to implement information hiding than other methods. The execution time of the maximum embedding capacity was around 1.90 s. Therefore, in future, we will improvise the method showcased in paper [22] to develop another multi-layer information hiding method, in order to achieve better information hiding computational capabilities, making it more competitive in the real-time application environment.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Anderson, R.J.; Petitcolas, F.A.P. On the limits of steganography. IEEE J. Sel. Areas Commun.
**1998**, 16, 474–481. [Google Scholar] [CrossRef] [Green Version] - Johnson, N.F.; Jajodia, S. Exploring steganography: Seeing the unseen. Computer
**1998**, 31, 26–34. [Google Scholar] [CrossRef] - Saha, A.; Halder, S.; Kollya, S. Image steganography using 24-bit bitmap images. In Proceedings of the 14th International Conference on Computer and Information Technology (ICCIT 2011), Dhaka, Bangladesh, 22–24 December 2011; pp. 56–60. [Google Scholar] [CrossRef]
- Pitropakis, N.; Lambrinoudakis, C.; Geneiatakis, D.; Gritzalis, D. A Practical Steganographic Approach for Matroska Based High Quality Video Files. In Proceedings of the 2013 27th International Conference on Advanced Information Networking and Applications Workshops, Barcelona, Spain, 25–28 March 2013; pp. 684–688. [Google Scholar] [CrossRef]
- Chan, C.K.; Cheng, L.M. Hiding Data in Images by Simple LSB Substitution. Pattern Recognit.
**2004**, 37, 469–474. [Google Scholar] [CrossRef] - Zhang, X.P.; Wang, S.Z. Efficient Steganographic Embedding by Exploiting Modification Direction. IEEE Commun. Lett.
**2006**, 10, 781–783. [Google Scholar] [CrossRef] - Lee, C.F.; Chen, H.L. A Novel Data Hiding Scheme Based on Modulus Function. J. Syst. Softw.
**2010**, 83, 832–843. [Google Scholar] [CrossRef] - Lee, C.F.; Weng, C.Y.; Chen, K.C. An Efficient Reversible Data Hiding with Reduplicated Exploiting Modification Direction Using Image Interpolation and Edge Detection. Multimed. Tools Appl.
**2017**, 76, 9993–10016. [Google Scholar] [CrossRef] - Chang, C.C.; Chou, Y.C.; Kieu, T.D. An Information Hiding Scheme Using Sudoku. In Proceedings of the 2008 3rd International Conference on Innovative Computing Information and Control, Dalian, China, 18–20 June 2008; pp. 17–22. [Google Scholar]
- Chang, C.C.; Liu, Y.; Nguyen, T.S. A Novel Turtle Shell Based Scheme for Data Hiding. In Proceedings of the 2014 Tenth International Conference on Intelligent Information Hiding and Multimedia Signal Processing, Kitakyushu, Japan, 27–29 August 2014; pp. 89–93. [Google Scholar] [CrossRef]
- Kurup, S.; Rodrigues, A.; Bhise, A. Data Hiding Scheme Based on Octagon Shaped Shell. In Proceeding of the 2015 International Conference on Advances in Computing, Communications and Informatics (ICACCI-2015), Kochi, India, 10–13 August 2015; pp. 1982–1986. [Google Scholar] [CrossRef]
- Leng, H.S.; Tseng, H.W. Maximizing the Payload of the Octagon-Shaped Shell-based Data Hiding Scheme. In Proceedings of the 2017 IEEE 8th International Conference on Awareness Science and Technology (iCAST), Taichung, Taiwan, 8–10 November 2017; pp. 45–49. [Google Scholar]
- Leng, H.S. Data Hiding Scheme Based on Regular Octagon-Shaped Shells. In Advances in Intelligent Information Hiding and Multimedia Signal Processing; Smart Innovation, Systems and Technologies; Springer: Cham, Switzerland, 2018; pp. 29–35. [Google Scholar]
- Lee, C.F.; Wang, Y.X. Secure Image Hiding Scheme Based on Magic Signet. J. Electron. Sci. Technol. (JEST)
**2020**, 18, 93–101. [Google Scholar] [CrossRef] - Zhang, M.; Zhang, S.; Harn, L. An Efficient and Adaptive Data-Hiding Scheme Based on secure random matrix. PLoS ONE
**2019**, 14, e0222892. [Google Scholar] [CrossRef] [PubMed] - Chang, C.C.; Liu, Y. Fast Turtle Shell-Based Data Embedding Mechanisms with Good Visual Quality. J. Real-Time Image Process.
**2019**, 16, 589–599. [Google Scholar] [CrossRef] - Nguyen, T.T.; Pan, J.S.; Ngo, T.G.; Dao, T.K. A Data Hiding Approach Based on Reference-Affected Matrix. In Advances in Intelligent Information Hiding and Multimedia Signal Processing; Smart Innovation, Systems and Technologies; Springer: Singapore, 2020; Volume 156, pp. 53–64. [Google Scholar]
- USC-SIPI Image Database. Available online: http://sipi.usc.edu/database/ (accessed on 28 July 2020).
- Xie, X.Z.; Lin, C.C.; Chang, C.C. Data Hiding Based on a Two-layer Turtle Shell Matrix. Symmetry
**2018**, 10, 47. [Google Scholar] [CrossRef] [Green Version] - Xia, B.B.; Wang, A.H.; Chang, C.C.; Liu, L. An Image Steganography Scheme Using 3D-Sudoku. J. Inf. Hiding Multimed. Signal Process.
**2016**, 7, 836–845. [Google Scholar] - True Color Kodak Images. Available online: http://r0k.us/graphics/kodak/ (accessed on 28 July 2020).
- Shen, J.J.; Lee, C.F.; Li, Y.H.; Agrawal, S. Image Steganographic Scheme Based on Doublelayer Magic Matrix. In Proceedings of the 2019 IEEE 10th International Conference on Awareness Science and Technology (iCAST), Morioka, Japan, 23–25 October 2019; pp. 1–6. [Google Scholar] [CrossRef]

**Figure 4.**Double-layer octagon-shaped shell reference matrix. (

**a**) The top-layer octagon type; (

**b**) Type attributes corresponds to matrix elements.

**Figure 8.**Six grayscale test images; they should be listed as (

**a**) Lena, (

**b**) Baboon, (

**c**) Peppers, (

**d**) Boat, (

**e**) Airplane, and (

**f**) House.

Images | Lena | Baboon | Peppers | Boat | Airplane | House | Average | |
---|---|---|---|---|---|---|---|---|

Magic signet [14] | PSNR | 44.14 | 44.14 | 44.14 | 44.02 | 44.17 | 46.13 | 44.12 |

EC(bpp) | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Secure random matrix [15] | PSNR | 46.38 | 46.37 | 46.37 | 46.38 | NA | 46.37 | 46.37 |

EC(bpp) | 2 | 2 | 2 | 2 | NA | 2 | 2 | |

Fast turtle shell-based matrix (Scheme 2) [16] | PSNR | 46.85 | 46.85 | 46.84 | NA | NA | NA | 46.84 |

EC(bpp) | 2 | 2 | 2 | NA | NA | NA | 2 | |

3D-sudoku [20] | PSNR | 41.31 | 41.25 | 41.30 | 41.23 | 41.28 | 41.26 | 41.27 |

EC(bpp) | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Regular-octagon shape [12] | PSNR | 43.00 | 42.98 | 42.99 | 43.01 | 42.98 | 43.01 | 43.01 |

EC(bpp) | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | |

Two-layer turtle-shape [19] | PSNR | 47.13 | 47.09 | 47.13 | 47.12 | 47.11 | 47.12 | 47.13 |

EC(bpp) | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | |

x-cross-shaped reference-affected matrix [17] | PSNR | 40.37 | 38.14 | 40.35 | NA | NA | NA | 39.62 |

EC(bpp) | 2.6134 | 2.6402 | 2.6211 | NA | NA | NA | 2.6249 | |

Maximizing the payload-octagon [13] | PSNR | 30.59 | 30.59 | 30.62 | NA | 30.68 | NA | 30.62 |

EC(bpp) | 3.5 | 3.5 | 3.5 | NA | 3.5 | NA | 3.5 | |

Proposed scheme | PSNR | 36.93 | 36.91 | 36.91 | 36.92 | 36.89 | 36.92 | 36.91 |

EC(bpp) | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 |

PSNR (PAYLOAD) | PSNR $(50\times {10}^{4})$ | PSNR $(60\times {10}^{4})$ | PSNR $(70\times {10}^{4})$ | PSNR $(80\times {10}^{4})$ | PSNR $(90\times {10}^{4})$ | |
---|---|---|---|---|---|---|

Images | ||||||

Lena | 39.5476 | 38.7697 | 38.0882 | 37.5272 | 36.9899 | |

Baboon | 39.5653 | 38.7780 | 38.0975 | 37.5150 | 37.0135 | |

Peppers | 39.5849 | 38.7789 | 38.0879 | 37.5354 | 37.0206 | |

Boat | 39.5528 | 38.7536 | 38.0963 | 37.5182 | 36.9911 | |

Airplane | 39.5442 | 38.7618 | 38.1041 | 37.5161 | 37.0127 | |

House | 39.5430 | 38.7628 | 38.0847 | 37.5231 | 37.0145 | |

Average | 39.56 | 38.77 | 38.09 | 37.52 | 37.01 |

Image | SSIM | EC (bpp) | PSNR (dB) |
---|---|---|---|

Lena | 0.9085 | 3.5 | 36.919709 |

Baboon | 0.9685 | 3.5 | 36.914033 |

Peppers | 0.9111 | 3.5 | 36.927763 |

Boat | 0.9334 | 3.5 | 36.897141 |

Airplane | 0.8986 | 3.5 | 36.919945 |

House | 0.9224 | 3.5 | 36.914813 |

**Table 4.**Comparison of

**EC**,

**PSNR**values, and embedding time among Chang et al.’s turtle shell (TS) in a single-layer embedding, Leng’s octagon-shaped shells scheme in a single-layer embedding, Xie et al.’s two-layer turtle shell matrix embedding, Shen et al.’s double-layer square magic matrix scheme, and the proposed method in a two-layer embedding using octagon-shaped shells.

Chang et al.’s TS Scheme, Single-Layer Embedding [10] | Leng and Tseng’s Octagon-Shaped Shells Scheme, Single-Layer Embedding [12] | Xie et al.’s Two-Layer Turtle Shell Matrix Embedding [19] | Shen et al.’s Square Magic Matrix, Two-Layer Embedding [22] | The Proposed Method, Two-Layer Embedding | ||||||
---|---|---|---|---|---|---|---|---|---|---|

EC = 1.5 | EC = 2.5 | EC = 2.5 | EC = 2.5 | EC = 3.5 | ||||||

PSNR | Time | PSNR | Time | PSNR | Time | PSNR | Time | PSNR | Time | |

Lena | 49.42 | 0.65 | 47.13 | 0.85 | 43.00 | 1.45 | 42.70 | 0.69 | 36.92 | 1.85 |

Airplane | 49.40 | 0.63 | 47.11 | 0.85 | 42.98 | 1.45 | 42.69 | 0.68 | 36.92 | 1.85 |

Boat | 49.40 | 0.69 | 47.12 | 0.89 | 43.01 | 1.49 | 42.68 | 0.70 | 36.90 | 1.89 |

Peppers | 49.40 | 0.70 | 47.13 | 0.92 | 42.99 | 1.42 | 42.69 | 0.71 | 36.93 | 1.82 |

Baboon | 49.39 | 0.78 | 47.09 | 0.92 | 42.98 | 1.42 | 42.68 | 0.75 | 36.91 | 1.82 |

House | 49.40 | 0.69 | 47.12 | 0.87 | 43.01 | 1.47 | 42.68 | 0.70 | 36.91 | 1.87 |

Average | 49.40 | 0.69 | 47.12 | 0.88 | 42.99 | 1.45 | 42.69 | 0.69 | 36.91 | 1.82 |

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## Share and Cite

**MDPI and ACS Style**

Lee, C.-F.; Shen, J.-J.; Agrawal, S.; Li, Y.-H.
High-Capacity Embedding Method Based on Double-Layer Octagon-Shaped Shell Matrix. *Symmetry* **2021**, *13*, 583.
https://doi.org/10.3390/sym13040583

**AMA Style**

Lee C-F, Shen J-J, Agrawal S, Li Y-H.
High-Capacity Embedding Method Based on Double-Layer Octagon-Shaped Shell Matrix. *Symmetry*. 2021; 13(4):583.
https://doi.org/10.3390/sym13040583

**Chicago/Turabian Style**

Lee, Chin-Feng, Jau-Ji Shen, Somya Agrawal, and Yen-Hsi Li.
2021. "High-Capacity Embedding Method Based on Double-Layer Octagon-Shaped Shell Matrix" *Symmetry* 13, no. 4: 583.
https://doi.org/10.3390/sym13040583