Table 2 Performance evaluation metrics used to test the robustness of our proposed approach

CC

Correlation coefficient (CC) indicates the degree of correlation between the enhanced and the original images; it ranges between ‏1 and − 1. ‏One and − 1 indicates that the parameters are perfectly correlated, while 0 indicates no relation between two parameters

\(CC=\frac{\sum_{i=1}^{n}({X}_{i}-\overline{X })({Y}_{i}-\overline{Y })}{\sqrt{\sum_{i=1}^{n}{({X}_{i}-\overline{X })}^{2}\sum_{i=1}^{n}{({Y}_{i}-\overline{Y })}^{2}}}\)(5)

where xi and yi are the pixel values of enhanced image

Higher value (close to + 1)

[51]

SAM

It determines the spectral similarity between the pixels and the vector of the reference and fused images by calculating the spectral angle between these two spectra. It is performed on a pixel-by-pixel base

\(SAM{(V,\widehat{V})}^{\Delta }=arccos\left[\frac{(V,\widehat{V})}{{\Vert V\Vert }_{2} {\Vert \widehat{V}\Vert }_{2}}\right]\)(6)

Lower value

[52]

SSIM

It calculates the structural similarity index between the reference and fused images, which is a value between − 1 and + 1. A value of + 1 indicates that the 2 given images are very similar or identical

\(SSIM= \frac{\left(2{\mu }_{{I}_{r}}{\mu }_{{I}_{f}}+{C}_{1}\right)(2{\sigma }_{{I}_{r}}{\mu }_{{I}_{f}}+{C}_{2})}{\left({{\mu }^{2}}_{{I}_{r}}+{{\mu }^{2}}_{{I}_{f}}+{C}_{1}\right)({{\sigma }^{2}}_{{I}_{r}}+{{\sigma }^{2}}_{{I}_{f}}+{C}_{2})}\)(7)

Higher value (close to + 1)

[53]

UQI

It is designed to provide a comparison of distortion information between the original image and the distorted image as a combination of three factors, namely loss of correlation, distortion of luminance, and distortion of contrast. UQI is a value between − 1 and + 1. A value of + 1 indicates that the 2 given images are very similar

\(UQI= \frac{4 {I}_{r}{I}_{f}({\mu }_{{I}_{r}}+ {\mu }_{{I}_{f}} )}{({\sigma }^{2}{I}_{r}+{\sigma }^{2}{I}_{f})({\mu }^{2}{I}_{r}+{\mu }^{2}{I}_{f})}\)(8)

Higher value (close to + 1)

[54]