# Panchromatic and Hyperspectral Image Fusion Using Ratio Residual Attention Networks
## 4 Results
### Evaluation Methods
#### 4.3.2 Objective Evaluation Methods  
Objective evaluation methods primarily involve the use of mathematical models to quantitatively analyze fused images. To objectively assess the advantages and disadvantages of the proposed fusion methods, several evaluation metrics are introduced. These include the Universal Image Quality Index (Q-index) [30], Spectral Angle Map (SAM) [31], Correlation Coefficient (CC) [8], Erreur Relative Global Adimensionnelle de Synthese (ERGAS) [32], Root Mean Squared Error (RMSE) [33], and runtime (Time) [7]. The Universal Image Quality Index comprehensively measures the correlation, luminance, and contrast between the fused image \(F\) and the reference image \(R\). Its calculation formula is as follows:  
\[Q=\frac{4\sigma_{R,F}\bar{R}\bar{F}}{\left(\sigma_{R}^{2}+\sigma_{F}^{2}\right) (\bar{R}^{2}+\bar{F}^{2})}. \tag{15}\]  
In this formula, \(\sigma\) represents the covariance calculation symbol, and \(\bar{R}\) and \(\bar{F}\) represent the mean values of the reference and fused images, respectively. The value of \(Q\) ranges from \([0,1]\), with an ideal value of the index being \(1\). Furthermore, the formula can be simplified as follows:  
\[Q=\frac{\sigma_{RF}}{\sigma_{R}\sigma_{F}}\cdot\frac{2\bar{R}\bar{F}}{\bar{R}^ {2}+\bar{F}^{2}}\cdot\frac{2\sigma_{R}\sigma_{F}}{\sigma_{R}^{2}+\sigma_{F} ^{2}}. \tag{16}\]  
The first term in this formula measures the linear correlation between the fused image and the reference image. The second term evaluates the average luminance information of both the fused and reference images, while the third term assesses the similarity in contrast. The Spectral Angle Mapper (SAM) quantifies the similarity between the reference image \(R\) and the fused image \(F\) by calculating the absolute angle between two spectral vectors constructed from the spectrum of each pixel in the images. Given the spectral vectors \(u_{R}\) and \(u_{F}\) for pixels in images \(R\) and \(F\), the formula for calculating the spectral angle is as follows:  
\[\text{SAM}=\text{arccos}\bigg{(}\frac{\langle u_{R},u_{F}\rangle}{|u_{R}||u_{F }|}\bigg{)}. \tag{17}\]  
In the fusion experiment analysis, the spectral angle is averaged over the pixel vectors of the entire image to globally measure the spectral distortion of the image. Generally, the smaller the spectral angle, the more similar the two spectral vectors are. For an ideal fused image, the spectral angle should be 0. The correlation coefficient measures the mutual correlation between the fused image and the reference image. The higher the correlation coefficient, the stronger the correlation between the two, with an ideal value of 1. It can be calculated by the following formula:  
\[\text{CC}^{k}=\frac{\sum_{i=1}^{M}\sum_{j=1}^{N}\left[\left(R_{i,j}^{k}-\bar{R }^{k}\right)\left(F_{i,j}^{k}-\bar{F}^{k}\right)\right]}{\sqrt{\sum_{i=1}^{M} \sum_{j=1}^{N}\left(R_{i,j}^{k}-\bar{R}^{k}\right)^{2}\sum_{i=1}^{M}\sum_{j=1} ^{N}\left(F_{i,j}^{k}-\bar{F}^{k}\right)^{2}}}, \tag{18}\]  
where \(k\) is the band number, \(\text{CC}^{k}\) is the correlation coefficient between the fused image and the reference image in the \(k\)-th band; \(M\) and \(N\) are the width and height in terms of the number of pixels of the evaluated image; \(\bar{R}^{k}\) and \(\bar{F}^{k}\) represent the mean values of the reference and fused images in the \(k\)-th band. The relative unitless global error measures the overall spectral quality of the fused image, and its definition is given by the following formula:  
\[\text{ERGAS}=100\frac{h}{l}\sqrt{\frac{1}{K}\sum_{k=1}^{K}\frac{\text{RMSE}^{2 }(k)}{\text{MEAN}^{2}(k)}}, \tag{19}\]  
where \(h\) and \(l\) represent the spatial resolution values of the high spatial resolution image and the low spatial resolution image, respectively; \(K\) is the number of spectral bands in the fused image; \(\text{MEAN}(k)\) represents the mean radiometric brightness of the \(k\)-th band in the reference image; and \(\text{RMSE}(k)\) is the root mean square error between the \(k\)-th band of the reference and fused images.