# A Self-Recovery Fragile Image Watermarking with Variable Watermark Capacity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. SVD Transform and Its Characteristics

**U**and

**V**(also known as left and right singular matrices), and a diagonal matrix

**S**which is also called the singular value matrix. Taking a 4 × 4 matrix $\mathit{A}={({a}_{ij})}_{4\times 4}$ for example, the formula of SVD transform can be expressed as:

**U**and

**V**, respectively; ${\lambda}_{i}\text{}(i=1,2,3,4)$ is the singular values in matrix

**S**, and ${\lambda}_{1}\ge {\lambda}_{2}\ge {\lambda}_{3}\ge {\lambda}_{4}\ge 0$. From Equation (1), we can see that the matrix

**A**can be expressed as the sum of four sub-images ${\mathit{u}}_{i}{\mathit{v}}_{i}^{\mathrm{T}}\text{}(i=1,2,3,4)$ and the singular values ${\lambda}_{i}\text{}(i=1,2,3,4)$ are their weight coefficients. Besides, the larger the singular values are, the greater the proportion of the sub-image on the matrix decomposition will be. To further analyze the roles of singular values and sub-images in matrix decomposition, we calculate the energy of the matrix by using the square of F-norm, as shown in Equation (2):

**A**. From Equation (2), we can draw a conclusion that the energy information of a matrix is concentrated on the singular values ${\lambda}_{i}\text{}(i=1,2,3,4)$, while the geometry and texture information are concentrated on its sub-images ${\mathit{u}}_{i}{\mathit{v}}_{i}^{\mathrm{T}}\text{}(i=1,2,3,4)$. To show the contributions of these sub-images for image texture information, Figure 1 gives the residual images formed by these sub-images, which takes image Lena as an example. From Figure 1, it can be observed that the residual image formed by the first sub-image ${\mathit{u}}_{1}{\mathit{v}}_{1}^{\mathrm{T}}$ contains the most texture information of image Lena, and only a little information is remained in other residual images. For other test images, the same conclusion can be obtained.

- Step 1: The host image is firstly divided into 2 × 2 image blocks, and then the SVD transform is conducted on each block.
- Step 2: For each block after SVD, the product of the left and right singular vectors (the first sub-image ${\mathit{u}}_{1}{\mathit{v}}_{1}^{\mathrm{T}}$) is calculated. For smooth blocks, the values in ${\mathit{u}}_{1}{\mathit{v}}_{1}^{\mathrm{T}}$ are approximately equal to 1/2 (0.5). Based on this point, two thresholds around 0.5, ${T}_{1}$ and ${T}_{2}$ are identified to label the pixels in ${\mathit{u}}_{1}{\mathit{v}}_{1}^{\mathrm{T}}$. For a pixel in ${\mathit{u}}_{1}{\mathit{v}}_{1}^{\mathrm{T}}$, if the pixel value falls in the range of ${T}_{1}$ to ${T}_{2}$, the pixel will be judged as a smooth pixel, else it is judged as a texture pixel.
- Step 3: Calculating the number of the smooth pixels in 2 × 2 block, if the number is larger than 3, the corresponding image block will be determined as a smooth block, else the block is determined as a texture block.

## 3. The Proposed Scheme

#### 3.1. Watermark Generation and Embedding

#### 3.1.1. Authentication Watermark

**B**, and 2-bit authentication watermark $\mathit{P}=({p}_{1},{p}_{2})$ is generated for each block, which can be expressed as:

#### 3.1.2. Recovery Watermark

- Step 1: To generate appropriate length recovery watermark, the image block $\mathit{C}={({c}_{ij})}_{2\times 2}$ after preprocessing is further adjusted by Equation (5), and then we get the processed image block ${\mathit{C}}^{\prime}={({c}_{ij}^{\prime})}_{2\times 2}$.$${c}_{ij}^{\prime}=\lfloor \frac{{c}_{ij}}{2}-8\rfloor ,\text{}i,j=1,2,$$
- Step 2: For each block ${\mathit{C}}^{\prime}$, the DCT transform is applied, and then we get the DCT coefficient matrix $\mathit{D}={({d}_{ij})}_{2\times 2}$:$$\mathit{D}=\left[\begin{array}{cc}{d}_{11}& {d}_{12}\\ {d}_{21}& {d}_{22}\end{array}\right],$$
- Step 3: To generate the recovery watermark, the DC coefficient ${d}_{11}$ is rounded and coded into 5 bits watermark including 1 bit sign flag and 4 bits coefficient encoding result, and the AC coefficient ${d}_{12}$ is coded into 4 bits watermark including 1 bit sign flag and 3 bits coefficient encoding result. It should be noted that if the coefficients are out of the coding range, they need to be modestly adjusted. For example, if the absolute value of ${d}_{11}$ is larger than 15, then it should be adjusted and make it equal to 15.

#### 3.1.3. Watermark Embedding

#### 3.2. Three-Level Tamper Detection

- (1)
- In the first level detection (① in Figure 3), the encrypted authentication watermark is firstly extracted from the image block, which can be expressed as ${\mathit{P}}_{\mathrm{e}}^{\prime}=({p}_{\mathrm{e}1}^{\prime},{p}_{\mathrm{e}2}^{\prime})$. With the secret key $key1$, a corresponding decryption process is conducted to the extracted watermark, and then we get the decrypted authentication watermark ${\mathit{P}}^{\mathbf{\prime}}=({p}_{1}^{\prime},{p}_{2}^{\prime})$. According to the generation process of authentication watermark in Section 3.1.1, a new authentication watermark for this block is regenerated, which is expressed as ${\mathit{P}}^{\mathbf{\u2033}}=({p}_{1}^{\u2033},{p}_{2}^{\u2033})$. If ${\mathit{P}}^{\mathbf{\prime}}\ne {\mathit{P}}^{\mathbf{\u2033}}$, then the detected block is marked as a tampered block, else it is marked as an authentic block.
- (2)
- In the second level detection (② in Figure 3), the recovery watermark of the block is firstly extracted from its mapping block generated by Equation (9), which is expressed as ${\mathit{R}}_{\mathrm{e}}^{\prime}$. Then, with the secret key $key2$, a decryption process is performed on ${\mathit{R}}_{\mathrm{e}}^{\prime}$, and the decrypted recovery watermark ${\mathit{R}}^{\mathbf{\prime}}$ is obtained. If the block is a smooth block, the length of ${\mathit{R}}^{\mathbf{\prime}}$ is 6 bits, else the length of ${\mathit{R}}^{\mathbf{\prime}}$ is equal to 10 bits. Accordingly, a new recovery watermark ${\mathit{R}}^{\mathbf{\u2033}}$ for the current block is regenerated, which has the same process as Section 3.1.2. At last, a comparison process is applied for the extracted recovery watermark ${\mathit{R}}^{\mathbf{\prime}}$ and the newly created watermark ${\mathit{R}}^{\mathbf{\u2033}}$. If ${\mathit{R}}^{\mathbf{\prime}}\ne {\mathit{R}}^{\mathbf{\u2033}}$, then the detected block is marked as a tampered block, else it is marked as an authentic block.
- (3)
- After the first two level detections, we get the preliminary tamper detection result. However, due to the fact that the tamper detection process is based on the image blocks, there might be a probability of misjudgment. In other words, an authentic image block might be falsely detected as a tampered block, and a truly tampered block might be detected as an authentic block. To further improve the tamper detection accuracy, the third level detection is applied (③ in Figure 3). This process is completed by using the block-neighborhood tampering characterization [27]. For a suspicious block
**A**shown in Figure 4, n is used to represent the number of the blocks that are detected as tampered in its block-neighborhood (the blocks in gray in Figure 4). If the central image block**A**is a valid block while the number of the tampered blocks in its block-neighborhood is more than 7 ($n\ge 7$), then the block**A**will also be determined as an invalid block. If the central image block**A**is a tampered block while the number n is less than 2 ($n<2$), then the image block**A**will be determined as a valid block.

#### 3.3. Image Recovery

- (1)
- For tampered smooth block, the recovery watermark is the average pixel value of original block. To reconstruct the block, the recovery watermark is first extracted from its mapping block. After decryption and the binary-to-decimal conversion, we get the final recovery data.
- (2)
- For invalid texture block, the recovery watermark is the quantized DCT coefficients. According to the inverse process of the watermark generation process (Steps 1–3) given in Section 3.1.2, the recovery data for texture block can be obtained.

## 4. Experimental Results and Comparison Analysis

#### 4.1. Imperceptibility Analysis

- (1)
- For the blocks whose 2 LSBs are embedded by 8 bits watermark information, the expectation value of MSE is ${1.5}^{2}$, then the PSNR’s expectation can be computed by:$$\mathrm{PSNR}=10{\mathrm{log}}_{10}(\frac{{255}^{2}}{{1.5}^{2}})=44.61\text{}(\mathrm{dB}),$$
- (2)
- For the blocks whose 3 LSBs are embedded by 12 bits watermark information, the expectation value of MSE is ${3.5}^{2}$, then the PSNR’s expectation is calculated by:$$\mathrm{PSNR}=10{\mathrm{log}}_{10}(\frac{{255}^{2}}{{3.5}^{2}})=37.25\text{}(\mathrm{dB}),$$

#### 4.2. Performance of Tamper Detection and Self-Recovery

#### 4.2.1. Text Addition Attack

#### 4.2.2. Copy-Move Attack

#### 4.2.3. Collage Attack

#### 4.2.4. Image Deletion Attack

#### 4.2.5. Content-Only Attack

#### 4.2.6. Large Area Tampering

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The residual images formed by sub-images: (

**a**) Original Lena image; (

**b**) residual image formed by ${\mathit{u}}_{1}{\mathit{v}}_{1}^{\mathrm{T}}$; (

**c**) residual image formed by ${\mathit{u}}_{2}{\mathit{v}}_{2}^{\mathrm{T}}$; (

**d**) residual image formed by ${\mathit{u}}_{3}{\mathit{v}}_{3}^{\mathrm{T}}$; (

**e**) residual image formed by ${\mathit{u}}_{4}{\mathit{v}}_{4}^{\mathrm{T}}$.

**Figure 5.**Test images: (

**a**) Car1; (

**b**) Car2; (

**c**) Clock; (

**d**) Airplane; (

**e**) Cameraman; (

**f**) Lena; (

**g**) Barbara; (

**h**) Venice; (

**i**) Boat; (

**j**) Goldhill.

**Figure 6.**Tamper detection and recovery performance for text addition attack: (

**a**) Original Lena image; (

**b**) watermarked Lena image; (

**c**) attacked image; (

**d**) tamper detection result using the method in [13]; (

**e**) tamper detection result using the method in [25]; (

**f**) tamper detection result of the proposed method; (

**g**) image recovery result using the method in [13]; (

**h**) image recovery result using the method in [25]; (

**i**) image recovery result of the proposed method.

**Figure 7.**Tamper detection and recovery performance for copy-move attack: (

**a**) Original Goldhill image; (

**b**) watermarked Goldhill image; (

**c**) attacked image; (

**d**) tamper detection result using the method in [13]; (

**e**) tamper detection result using the method in [25]; (

**f**) tamper detection result of the proposed method; (

**g**) image recovery result using the method in [13]; (

**h**) image recovery result using the method in [25]; (

**i**) image recovery result of the proposed method.

**Figure 8.**Tamper detection and recovery performance for collage attack: (

**a**) Watermarked Car1 image; (

**b**) watermarked Car2 image; (

**c**) attacked image; (

**d**) tamper detection result using the method in [13]; (

**e**) tamper detection result using the method in [25]; (

**f**) tamper detection result of the proposed method; (

**g**) image recovery result using the method in [13]; (

**h**) image recovery result using the method in [25]; (

**i**) image recovery result of the proposed method.

**Figure 9.**Tamper detection and recovery performance for image deletion attack: (

**a**) Original Venice image; (

**b**) watermarked Venice image; (

**c**) attacked image; (

**d**) tamper detection result using the method in [13]; (

**e**) tamper detection result using the method in [25]; (

**f**) tamper detection result of the proposed method; (

**g**) image recovery result using the method in [13]; (

**h**) image recovery result using the method in [25]; (

**i**) image recovery result of the proposed method.

**Figure 10.**Partial enlarged details of the restored images: (

**a**) Original ship image; (

**b**) restored ship image using the method in [13]; (

**c**) restored ship image using the method in [25]; (

**d**) restored ship image of the proposed method. To highlight the differences among these restored images, some regions in restored images are selected and circled in red.

**Figure 11.**Tamper detection and recovery performance for content-only attack: (

**a**) Original Clock image; (

**b**) watermarked Clock image; (

**c**) attacked Clock image; (

**d**) tamper detection result using the method in [13]; (

**e**) tamper detection result using the method in [25]; (

**f**) tamper detection result of the proposed method; (

**g**) image recovery result using the method in [13]; (

**h**) image recovery result using the method in [25]; (

**i**) image recovery result of the proposed method.

**Figure 12.**Partial enlarged details of the restored images: (

**a**) Original image; (

**b**) restored image using the method in [25]; (

**c**) restored image of the proposed method. To highlight the differences between the restored images, some regions in restored images are selected and circled in red.

**Figure 13.**Large area tampering test: (

**a**) Left 50%; (

**b**) tamper detection result; (

**c**) recovered image (PSNR = 32.05 dB); (

**d**) middle 80%; (

**e**) tamper detection result; (

**f**) recovered image (PSNR = 19.69 dB).

Parameters | Value Ranges | Functions |
---|---|---|

$M\times M$ | $256\times 256$ | Image size of the test images used in this paper |

$N$ | $(M\times M)/(2\times 2)$ | Total image blocks in host image |

${T}_{1}$, ${T}_{2}$ | ${T}_{1}=0.48$, ${T}_{2}=0.52$ | Thresholds used for block classification in this paper |

$k$ | A prime number & $k\in [1,N-1]$ | Secret key used to generate the block mapping sequence |

$key1$, $key2$ | Non-negative integers | Secret keys used to generate the pseudo-random sequences |

Test Images | Smooth Blocks | Texture Blocks | Watermark Capacity (bpp) | PSNR (dB) |
---|---|---|---|---|

Car1 | 10,516 | 5868 | 2.36 | 40.41 |

Car2 | 13,572 | 2812 | 2.17 | 42.20 |

Clock | 12,366 | 4018 | 2.25 | 41.22 |

Airplane | 10,946 | 5438 | 2.33 | 40.56 |

Cameraman | 10,535 | 5849 | 2.36 | 40.65 |

Lena | 8561 | 7823 | 2.48 | 39.82 |

Barbara | 7321 | 9063 | 2.55 | 39.52 |

Venice | 6062 | 10,322 | 2.63 | 39.15 |

Boat | 6029 | 10,355 | 2.63 | 38.87 |

Goldhill | 5038 | 11,346 | 2.69 | 38.91 |

Algorithm | FPR | FNR | PSNR (dB) of Recovered Image |
---|---|---|---|

Tong et al. [13] | 0.41% | 5.41% | 35.01 |

Chen et al. [25] | 0.66% | 0.93% | 40.85 |

The proposed method | 0.67% | 0 | 45.52 |

Algorithm | FPR | FNR | PSNR (dB) of Recovered Image |
---|---|---|---|

Tong et al. [13] | 0.09% | 26.13% | 25.90 |

Chen et al. [25] | 0.16% | 3.25% | 35.05 |

The proposed method | 0.25% | 0.44% | 36.13 |

Algorithm | FPR | FNR | PSNR (dB) of Recovered Image |
---|---|---|---|

Tong et al. [13] | 0.14% | 97.33% | 24.76 |

Chen et al. [25] | 0.27% | 0.14% | 47.17 |

The proposed method | 0.28% | 0 | 47.79 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, C.; Zhang, H.; Zhou, X.
A Self-Recovery Fragile Image Watermarking with Variable Watermark Capacity. *Appl. Sci.* **2018**, *8*, 548.
https://doi.org/10.3390/app8040548

**AMA Style**

Wang C, Zhang H, Zhou X.
A Self-Recovery Fragile Image Watermarking with Variable Watermark Capacity. *Applied Sciences*. 2018; 8(4):548.
https://doi.org/10.3390/app8040548

**Chicago/Turabian Style**

Wang, Chengyou, Heng Zhang, and Xiao Zhou.
2018. "A Self-Recovery Fragile Image Watermarking with Variable Watermark Capacity" *Applied Sciences* 8, no. 4: 548.
https://doi.org/10.3390/app8040548