# Nonlocal Means Two Dimensional Histogram-Based Image Segmentation via Minimizing Relative Entropy

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## Abstract

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## 1. Introduction

## 2. Non-Local Mean Two Dimensional Histogram (NLMTDH)

#### 2.1. Non-Local Mean Filter

#### 2.2. Construction of NLMTDH

## 3. Image Thresholding Based on NLMTDH Using Relative Entropy

#### 3.1. Relative Entropy

#### 3.2. Threshold Selection Based on NLMTDH Using Relative Entropy

## 4. Experimental Results and Discussion

#### 4.1. Results

#### 4.2. Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of Equation (10)

## References

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**Figure 3.**Thresholding results of test image using different methods. From left to right, the results are obtained by Otsu, Kapur, MCE,2DMCE and the proposed method.

Methods | Advantages | Disadvantages |
---|---|---|

superpixel [5,6] | reduce redundant information; less complexity | cannot locate the edges accurately |

watershed [7,8] | simple and intuition | usually result in over segmention |

active contour models [9,10] | rigorous mathematical base; | sensitive noise; high computation complexity |

clustering [11,12] | intensive value is enough; simple | the number of cluster cannot be determined automatically; spatial information is ignored; |

deep learning [13,14] | high segmentation accuracy | large computation burden |

thresholding [15,16] | simple, easy to be implemented | ignore spatial information |

Image | The Proposed | 2DMCE | MCE | OTSU | KAPUR | |
---|---|---|---|---|---|---|

ant | threshold | 52 52 | 44 45 | 69 | 84 | 183 |

ME | 0.0344 | 0.0379 | 0.0481 | 0.0829 | 0.8852 | |

bacteria | threshold | 65 65 | 44 45 | 98 | 99 | 70 |

ME | 0.0101 | 0.0605 | 0.4266 | 0.4398 | 0.0221 | |

block | threshold | 26 26 | 36 36 | 38 | 120 | 88 |

ME | 0.0183 | 0.0621 | 0.0679 | 0.2861 | 0.2613 | |

geometric | threshold | 36 36 | 36 36 | 41 | 70 | 126 |

ME | 0.0324 | 0.0347 | 0.0381 | 0.0986 | 0.2323 | |

junk | threshold | 187 186 | 209 206 | 129 | 134 | 158 |

ME | 0.0072 | 0.0090 | 0.0633 | 0.0492 | 0.0166 | |

mask | threshold | 23 24 | 28 30 | 30 | 57 | 116 |

ME | 0.0016 | 0.0129 | 0.0136 | 0.1134 | 0.2860 | |

casting13 | threshold | 144 144 | 134 127 | 74 | 80 | 114 |

ME | 0.0115 | 0.0128 | 0.1046 | 0.0795 | 0.0170 | |

casting18 | threshold | 154 153 | 158 154 | 92 | 138 | 114 |

ME | 0.0050 | 0.0063 | 0.2200 | 0.0074 | 0.0611 |

Image | The Propsed | 2DMCE | MCE | Otsu | Kapur |
---|---|---|---|---|---|

ant | 170.3123 | 21.1812 | 0.0317 | 0.0032 | 0.0067 |

bacteria | 163.7921 | 23.9648 | 0.0097 | 0.0031 | 0.0081 |

block | 94.4108 | 21.3898 | 0.0079 | 0.0027 | 0.0077 |

casting13 | 56.5179 | 9.2832 | 0.0066 | 0.0023 | 0.0043 |

casting14 | 56.6691 | 10.2639 | 0.0068 | 0.0027 | 0.0051 |

geometric | 62.0219 | 18.1187 | 0.0073 | 0.0020 | 0.0059 |

junk | 116.4295 | 14.0563 | 0.0061 | 0.0019 | 0.0045 |

mask | 99.1112 | 22.3298 | 0.0088 | 0.0023 | 0.0057 |

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**MDPI and ACS Style**

Jiang, C.; Yang, W.; Guo, Y.; Wu, F.; Tang, Y.
Nonlocal Means Two Dimensional Histogram-Based Image Segmentation via Minimizing Relative Entropy. *Entropy* **2018**, *20*, 827.
https://doi.org/10.3390/e20110827

**AMA Style**

Jiang C, Yang W, Guo Y, Wu F, Tang Y.
Nonlocal Means Two Dimensional Histogram-Based Image Segmentation via Minimizing Relative Entropy. *Entropy*. 2018; 20(11):827.
https://doi.org/10.3390/e20110827

**Chicago/Turabian Style**

Jiang, Chundi, Wei Yang, Yu Guo, Fei Wu, and Yinggan Tang.
2018. "Nonlocal Means Two Dimensional Histogram-Based Image Segmentation via Minimizing Relative Entropy" *Entropy* 20, no. 11: 827.
https://doi.org/10.3390/e20110827