# SVD-Based Image Watermarking Using the Fast Walsh-Hadamard Transform, Key Mapping, and Coefficient Ordering for Ownership Protection

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## Abstract

**:**

## 1. Introduction

- A blind image watermarking method is proposed that is highly robust and secured against numerous attacks while providing good quality watermarked images;
- To safeguard the unauthorized detection, the Gaussian mapping is used to scramble the watermark;
- To facilitate authentic and errorless extraction of the watermark image by generating the keys from the singular values the FWHT blocks of the cover image;
- It provides a good trade-off among robustness, security, and imperceptibility.

## 2. Background Information

#### 2.1. Singular Value Decomposition

#### 2.2. Fast Walsh-Hadamard Transform

## 3. Proposed Method

#### 3.1. Watermark Preprocessing

**Step 1.**- The watermark image W is reshaped into a one-dimensional sequence Q = {q(r), $1\le r\le N\times N$}.
**Step 2.**- Initially, a reference pattern $P=\left\{p\left(r\right),1\le r\le N\times N\right\}$ is generated using a Gaussian map, which is defined in Equation (4).$$p\left(r\right)=\mathrm{exp}\left\{-\mathrm{a}\times {\left(p(r+1)\right)}^{2}\right\}+\mathrm{b}$$
**Step 3.**- Then, the binary reference pattern $Z=\left\{z\left(r\right),1\le r\le N\times N\right\}$ is calculated using the following equation:$$z(r)=\{\begin{array}{ll}1& ifp(r)T\\ 0& otherwise\end{array}$$
**Step 4.**- Finally, the watermark sequence q(r) is scrambled with z(r) using Equation (6):$$u(r)=z(r)\oplus q(r),1\le r\le N\times N$$

#### 3.2. Watermark Embedding Process

**Step 1.**- The original host image $X$ is first divided into three channels ${X}_{red},{X}_{green},\mathrm{and}{X}_{blue}$, where ${X}_{red},{X}_{green},\mathrm{and}{X}_{blue}$represent the red, green, and blue channels of the original image, respectively. Then, the mean of the pixel values of each channel is calculated using Equation (7).$$\mu ({X}_{red})={\displaystyle \sum _{i=1}^{M}{\displaystyle \sum _{j=1}^{M}\frac{{X}_{red}}{255}}}{,}_{}^{}\mu ({X}_{green})={\displaystyle \sum _{i=1}^{M}{\displaystyle \sum _{j=1}^{M}\frac{{X}_{green}}{255}}}{,}_{}^{}\mu ({X}_{blue})={\displaystyle \sum _{i=1}^{M}{\displaystyle \sum _{j=1}^{M}\frac{{X}_{blue}}{255}}}$$
**Step 2.**- The selected channel ${X}_{min}$ is further divided into m × m non-overlapping blocks, $H=\{{H}_{i};1\le i\le $ n}, where $i$ is the block number and m is the length of the row and column of each block.
**Step 3.**- FWHT is applied in each block ${H}_{i}$ to obtain the transformed block ${R}_{i}$, where R
_{i}contains the FWHT coefficients. **Step 4.**- Among all the$n$blocks, each set of four consecutive blocks R
_{i}, R_{i+}_{1}, R_{i}_{+2}, and R_{i+}_{3}is selected to embed a watermark bit. The main idea of the embedding process is to sort the coefficients of the first row represented by $C\left({R}_{i:i+3}^{}\right),\mathrm{where}\left\{i:i+3\right\}\mathrm{indicates}\left\{i,i+1,i+2,i+3\right\}$ of each set of selected blocks R_{i}, R_{i}_{+1}, R_{i+}_{2}, and R_{i+3}except the DC value. If the watermark bit is 1, the selected low-frequency coefficients $C\left({R}_{i:i+3}^{}\right)$ are sorted in descending order; otherwise, they are sorted in ascending order. The concept of embedding the watermark bit in ascending and descending order with a block size of 4 × 4 where, m = 4, is shown in Figure 2.

**:**In this section, the process of mapping keys k1 and k2 is explained, which is defined in Equation (8). This step is introduced to strengthen the proposed algorithm under severe attack. To map the keys initially, SVD is applied in each block ${H}_{i}$to generate the necessary information. To perform the operation, the following steps are used:

- (1)
- Each block ${H}_{i}$ of the selected channel is decomposed into three matrices:${U}_{i},{D}_{i},\mathrm{and}{V}_{i}^{}$using Equation (9).$${H}_{i}={U}_{i}{D}_{i}{V}_{i}^{T}$$
_{i}of each block ${H}_{i}$. These singular values are unique for each block H_{i}. The keys k1 and k2 are calculated using these singular values. Thus, unauthorized people could not map the keys without the host image to prove fake ownership. To do this, initially, a null key $k1$ is defined. Then,$k1$ is generated using these singular values as defined in Equation (10) below:$$k1=append\left(k1,\left(asc\left({\lambda}_{ij}\right)\right)\right),u\left(r\right)=0\phantom{\rule{0ex}{0ex}}k1=append\left(k1,\left(desc\left({\lambda}_{ij}\right)\right)\right),u\left(r\right)=1$$ - (2)
- Finally, k1 is converted into a one-dimensional sequence of length L = $n\times m$, where n is the total number of blocks and m is the total number of singular values in each block.
- (3)
- To generate key k2, define a null key $k2$ with length S, where S = n/m. Then, $k2$is generated from key $k1$using the following Equation (11):$$k2=append\left(k2,\mu \left(k{1}_{h:h+t}\right)\right)where1\le h\le L$$

**Step 5.**- Inverse FWHT is applied to each transformed block${R}_{i}^{\prime}$ and the watermarked blocks ${H}_{i}^{\prime}$ are found.
**Step 6.**- Finally, three watermarked channels${X}_{red}^{\prime},{X}_{green}^{\prime},\mathrm{and}{X}_{blue}^{\prime}$ are combined to generate the watermarked image ${X}^{\prime}$.

Algorithm 1: Watermark Insertion |

Variable Declaration: X: Host image $\mu $: Mean intensity value of each channel of host image (Lena) ${X}_{min}$: Channel with minimum mean ${H}_{i}$: Non-overlapping blocks of ${X}_{min}$ (size 4 × 4) FWHT, SVD: Transformation and decomposition used in the algorithm ${R}_{i}$: FWHT transformed block of ${H}_{i}$ $C\left({R}_{i:i+3}\right)$: Three coefficients of first row (except DC value) of the consecutive transformed block $C\left({{R}^{\prime}}_{i:i+3}\right)$: Coefficients in ascending or descending order W: Watermark image u: Scrambled watermark sequence Watermark Embedding Procedure: 1. Watermark preprocess: scramble W to obtain u using Gaussian mapping 2. Read the host image and calculate$\mu $ of each channel (Red, Green, Blue) X.bmp (host image with size of 256 × 256) W.bmp (watermark image with size of 32 × 32) 3. Select channel ${X}_{min}$ and divide it into 4 × 4 ${H}_{i}$ blocks 4. Apply FWHT to each block ${H}_{i}$ and found ${R}_{i}$ 5. Watermark Insertion
$$C\left({R}_{i:i+3}^{\prime}\right)=asc\left(C\left({R}_{i:i+3}^{}\right)\right);mappingkeyk1andk2,whenu\left(r\right)=0$$
$$C\left({R}_{i:i+3}^{\prime}\right)=desc\left(C\left({R}_{i:i+3}^{}\right)\right);mappingkeyk1andk2,whenu\left(r\right)=1$$
// Use SVD to map keys $k1\mathrm{and}k2$ 6. Perform inverse FWHT and combine the channels to get the Watermarked Image |

#### 3.3. Watermark Detection Process

**Step 1.**- The attacked watermarked image ${X}^{\ast}$ is first divided into three channels, {${X}_{red}^{\ast},{X}_{green}^{\ast},and{X}_{blue}^{\ast}$}. Then, the mean value of the pixels of the red, green, and blue channels represented by $\mu ({X}_{red}^{\ast}),\mu ({X}_{green}^{\ast}),\mathrm{and}\mu ({X}_{blue}^{\ast})$ are calculated. Thereafter that, the channel with minimum mean ${X}_{min}^{\ast}$is selected for extracting the watermark.
**Step 2.**- The selected channel ${X}_{min}^{\ast}$ is further divided into m × m non-overlapping blocks${H}_{i}^{\ast}$, where $i$ is the block number.
**Step 3.**- FWHT is carried out on each block${H}_{i}^{\ast}$. After applying this operation, the transformed blocks${R}_{i}^{\ast}$ are found.
**Step 4.**- The degree of ascendant/descendant denoted by dof is calculated for four consecutive transformed blocks $\{{R}_{i}^{\ast},{R}_{i+1}^{\ast},{R}_{i+2}^{\ast},{R}_{i+3}^{\ast}$}. Therefore, dof(asc) represents the number of times that the low-frequency coefficients in the first row${C}^{\ast}\left({R}_{i:i+3}^{\ast}\right)$of each transformed block except for the DC value are in ascending order. Similarly, the dof(desc) represents the number of times that the low-frequency coefficients in the first row${C}^{\ast}\left({R}_{i:i+3}^{\ast}\right)$ of each transformed block except the DC value are in descending order.

**Authenticate**$\mathit{k}1$

**with**$\mathit{k}2$

**:**This operation is carried out to authenticate key $k1$ using k2. For this purpose, the average of the consecutive t values of $k1$ is calculated and compared with one value of $k2$. This operation is represented using Equations (16) and (17) given below:

**Step 5.**- The hidden binary sequence is found using the following rule:
**If**$do{f}_{h}\left(asc\right)do{f}_{h}\left(desc\right)andk1\leftarrow k2$

then$u\left(r\right)=0$**else If**$do{f}_{h}\left(asc\right)>do{f}_{h}\left(desc\right)andk1\leftarrow k2$

then$u\left(r\right)=1$ **Step 6.**- The binary watermark sequence q*(r) is extracted with key k3 using the following equation:$${q}^{\ast}(r)=z(r)\oplus u(r),1\le r\le N\times N.$$

Algorithm 2: Watermark Extraction |

Variable Declaration: ${X}^{\ast}$: Attacked watermarked image $\mu $: Mean intensity value of each channel of ${X}^{\ast}$ ${X}_{min}^{\ast}$: Channel with minimum mean ${H}_{i}^{\ast}$: Non-overlapping blocks of ${X}_{min}^{\ast}$ (size 4 × 4) FWHT: Transformations used in the algorithm ${R}_{i}^{\ast}$: FWHT transformed block of ${H}_{i}^{\ast}$ ${C}^{\ast}\left({R}_{i:i+3}^{\ast}\right)$: Three coefficients of first row (except DC value) of four consecutive transformed block W: Watermark image u: Scrambled watermark sequence dof(asc/desc): The number of times low-frequency coefficients in the first row ${C}^{\ast}\left({R}_{i:i+3}^{\ast}\right)$ of each transformed block except the DC value are in ascending/descending order. Watermark Extraction Procedure: 1. Read ${X}^{\ast}$ and calculate $\mu $ of each channel (Red, Green, Blue) 2. Select channel ${X}_{min}^{\ast}$ and divide into 4 × 4 ${H}_{i}^{\ast}$ blocks 3. Apply FWHT to each block ${H}_{i}^{\ast}$ and found ${R}_{i}^{\ast}$ 4. Watermark extraction (a) Modifying dof(asc/desc) into dof′(asc/desc) with key $k1$
$$do{f}^{\prime}\left(asc\right)=do{f}^{\prime}\left(asc\right)+1;ifk{1}_{h}k{1}_{h+1}$$
$$do{f}^{\prime}\left(desc\right)=do{f}^{\prime}\left(desc\right)+1;fk{1}_{h}k{1}_{h+1}$$
$$do{f}_{h}\left(asc\right)=dof\left(asc\right)+do{f}^{\prime}\left(asc\right)$$
$$do{f}_{h}\left(desc\right)=dof\left(desc\right)+do{f}^{\prime}\left(desc\right)$$
$$if\left(\mu \left(k{1}_{h:h+t}\right)\right)=k{2}_{h};k1\leftarrow k2$$
$$if\left(\mu \left(k{1}_{h:h+t}\right)\right)!=k{2}_{h};!k1\leftarrow k2$$
// Consecutive t values of the key are considered each time for extracting a one-bit watermark, where$\frac{L}{t}={N}^{2}$ and $\mu \left(k{1}_{h:h+t}\right)$ means mean of these t values (c) Watermark extraction If $do{f}_{h}\left(asc\right)>do{f}_{h}\left(desc\right)andk1\leftarrow k2$ then$u\left(r\right)=0$ else If $do{f}_{h}\left(asc\right)>do{f}_{h}\left(desc\right)andk1\leftarrow k2$ then$u\left(r\right)=1$ where, $1\le r\le 32\times 32$ (d) Re-scramble u to get W |

## 4. Experimental Results and Discussions

_{i}is 4 × 4. Therefore, the total number blocks is 4096. Thus, the length of the key k1 is 16384. The main reason for selecting a smaller value for m to embed the watermark bit is that sorting larger blocks causes greater degradation in the quality of the watermarked image.

**Imperceptibility test:**The imperceptibility of the watermarked images can be evaluated in terms of the peak signal-to-noise ratio (PSNR), as given in Equation (19).

**Security analysis:**For a secured watermarking method, how it performs against various attacks is very important. The proposed method utilizes a Gaussian map to enhance the security. To encrypt the watermark image, some predefined constants are used such as a, b, and $p\left(1\right)$, which are considered as secret key k3. If the selected value for a, b, and $p\left(1\right)$ are wrong, in that case, the watermark will not be extracted properly. Further, in order to make the watermarking method more secured, the two keys k1 and k2 are used. The key k1 is generated from the singular values of each block ${H}_{i}$ of the selected channel of host image. Moreover, it is observed that these singular values are floating point numbers, and it is not possible to find out these singular values without the host image. Therefore, it is not possible to generate key k1 without the host image. The key k2 is generated from key k1, which is used to authenticate the key k1 in the watermark extraction process. Therefore, it is not possible to generate the key k2 without k1. These keys (k1, k2, k3) are used in the watermark detection process to extract the embedded watermark. The correct watermark can be extracted when all the keys (k1, k2, and k3) are correct. In other words, if any one of the keys is wrong, then the watermark will not be extracted correctly. This phenomenon is illustrated in Figure 6. Moreover, the size of each block H

_{i}of the selected channel of the host image is 4 × 4; therefore, the total number of blocks in each host image is 4096. Thus, the length of the key k1 is 4096 × 4 = 16,384, and the length of the key k2 is (4096/4) + 1 = 1025, which are quite long, indicating that the key space is large enough. As the key k1, k2, and k3 are floating point numbers, therefore, the value of these keys cannot be determined. Hence, the probability of extracting the right watermark is near to 0. Therefore, the attacker cannot detect the correct watermark without the right key, which enhances the security of the proposed watermarking method.

**Robustness test:**To measure the robustness of the proposed algorithm, the normalized correlation (NC) is calculated between the original watermark image and the extracted watermark image. The NC value is calculated using Equation (20):

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**) Sorting in ascending order to embed 0 bits and (

**b**) sorting in descending order to embed 1 bit.

**Figure 5.**Watermark images: (

**a**) original, (

**b**) scrambled with a = 10, b = 0.05, and y0 = 20, and (

**c**) scrambled with a = 30, b = 0.01, and y0 = 10.

**Figure 7.**Analysis of proposed method under No attack, Gaussian noise (0.01), Speckle noise, and Salt and Pepper noise (0.01). NC: normalized correlation.

**Figure 8.**Analysis of proposed method under Adjustment, Cropping (50%), Sharpening (0.1), and Weiner filtering.

**Figure 9.**Analysis of the proposed method under Poison noise, Median filtering, Compression (quality factor: 50%), and Rotation.

**Table 1.**Comparison between the proposed and recent methods in terms of peak signal-to-noise ratio (PSNR).

Watermarked Images | Proposed Method | Ahmed et al. [23] | Patvardhar et al. [24] | Su et al. [13] |
---|---|---|---|---|

50.04 | 54.2577 | 39.4428 | ||

49.78 | 47.1961 | 40.8216 | ||

51.56 | 47.1836 | 54.3499 | ||

52.64 |

No | Attack Type | Lena | Peppers | Baboon | Fruit |
---|---|---|---|---|---|

1 | Gaussian (0.01) | 1.0 | 1.0 | 1.0 | 1.0 |

2 | Speckle (0.01) | 1.0 | 1.0 | 1.0 | 1.0 |

3 | Adjustment | 1.0 | 1.0 | 1.0 | 1.0 |

4 | Cropping (50%) | 1.0 | 1.0 | 1.0 | 1.0 |

5 | Sharpening (tol = 0.1) | 1.0 | 1.0 | 1.0 | 1.0 |

6 | Rotation (40^{0}) | 1.0 | 1.0 | 1.0 | 1.0 |

7 | Wiener filtering | 1.0 | 1.0 | 1.0 | 1.0 |

8 | Poison noise | 1.0 | 1.0 | 1.0 | 1.0 |

9 | Salt and pepper noise (0.01) | 1.0 | 1.0 | 1.0 | 1.0 |

10 | Median filtering | 1.0 | 1.0 | 1.0 | 1.0 |

11 | Compression (quality factor = 50%) | 1.0 | 1.0 | 1.0 | 1.0 |

No | Attack Type | Lena | Peppers | Baboon | Fruit |
---|---|---|---|---|---|

1 | Gaussian (0.1) | 0.9997 | 0.9823 | 1.0 | 0.9351 |

2 | Speckle (0.01) | 0.8835 | 0.9292 | 0.9068 | 0.9349 |

3 | Adjustment | 0.9543 | 0.7544 | 0.9014 | 0.6137 |

4 | Cropping (50%) | 0.7919 | 0.7821 | 0.7912 | 0.7866 |

5 | Sharpening (tol = 0.1) | 0.9578 | 0.9335 | 0.9241 | 0.8594 |

6 | Rotation (40^{0}) | 0.5160 | 0.5132 | 0.5194 | 0.5193 |

7 | Wiener filtering | 0.6753 | 0.6785 | 0.6884 | 0.6771 |

8 | Poison noise | 0.9950 | 0.9963 | 0.9992 | 0.9990 |

9 | Salt and pepper noise (0.01) | 0.9945 | 0.9931 | 0.9956 | 0.9944 |

10 | Median filtering | 0.9762 | 0.9541 | 0.9896 | 0.9459 |

11 | Compression (quality factor = 50%) | 0.5775 | 0.5936 | 0.5912 | 0.5676 |

No | Attack Type | Ahmed et al. [23] | Patvardhar et al. [24] | Su et al. [13] | Proposed |
---|---|---|---|---|---|

1 | Gaussian noise (0.1) | 0.9625 | 0.9885 | 0.9131 | 1.0 |

2 | Speckle noise (0.01) | 0.9601 | -- | -- | 1.0 |

3 | Contrast Adjustment | -- | 0.9491 | -- | 1.0 |

4 | Cropping (50%) | -- | 0.9947 | 0.9604 | 1.0 |

5 | Sharpening | 0.9388 | -- | 0.9999 | 1.0 |

6 | Rotation (25°) | 0.7991 | 0.9989 | -- | 1.0 |

7 | Poison noise | 0.9884 | -- | -- | 1.0 |

8 | Salt and pepper noise (0.01) | 0.9117 | 0.9807 | 0.9902 | 1.0 |

9 | Median filtering | 0.9908 | 0.9989 | 0.8814 | 1.0 |

10 | JPEG compression (quality factor = 20%) | 0.9784 | 0.9895 | 0.8469 | 1.0 |

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**MDPI and ACS Style**

Khanam, T.; Dhar, P.K.; Kowsar, S.; Kim, J.-M.
SVD-Based Image Watermarking Using the Fast Walsh-Hadamard Transform, Key Mapping, and Coefficient Ordering for Ownership Protection. *Symmetry* **2020**, *12*, 52.
https://doi.org/10.3390/sym12010052

**AMA Style**

Khanam T, Dhar PK, Kowsar S, Kim J-M.
SVD-Based Image Watermarking Using the Fast Walsh-Hadamard Transform, Key Mapping, and Coefficient Ordering for Ownership Protection. *Symmetry*. 2020; 12(1):52.
https://doi.org/10.3390/sym12010052

**Chicago/Turabian Style**

Khanam, Tahmina, Pranab Kumar Dhar, Saki Kowsar, and Jong-Myon Kim.
2020. "SVD-Based Image Watermarking Using the Fast Walsh-Hadamard Transform, Key Mapping, and Coefficient Ordering for Ownership Protection" *Symmetry* 12, no. 1: 52.
https://doi.org/10.3390/sym12010052