# Rational Spline Image Upscaling with Constraint Parameters

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## Abstract

**:**

## 1. Introduction

## 2. A Bivariate Rational Interpolation

## 3. Basic Algorithms

#### 3.1. Image Non-Smooth Areas Detection

#### 3.2. Image Interpolation

#### 3.3. Parameters Optimization

## 4. Experiments

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Images used for quantitative comparison. (

**a**) Light-tower; (

**b**) Dollar; (

**c**) Cliff; (

**d**) Barbara.

**Table 1.**Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index (SSIM) results of the reconstructed high-resolution (HR) images by different methods.

NEDI | DFDF | SAI | Our Method | |||||
---|---|---|---|---|---|---|---|---|

PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | |

Light-tower | 22.78 | 0.7937 | 23.25 | 0.7971 | 22.86 | 0.8005 | 23.37 | 0.8009 |

Dollar | 19.10 | 0.8084 | 19.21 | 0.8066 | 19.24 | 0.8055 | 19.36 | 0.8118 |

Cliff | 25.08 | 0.7115 | 25.05 | 0.7184 | 25.16 | 0.7233 | 25.22 | 0.7268 |

Barbara | 22.35 | 0.8513 | 23.64 | 0.8766 | 23.54 | 0.8635 | 24.12 | 0.8801 |

Milkdrop | 30.97 | 0.9156 | 34.36 | 0.9196 | 32.39 | 0.9176 | 34.48 | 0.9216 |

Couple | 28.65 | 0.9391 | 29.06 | 0.9413 | 29.32 | 0.9443 | 29.14 | 0.9420 |

Goldhill | 26.60 | 0.7645 | 26.69 | 0.7678 | 26.92 | 0.7772 | 26.92 | 0.7750 |

Door | 33.12 | 0.9446 | 33.08 | 0.9447 | 31.16 | 0.9467 | 33.20 | 0.9478 |

Sky | 28.41 | 0.9154 | 28.95 | 0.8608 | 29.05 | 0.9364 | 28.96 | 0.9378 |

Boat | 25.82 | 0.8941 | 25.54 | 0.8378 | 25.43 | 0.9120 | 25.61 | 0.8973 |

Average | 25.22 | 0.8207 | 25.74 | 0.8225 | 25.58 | 0.8326 | 25.98 | 0.8384 |

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**MDPI and ACS Style**

Yao, X.; Zhang, Y.; Bao, F.; Zhang, C.
Rational Spline Image Upscaling with Constraint Parameters. *Math. Comput. Appl.* **2016**, *21*, 48.
https://doi.org/10.3390/mca21040048

**AMA Style**

Yao X, Zhang Y, Bao F, Zhang C.
Rational Spline Image Upscaling with Constraint Parameters. *Mathematical and Computational Applications*. 2016; 21(4):48.
https://doi.org/10.3390/mca21040048

**Chicago/Turabian Style**

Yao, Xunxiang, Yunfeng Zhang, Fangxun Bao, and Caiming Zhang.
2016. "Rational Spline Image Upscaling with Constraint Parameters" *Mathematical and Computational Applications* 21, no. 4: 48.
https://doi.org/10.3390/mca21040048