# An Improved Integer Transform Combining with an Irregular Block Partition

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Weng et al.’s Method

#### 2.2. Alattar’s Method

## 3. The Proposed Scheme

#### 3.1. Irregular Block Partition

#### 3.2. Performance Analysis

#### 3.3. Block Selection

#### 3.4. Two-Layer Embedding

## 4. Embedding and Extraction Procedures

#### 4.1. Embedding Procedure

**(1)****One-layer embedding****Data bits embedding**if $\mathbf{x}\in {O}_{s1}$$\mathbf{y}$=$\mathbf{x}$;elseif $\mathbf{x}\in {E}_{s}$endFor the blocks without adjacent $r+s+1$ pixels, they are ignored in the embedding procedure to ensure reversibility. We employ $W{M}_{<p{T}_{h}}$ to describe the number of data bits embedded into the host image, which is equivalent to the number of difference values belonging to $\left[-p{T}_{h},p{T}_{h}\right)$.**Overhead information embedding**The overhead information is obtained according to the description above. Suppose ${P}_{C}$ denotes the required payload, and it is partitioned into two parts which correspond to the first and second embedding layers, respectively. ${P}_{L}$ stands for the to-be-embedded payload of the current layer, while $W{M}_{<p{T}_{h}}-{L}_{\sum}$ represents the maximal embedding capacity. Firstly, ${P}_{L}$ is embedded into the blocks in ${E}_{s}$ according to the step of data bits embedding. Secondly, for the first ${L}_{\sum}$ modified pixels, we collect their LSBs (least significant binary) and append them to the payload ${P}_{L}$. In this way, the locations of their LSBs are vacant so that they can be occupied by the overhead information. Finally, the rest of the payload ${P}_{L}$ along with ${L}_{\sum}$ LSBs are embedded into the remaining blocks in ${E}_{s}$ according to the step of data bits embedding.

**(2)****Watermarked image obtaining**- If ${P}_{C}\le W{M}_{<p{T}_{h}}-{L}_{\sum}$The payload ${P}_{C}$ can be satisfied by one-layer embedding. Therefore, a watermarked image ${I}_{w}$ is created after (1) is performed.elseif ${P}_{C}>W{M}_{<p{T}_{h}}-{L}_{\sum}$Two-layer embedding is adopted to achieve required payload ${P}_{C}$. The remaining payload is defined as ${P}_{L}={P}_{C}-(W{M}_{<p{T}_{h}}-{L}_{\sum})$. Suppose ${P}_{C}={P}_{L}$, then we repeat (1) for the second-layer embedding.end

#### 4.2. Extraction Procedure

**Step 1: Overhead information extraction**

**Step 2: Data extraction and original image recovery**

## 5. Experimental Results

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Taking a $4\times 4$-sized block for example, a two-dimensional image block is arranged into a one-dimensional pixel list according to the arrow direction.

**Figure 3.**An $r\times s$-sized block is marked in black, and its neighborhood is composed of $r+s+1$ pixels marked in green, where $n=r\times s$.

**Figure 5.**Comparison of capacity-distortion performance among six reversible data hiding (RDH) schemes on six images: Airplane, Baboon, Barbara, Goldhill, Sailboat and Lena.

**Table 1.**Performance comparison between one-layer embedding and two-layer embedding on Lena and Barbara.

One-Layer Embedding | Two-Layer Embedding | |||
---|---|---|---|---|

Image | Lena | Barbara | Lena | Barbara |

$v{T}_{h1}$ | 8 | 8 | 5 | 5 |

$p{T}_{h1}$ | 7 | 9 | 6 | 6 |

$v{T}_{h2}$ | 0 | 0 | 10 | 8 |

$p{T}_{h2}$ | 0 | 0 | 1 | 2 |

Payload (proposed, in bpp) | 0.5 | 0.4 | 0.5 | 0.4 |

PSNR (proposed, in dB) | 41.20 | 41.89 | 41.65 | 42.88 |

**Table 2.**Size of the compressed location map ${L}_{S}$ under different payload size for Lena and Baboon.

Lena | Baboon | ||
---|---|---|---|

Payload (in bpp) | ${\mathit{L}}_{\mathit{S}}$ (in bits) | Payload (in bpp) | ${\mathit{L}}_{\mathit{S}}$ (in bits) |

0.1 | 40 | 0.1 | 40 |

0.2 | 40 | 0.2 | 40 |

0.3 | 40 | 0.3 | 40 |

0.4 | 40 | 0.4 | 40 |

0.5 | 40 | 0.5 | 64 |

0.6 | 40 | 0.6 | 80 |

0.7 | 40 | 0.7 | 136 |

0.8 | 40 | - | - |

0.9 | 40 | - | - |

1.0 | 40 | - | - |

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**MDPI and ACS Style**

Weng, S.; Chen, Y.; Hong, W.; Pan, J.-S.; Chang, C.-C.; Liu, Y.
An Improved Integer Transform Combining with an Irregular Block Partition. *Symmetry* **2019**, *11*, 49.
https://doi.org/10.3390/sym11010049

**AMA Style**

Weng S, Chen Y, Hong W, Pan J-S, Chang C-C, Liu Y.
An Improved Integer Transform Combining with an Irregular Block Partition. *Symmetry*. 2019; 11(1):49.
https://doi.org/10.3390/sym11010049

**Chicago/Turabian Style**

Weng, Shaowei, Yi Chen, Wien Hong, Jeng-Shyang Pan, Chin-Chen Chang, and Yijun Liu.
2019. "An Improved Integer Transform Combining with an Irregular Block Partition" *Symmetry* 11, no. 1: 49.
https://doi.org/10.3390/sym11010049