# The Tradeoff Analysis for Remote Sensing Image Fusion Using Expanded Spectral Angle Mapper

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Tradeoff Analysis Based on RMSE

_{1}, u

_{2},⋯, u

_{n}} and {v

_{1}, v

_{2},⋯, v

_{n}} denote the spectral vectors of U and V, respectively. The greater the RMSE

_{UV}, the higher the difference between U and V will be. Before the analysis, the LRMI is first resampled to the same spatial resolution of the HRPI and HRMI. Let T, P, and F denote the LRMI, HRPI, and HRMI. Then based on RMSE, let us consider the following minimization problem:

_{TF}and RMSE

_{FP}are used to estimate the spectral and spatial quality of the HRMI, respectively. The subsequent result is:

_{i},p

_{i}, f

_{i}represent the pixel values of the LRMI, the HRPI and the HRMI. The solution for (2) is that the HRMI is equal to the (LRMI+HRPI)/2.

_{u}and μ

_{v}are the mean values of U and V.

^{n}}, V={v+1, v−1, ⋯, v+(−l)

^{n}}, and u≠v, the value of the CC is 1, but the value of the ESAM is less than 1. Therefore, the ESAM is more informative than the SAM and CC in terms of measuring how close the pixel values of the two images are.

## 3. Experiments

^{-1/2}(1/16, 1/4, 3/8, 1/4, 1/16), together with a decomposition level of two, is employed to abstract the high frequency information of the HRPI. Fused HRMIs using different algorithms are shown in Figures l(c)-(f).

#### 3.1. Visual inspection

#### 3.2. Quantitative analysis

_{TF}, followed by the OWD and MAIM methods, the AW method has the lowest AE

_{TF}. On the other hand, the IHS method has the lowest AE

_{FP}, followed by the MAIM and AW methods, the OWD method has the highest AE

_{FP}. The higher the AE, the lower the similarity of two images is. Therefore, in terms of transferring details, the performances of the IHS, MAIM, AW, and OWD methods decrease; in terms of preserving spectral property, the order is the AW, MAIM, OWD, and IHS methods.

_{TFP}is the sum of the RMSE

_{TF}and the RMSE

_{FP}; the $\mathit{\text{RMS}}{E}_{\mathit{\text{TP}}}/\sqrt{2}$ is the minimum RMSE value for any fused HRMI. From the fifth and sixth columns, it can be found that all the RMSE

_{TFP}values of the HRMIs exceed the corresponding $\mathit{\text{RMS}}{E}_{\mathit{\text{TP}}}/\sqrt{2}$ values. Based on the RMSE

_{TF}, the grade of the extent that the HRMI is close to the LRMI is the AW, MAIM, OWD, and IHS methods. Based on the RMSE

_{FP}, the order of the extent that the HRMI is close to the HRPI is the IHS, AW, MAIM, and OWD methods. This indicates that the tradeoff property exists in terms of the RMSE index.

## 4. Conclusions

## Supplementary Material

sensors-08-00520-s001.pdf## Acknowledgments

## References

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**Figure 1.**Fusion results of different methods. (a) the LRMIs as a colour composite; (b) The HRPI; (c) The HRMIs produced by the IHS method; (d) The HRMIs produced by the OWD method; (e) The HRMIs produced by the AW method; (f) The HRMIs produced by the MAIM method.

IHS | OWD | AW | MAIM | |
---|---|---|---|---|

AE_{TF}16×16 | 22.71° | 13.34° | 9.59° | 13.28° |

AE_{TF}32×32 | 21.74° | 13.28° | 8.79° | 12.33° |

AE_{TF}64×64 | 20.83° | 13.29° | 8.19° | 11.57° |

AE_{TF}128×128 | 20.09° | 13.41° | 7.85° | 10.97° |

AE_{FP} 16×16 | 4.99° | 20.14° | 17.84° | 16.23° |

AE_{FP}32×32 | 4.57° | 19.00° | 16.87° | 15.63° |

AE_{FP}64×64 | 4.21° | 17.93° | 15.92° | 15.00° |

AE_{FP} 128×128 | 3.87° | 16.95° | 15.01° | 14.29° |

RMSE_{TF} | RMSE_{FP} | RMSE_{TFP} | ${\text{RMSE}}_{TP}/\sqrt{2}$ | ||
---|---|---|---|---|---|

IHS | B_{1} | 49.82 | 13.52 | 63.34 | 34.77 |

B_{2} | 49.44 | 12.24 | 61.69 | 30.47 | |

B_{3} | 48.56 | 17.38 | 65.94 | 28.28 | |

B_{4} | 34.31 | 33.83 | 68.13 | 28.61 | |

OWD | B_{1} | 32.27 | 47.09 | 79.36 | 34.77 |

B_{2} | 31.65 | 41.32 | 72.97 | 30.47 | |

B_{3} | 31.74 | 38.33 | 70.07 | 28.28 | |

B_{4} | 21.62 | 36.78 | 58.40 | 28.61 | |

AW | B_{1} | 17.13 | 44.94 | 62.08 | 34.77 |

B_{2} | 17.43 | 37.98 | 55.40 | 30.47 | |

B_{3} | 17.17 | 36.69 | 53.86 | 28.28 | |

B_{4} | 14.86 | 31.24 | 46.10 | 28.61 | |

MAIM | B_{1} | 19.93 | 45.87 | 65.79 | 34.77 |

B_{2} | 19.25 | 39.96 | 59.21 | 30.47 | |

B_{3} | 18.51 | 37.52 | 56.03 | 28.28 | |

B_{4} | 22.99 | 35.69 | 58.69 | 28.61 |

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**MDPI and ACS Style**

Chen, S.; Su, H.; Zhang, R.; Tian, J.; Yang, L.
The Tradeoff Analysis for Remote Sensing Image Fusion Using Expanded Spectral Angle Mapper. *Sensors* **2008**, *8*, 520-528.
https://doi.org/10.3390/s8010520

**AMA Style**

Chen S, Su H, Zhang R, Tian J, Yang L.
The Tradeoff Analysis for Remote Sensing Image Fusion Using Expanded Spectral Angle Mapper. *Sensors*. 2008; 8(1):520-528.
https://doi.org/10.3390/s8010520

**Chicago/Turabian Style**

Chen, Shaohui, Hongbo Su, Renhua Zhang, Jing Tian, and Lihu Yang.
2008. "The Tradeoff Analysis for Remote Sensing Image Fusion Using Expanded Spectral Angle Mapper" *Sensors* 8, no. 1: 520-528.
https://doi.org/10.3390/s8010520