# Double-Matrix Decomposition Image Steganography Scheme Based on Wavelet Transform with Multi-Region Coverage

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Image Wavelet Transform

#### 2.2. Arnold Transformation

_{n+}

_{1}, y

_{n+}

_{1}represent the pixel position after transformation; a, b are parameters and both are positive integers; n represents the number of current transformations; N is the length or width of the image, and mod is Modular operation.

#### 2.3. Hessenberg Matrix Decomposition

_{ij})

_{n×n}$\in $R

^{n×n}satisfy h

_{ij}= 0(j > i + 1), then H is called the upper Hessenberg matrix, and its specific form is:

_{ij})$\in $R

^{M×N}, then there are orthogonal matrices Q

_{1}, Q

_{2}, …, Q

_{n}

_{−2}, so that X is transformed into an upper Hessenberg matrix through orthogonal similar transformation, that is Q

_{n−}

_{2}, …, Q

_{2,}Q

_{1}X Q

_{1}, Q

_{2}, …, Q

_{n}

_{−2}= H. From the above, we can acquire the calculation formula of Hessenberg matrix decomposition as follows:

_{i,j}of H is 0.

_{n}is the identity matrix of n×n.

#### 2.4. Singular-Value Decomposition

#### 2.5. Logistic Chaotic System

_{0}(0 < 𝑥

_{0}< 1). When 3.5699 < 𝜇 < 4, the system goes into chaos.

## 3. A Steganography Scheme of Dual-Matrix Decomposition Image with Multi-Region Coverage

#### 3.1. Hidden Region Selection in Wavelet Domain

#### 3.2. Steganographic Embedding Algorithm for Secret Image

Algorithm 1: The embedding algorithm |

Input: Cover image C, Secret image S, embedding factor α. |

Output: Steganographic image C*, Encrypted key matrix U, _{w}_{1′}V._{w}_{1}‘ |

1: [LL2, HL2, LH2, HH2, HL, LH, HH] = DWT2level(C) |

2: RS = [HH2, HL(1:128,129:256);LH(129:256,1:128), HH(1:128,1:128)] 3: RS’ = Arnold (RS) 4: P H P ^{T} = Hessenberg (RS’)5: HU _{w} ⋅ HS_{w} $\cdot $ HV_{w} = SVD (H)6: S’ = Arnold (S) 7: U _{w}_{1} S_{w}_{1} V_{w}_{1}^{T} = SVD (S’)$8:\text{}{{HS}_{w}}_{1}*={HS}_{w}+\alpha $S _{w}_{1} 9: H* = HU _{w} HS_{w}_{1}* HV_{w}10: RSS’ = P H*P ^{T}11: RSS = Rearnold(RSS’)//The hidden area after embedding secret information is shown in the Figure 5. In Figure 5, HL*, LH*, HH* and HH2* are HL, LH, HH, and HH2 embedded with secret information respectively. 12: HH2* = RSS (1:128,1:128) 13: HL* = HL, LH* = LH, HH* = HH 14: HL*(1:128,129:256) = RSS (129:256,1:128) 15: LH*(129:256,1:128) = RSS (1:128,129:256) 16: HH*(1:128,1:128) = RSS (129:256,129:256) 17: C* = IDWT2level [LL2, HL2, LH2, HH2*, HL*, LH*, HH*] 18: (U _{w}_{1′}, V_{w}_{1}‘) = ChaoticEncrypt ( U_{w}_{1}, V_{w}) |

_{w}

_{1}and V

_{w}

_{1}are the keys when extracting secret images. To ensure the security of the keys, the encryption algorithm of logistics chaos system is used to encrypt them to obtain U

_{w}

_{1′}and V

_{w}

_{1}‘.

#### 3.3. Steganographic Extraction Algorithm of Secret Image

_{w}

_{1}, V

_{w}

_{1}, and the output is the extracted secret image—extracted image. The overall process of the extraction algorithm is shown in Figure 6, and the specific extracting algorithm is Algorithm 2, as follows:

Algorithm 2: The extracting algorithm |

Input: Steganographic image C*, Encrypted key matrix U._{w}_{1′,}V_{w}_{1}′ |

Output: The extracted Secret image C’. |

1: [LL_{w}_{2}, HL_{w}_{2}, LH_{w}_{2}, HH_{w}_{2}, HL_{w}, LH_{w}, HH_{w}] = DWT2level (C*)2: RS _{w} = [HH_{w}_{2}, HL_{w}(1:128,129:256); LH_{w}(129:256,1:128), HH_{w}(1:128,1:128)]2: RS _{w}* = Arnold (RS_{w})3: (U _{w1,} V_{w1}) = ChaoticDecrypt (U_{w}_{1′}, V_{w}_{1′})4: P _{w}H_{w}P_{w}^{T} = Hessenberg (RS_{w}*)5: HU _{w}*HSb_{w}*HV_{w}*^{T} = SVD (H_{w})6: S _{w1}* = (HSb_{w1}*-HS_{w})/α7: C1′ = U _{w}_{1} S_{w}_{1}* V_{w}_{1}^{T}8: C’ = Rearnold (C1′) |

#### 3.4. Key Encryption Algorithm of Logistic Chaotic Map

## 4. Simulation Results and Analysis

#### 4.1. Evaluation Indicators

_{max}is the maximum pixel value in the cover image.

_{1}and d

_{2}are two variables which are uesd to stabilize the division with a weak denominator.

#### 4.2. Parameter Setting

#### 4.3. Concealment Analysis

#### 4.4. Robustness Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Second-order discrete wavelet transform. (

**a**) Original image of Lena; (

**b**) Multi-wavelet transform Lena; (

**c**) Schematic diagram of component position.

**Figure 9.**(

**a**) Steganographic image and Value of PSNR, SSIM (

**b**) Extracted secret image and Value of NC.

**Figure 11.**Steganographic images. (

**a**) Steganographic image Baboon; (

**b**) Steganographic image Airplane; (

**c**) Steganographic image Man; (

**d**) Steganographic image Peppers; (

**e**) Steganographic image Tiffany; (

**f**) Steganographic image Barbara.

Gaussian Noise | Salt and Pepper Noise | Speckle Noise | Compression Attack | Histogram Equalization | Motion Blur | Sharpen Attack |
---|---|---|---|---|---|---|

0.001 | 0.001 | 0.001 | JEPG | — | — | Sharpen 0.2 |

JEPG2000 |

Steganographic Image | Embedding Factor $\mathit{\alpha}$ | PSNR | SSIM | NC |
---|---|---|---|---|

Lena | 0.01 | 49.5106 dB | 0.99389 | 0.99955 |

Baboon | 0.01 | 49.488 dB | 0.99792 | 0.99966 |

Airplane | 0.02 | 49.1286 dB | 0.99494 | 0.99897 |

Man | 0.01 | 57.448 dB | 0.99938 | 0.99376 |

Peppers | 0.01 | 49.525 dB | 0.99792 | 0.99944 |

Tiffany | 0.02 | 49.3406 dB | 0.99317 | 0.99961 |

Barbara | 0.01 | 49.668 dB | 0.99637 | 0.99938 |

Scheme | Cover Image | PSNR | SSIM |
---|---|---|---|

Proposed Scheme | Lena | 49.5106 dB | 0.99389 |

Baboon | 49.488 dB | 0.99792 | |

Peppers | 49.525 dB | 0.99787 | |

Traditional DWT-SVD | Lena | 34.0904 dB | 0.99566 |

Baboon | 34.1014 dB | 0.99879 | |

Pepper | 34.1188 dB | 0.99738 |

Algorithm | Size of Secret | PSNR | SSIM | NC |
---|---|---|---|---|

R. Thabit et al. [44]. | 49,152 bits | 43.29 dB | — | — |

Sajasi et al. [45] | 256 × 256 | 47.78 dB | — | — |

Kanan et al. [46] | 256 × 256 | 45.12 dB | — | — |

Gulave et al. [47] | 78.7 KB | 39.84 dB | 0.953 | — |

Subhedar et al. [48] | 256 × 256 | 49.0369 dB | 0.9963 | — |

Proposed algorithm | 256 × 256 70 KB | 49.5106 dB | 0.99389 | 0.99955 |

Attack Type | Baboon | Pepper | Lena | |||
---|---|---|---|---|---|---|

NC | NC | NC | ||||

Two-Level Matrix Decomposition | Single Singular-Value Decomposition | Two-Level Matrix Decomposition | Single Singular-Value Decomposition | Two-Level Matrix Decomposition | Single Singular-Value Decomposition | |

Gaussian Nosie | 0.98968 | 0.921 | 0.97797 | 0.97565 | 0.97996 | 0.95773 |

Salt and pepper noise | 0.98997 | 0.96221 | 0.98466 | 0.9469 | 0.98663 | 0.93113 |

Speckle noise | 0.98315 | 0.96367 | 0.96114 | 0.94123 | 0.96599 | 0.94122 |

JPEG compression | 0.95727 | 0.94321 | 0.94917 | 0.92172 | 0.94866 | 0.89297 |

JPEG2000 compression | 0.88304 | 0.86265 | 0.98098 | 0.96023 | 0.9815 | 0.88772 |

Sharpening attack | 0.91732 | 0.89983 | 0.9732 | 0.92113 | 0.96797 | 0.93211 |

Histogram equalization | 0.85805 | 0.83122 | 0.9384 | 0.91111 | 0.88757 | 0.86877 |

Motion blur | 0.51954 | 0.48999 | 0.50069 | 0.4902 | 0.561 | 0.49222 |

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**MDPI and ACS Style**

Pan, P.; Wu, Z.; Yang, C.; Zhao, B.
Double-Matrix Decomposition Image Steganography Scheme Based on Wavelet Transform with Multi-Region Coverage. *Entropy* **2022**, *24*, 246.
https://doi.org/10.3390/e24020246

**AMA Style**

Pan P, Wu Z, Yang C, Zhao B.
Double-Matrix Decomposition Image Steganography Scheme Based on Wavelet Transform with Multi-Region Coverage. *Entropy*. 2022; 24(2):246.
https://doi.org/10.3390/e24020246

**Chicago/Turabian Style**

Pan, Ping, Zeming Wu, Chen Yang, and Bing Zhao.
2022. "Double-Matrix Decomposition Image Steganography Scheme Based on Wavelet Transform with Multi-Region Coverage" *Entropy* 24, no. 2: 246.
https://doi.org/10.3390/e24020246