#
Image Zooming Based on Two Classes of C^{1}-Continuous Coons Patches Construction with Shape Parameters over Triangular Domain

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Rational Quadratic Trigonometric Hermite Functions with Shape Parameters

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

- ${T}_{i}\left(t\right)\ge 0,\left(i=0,1,2\right)$ and ${T}_{3}\left(t\right)\le 0$,
- Monotonicity: For fixed $t\in \left[0,1\right]$, ${T}_{0}\left(t\right)$, ${T}_{2}\left(t\right)$, and ${T}_{3}\left(t\right)$, are monotonically increasing for $\frac{\alpha}{\beta}$; ${T}_{1}\left(t\right)$ is monotonically decreasing for $\frac{\alpha}{\beta}$,
- End-point properties:$${T}_{0}(0)=1,{T}_{1}(0)=0,{T}_{2}(0)=0,{T}_{3}(0)=0,$$$${T}_{0}(1)=0,{T}_{1}(1)=1,{T}_{2}(1)=0,{T}_{3}(1)=0,$$$${T}_{0}{}^{\prime}(0)=0,{T}_{1}{}^{\prime}(0)=0,{T}_{2}{}^{\prime}(0)=1,{T}_{3}{}^{\prime}(0)=0,$$$${T}_{0}{}^{\prime}(1)=0,{T}_{1}{}^{\prime}(1)=0,{T}_{2}{}^{\prime}(1)=0,{T}_{3}{}^{\prime}(1)=1.$$

#### 2.2. Two Classes of ${C}^{1}$ Coons Patches Constructions over Triangular Domain

#### 2.2.1. Relationship between Barycentric Coordinates and Cartesian Coordinates

#### 2.2.2. Coons Patch Construction Based on Side–Side Method

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

#### 2.2.3. Coons Patch Construction Based on Side–Vertex Method

**Theorem**

**2.**

**Proof**

**of Theorem 2.**

#### 2.3. Region Control of Shape Parameters

#### 2.4. Image Zooming Based on Two Classes of Coons Patches Construction over Triangular Domain

#### Method of Image Zooming by Coons Patch Construction

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PSNR | Peak Signal to Noise Ratio |

SSIM | Structural Similarity |

FSIM | Feature Similarity |

MS-SSIM | Multiscale Structural Similarity |

SS | Side–side Method Based on the Rational Quadratic Trigonometric Hermite Functions |

SV | Side-vertex Method Based on the Rational Quadratic Trigonometric Hermite Functions |

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**Figure 1.**Coons patches constructed by SS and SV. (

**a**) SS with ${\alpha}_{1}={\beta}_{1}={\alpha}_{2}={\beta}_{2}={\alpha}_{3}={\beta}_{3}=1$. (

**b**) SS with ${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}=1,\phantom{\rule{3.33333pt}{0ex}}{\beta}_{1}=10,\phantom{\rule{3.33333pt}{0ex}}{\beta}_{2}=15,{\beta}_{3}=20$. (

**c**) SS with ${\alpha}_{1}=10,{\alpha}_{2}=15,{\alpha}_{3}=20,{\beta}_{1}={\beta}_{2}={\beta}_{3}=1$. (

**d**) SV with ${\alpha}_{1}={\beta}_{1}={\alpha}_{2}={\beta}_{2}={\alpha}_{3}={\beta}_{3}=1$. (

**e**) SV with ${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}=1,{\beta}_{1}=1.1,{\beta}_{2}=1.2,{\beta}_{3}=1.3$. (

**f**) S with ${\alpha}_{1}=1.1,{\alpha}_{2}=1.2,{\alpha}_{3}=1.3,{\beta}_{1}={\beta}_{2}={\beta}_{3}=1$.

**Figure 3.**Splice of Coons patches constructed by side–side (SS) and side–vertex (SV). (

**a**) SS with ${\alpha}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\alpha}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}1$. (

**b**) SS with ${\alpha}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\alpha}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}1,\phantom{\rule{4pt}{0ex}}{\beta}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}{\beta}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}15,\phantom{\rule{4pt}{0ex}}{\beta}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}20$. (

**c**) SS with ${\alpha}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}15,\phantom{\rule{4pt}{0ex}}{\alpha}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}20,\phantom{\rule{4pt}{0ex}}{\beta}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}1$. (

**d**) SV with ${\alpha}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\alpha}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}1$. (

**e**) SV with ${\alpha}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\alpha}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}1,\phantom{\rule{4pt}{0ex}}{\beta}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}{\beta}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}13,\phantom{\rule{4pt}{0ex}}{\beta}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}1.2$. (

**f**) SV with ${\alpha}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=13,\phantom{\rule{4pt}{0ex}}{\alpha}_{3}=1.2,\phantom{\rule{4pt}{0ex}}{\beta}_{1}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\beta}_{3}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}1$.

**Figure 4.**Image zooming and edge detection results on image ‘pepper’ by different algorithm (Factor 4). (

**a**) Original. (

**b**) Bilinear. (

**c**) Bicubic. (

**d**) Iterative curvature-based interpolation (ICBI). (

**e**) Novel edge orientation adaptive interpolation scheme for resolution enhancement of still images (NEDI). (

**f**) Super-resolution using iterative Wiener filter based on nonlocal means (SR-NLM). (

**g**) Rational ball cubic B-spline (RBC). (

**h**) SS with ${\alpha}_{1}=0.9,{\beta}_{1}=1,{\alpha}_{2}=2,{\beta}_{2}=3,{\alpha}_{3}=1,{\beta}_{3}=4$. (

**i**) SV with ${\alpha}_{1}=1,{\beta}_{1}=1.01,{\alpha}_{2}=1,{\beta}_{2}=1.06,{\alpha}_{3}=1,{\beta}_{3}=1.05$.

**Figure 5.**Image zooming and edge detection results on image ‘plane’ by different algorithm (Factor 4). (

**a**) Original. (

**b**) Bilinear. (

**c**) Bicubic. (

**d**) ICBI. (

**e**) NEDI. (

**f**) SR-NLM. (

**g**) RBC. (

**h**) SS with ${\alpha}_{1}=0.9,{\beta}_{1}=1,{\alpha}_{2}=2,{\beta}_{2}=3,{\alpha}_{3}=1,{\beta}_{3}=4$. (

**i**) SV with ${\alpha}_{1}=1,{\beta}_{1}=1.01,{\alpha}_{2}=1,{\beta}_{2}=1.06,{\alpha}_{3}=1,{\beta}_{3}=1.05$.

**Figure 6.**Sensitivity analysis of SS with $r=\frac{{\alpha}_{1}}{{\beta}_{1}}=\frac{{\alpha}_{2}}{{\beta}_{2}}=\frac{{\alpha}_{3}}{{\beta}_{3}}$ (Factor 4). (

**a**) Peak signal to noise ratio (PSNR). (

**b**) Structural similarity (SSIM). (

**c**) Feature similarity (FSIM). (

**d**) Multiscale structural similarity (MS-SSIM).

**Figure 7.**Sensitivity analysis of SV with $r=\frac{{\alpha}_{1}}{{\beta}_{1}}=\frac{{\alpha}_{2}}{{\beta}_{2}}=\frac{{\alpha}_{3}}{{\beta}_{3}}$ (Factor 4). (

**a**) PSNR. (

**b**) SSIM. (

**c**) FSIM. (

**d**) MS-SSIM.

Parameter | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\beta}}_{3}$ |
---|---|---|---|---|---|---|

$S{S}_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 |

$S{V}_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 |

$S{S}_{2}$ | 0.9 | 1 | 2 | 3 | 1 | 4 |

$S{V}_{2}$ | 1 | 1.01 | 1 | 1.06 | 1 | 1.05 |

**Table 2.**Comparison of different image zooming methods in terms of peak signal to noise ratio (PSNR).

Image | Bilinear | Bicubic | ICBI | NEDI | SR-NLM | RBC | ${\mathit{SS}}_{1}$ | ${\mathit{SS}}_{2}$ | ${\mathit{SV}}_{1}$ | ${\mathit{SV}}_{2}$ |
---|---|---|---|---|---|---|---|---|---|---|

Pepper | $23.06$ | $22.70$ | $26.82$ | $22.69$ | $22.79$ | $22.98$ | $26.75$ | $27.05$ | $26.24$ | $26.25$ |

Plane | $22.23$ | $21.88$ | $25.05$ | $24.88$ | $25.29$ | $25.79$ | $25.58$ | $25.76$ | $24.34$ | $24.35$ |

Flower | $25.55$ | $25.22$ | $29.84$ | $28.59$ | $28.97$ | $29.64$ | $29.59$ | $29.72$ | $28.45$ | $28.46$ |

Average | $23.62$ | $23.27$ | $27.23$ | $25.39$ | $25.68$ | $26.13$ | $27.30$ | $27.51$ | $26.34$ | $26.35$ |

**Table 3.**Comparison of different image zooming methods in terms of Structural similarity index (SSIM).

Image | Bilinear | Bicubic | ICBI | NEDI | SR-NLM | RBC | ${\mathit{SS}}_{1}$ | ${\mathit{SS}}_{2}$ | ${\mathit{SV}}_{1}$ | ${\mathit{SV}}_{2}$ |
---|---|---|---|---|---|---|---|---|---|---|

Pepper | $0.717$ | $0.705$ | $0.784$ | $0.669$ | $0.666$ | $0.699$ | $0.780$ | $0.784$ | $0.780$ | $0.781$ |

Plane | $0.734$ | $0.731$ | $0.812$ | $0.806$ | $0.810$ | $0.823$ | $0.821$ | $0.824$ | $0.837$ | $0.837$ |

Flower | $0.806$ | $0.801$ | $0.884$ | $0.789$ | $0.797$ | $0.812$ | $0.891$ | $0.892$ | $0.864$ | $0.864$ |

Average | $0.752$ | $0.746$ | $0.827$ | $0.803$ | $0.806$ | $0.829$ | $0.831$ | $0.833$ | $0.827$ | $0.827$ |

Image | Bilinear | Bicubic | ICBI | NEDI | SR-NLM | RBC | ${\mathit{SS}}_{1}$ | ${\mathit{SS}}_{2}$ | ${\mathit{SV}}_{1}$ | ${\mathit{SV}}_{2}$ |
---|---|---|---|---|---|---|---|---|---|---|

Pepper | $0.875$ | $0.876$ | $0.932$ | $0.958$ | $0.963$ | $0.970$ | $0.934$ | $0.935$ | $0.937$ | $0.937$ |

Plane | $0.734$ | $0.731$ | $0.906$ | $0.943$ | $0.946$ | $0.952$ | $0.910$ | $0.911$ | $0.914$ | $0.914$ |

Flower | $0.897$ | $0.897$ | $0.952$ | $0.936$ | $0.944$ | $0.954$ | $0.952$ | $0.953$ | $0.942$ | $0.940$ |

Average | $0.871$ | $0.871$ | $0.930$ | $0.956$ | $0.951$ | $0.959$ | $0.932$ | $0.933$ | $0.931$ | $0.931$ |

**Table 5.**Comparison of different image zooming methods in terms of multiscale structural similarity (MS-SSIM).

Image | Bilinear | Bicubic | ICBI | NEDI | SR-NLM | RBC | ${\mathit{SS}}_{1}$ | ${\mathit{SS}}_{2}$ | ${\mathit{SV}}_{1}$ | ${\mathit{SV}}_{2}$ |
---|---|---|---|---|---|---|---|---|---|---|

Pepper | $0.836$ | $0.836$ | $0.899$ | $0.869$ | $0.876$ | $0.881$ | $0.903$ | $0.905$ | $0.908$ | $0.908$ |

Plane | $0.813$ | $0.813$ | $0.892$ | $0.718$ | $0.930$ | $0.940$ | $0.894$ | $0.900$ | $0.904$ | $0.905$ |

Flower | $0.844$ | $0.844$ | $0.931$ | $0.915$ | $0.919$ | $0.926$ | $0.931$ | $0.932$ | $0.923$ | $0.924$ |

Average | $0.831$ | $0.831$ | $0.909$ | $0.834$ | $0.908$ | $0.916$ | $0.909$ | $0.912$ | $0.912$ | $0.912$ |

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## Share and Cite

**MDPI and ACS Style**

Tang, Y.; Zhu, Y.
Image Zooming Based on Two Classes of *C*^{1}-Continuous Coons Patches Construction with Shape Parameters over Triangular Domain. *Symmetry* **2020**, *12*, 661.
https://doi.org/10.3390/sym12040661

**AMA Style**

Tang Y, Zhu Y.
Image Zooming Based on Two Classes of *C*^{1}-Continuous Coons Patches Construction with Shape Parameters over Triangular Domain. *Symmetry*. 2020; 12(4):661.
https://doi.org/10.3390/sym12040661

**Chicago/Turabian Style**

Tang, Yunyi, and Yuanpeng Zhu.
2020. "Image Zooming Based on Two Classes of *C*^{1}-Continuous Coons Patches Construction with Shape Parameters over Triangular Domain" *Symmetry* 12, no. 4: 661.
https://doi.org/10.3390/sym12040661