## 2 HGF Model
### Fusion Algorithm of Hyperspectral Images  
The HA of a spectrum can be approximated as a representation of a spectral curve as a superposition of sine (cosine) curves [11]. In particular, the HA of a spectrum is the decomposition of a spectral curve into several spectra of different frequencies, which are multiples of a basic frequency, that are then superimposed to represent the original spectrum. In other words, HA first decomposes the spectrum and then reconstructs it. Based on the Fourier expansion of a periodic waveform, for the image element spectral curves in hyperspectral remote sensing images, the transform expansion can be expressed as a Fourier series:  
\[\begin{array}{l}x(n)=\frac{A_{0}}{2}+\sum\limits_{h=1}^{\infty}\left[A_{h} \cos(2h\tau n/L)+B_{h}\sin(2h\tau n/L)\right]\\ =\frac{A_{0}}{2}+\sum\limits_{h=1}^{\infty}C_{h}\sin(2h\tau n/L+\varphi_{h}) \end{array} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **GF-1: Band 4** & **GF-5** & **Sentinel-2B: Band 4** & **ZY1-02D** \\ \hline Acquisition time & 25 May 2019 & 29 May 2019 & 20 June 2020 & 28 June 2020 \\ Pixel & 8 m & 30 m & 10 m & 30 m \\ Spectral resolution & – & VNIR: 5 nm SWIR: 10 nm & – & VNIR: 10 nm SWIR: 20 nm \\ Dims & \(510\times 455\times 1\) & 136 \(\times\) 121 \(\times\) 303 & \(420\times 354\times 1\) & 140 \(\times\) 118 \(\times\) 151 \\ \hline \end{tabular}
\end{table}
Table 1: The basic information of the images used in this study.  
Figure 1: HGF spatial–spectral fusion model.  
where \(C_{h}\)sin(\(2h\pi n/L+\varphi_{h}\)) is the \(h^{th}\) harmonic component, and \(A_{h}\), \(B_{h}\), \(C_{h}\), and \(\varphi_{h}\) are calculated as follows:  
\[\left\{\begin{array}{l}A_{h}=\frac{2}{L}\sum\limits_{n=1}^{L}x(n)\cos(2h\pi n /L)\\ B_{h}=\frac{2}{L}\sum\limits_{n=1}^{L}x(n)\sin(2h\pi n/L)\\ C_{h}=\sqrt{{A_{h}}^{2}+{B_{h}}^{2}}\\ \varphi_{h}=\arctan(-\,A_{h}/\,B_{h})\end{array}\right. \tag{2}\]  
where \(x(n)\) is the discrete spectral curve, \(n\) is the band number, \(L\) is the total number of bands, \(A_{0}/2\) represents the harmonic residual term, \(h\) represents the number of decompositions and its value range is 0-151, \(C_{h}\) represents the amplitude of the \(h\)th harmonic component, \(\varphi_{h}\) represents the phase of the \(h\)th harmonic component, and the spectral curve dimension equals \(W=2h+1\) after harmonic transformation.  
From the above equation, it can be observed that each element spectrum is a superposition of a series of sine (cosine) component curves, and each sine (cosine) curve is composed of the harmonic residual term \(A_{0}/2\), amplitude \(C_{h}\), and phase \(\varphi_{h}\). According to the Equation (1), \(A_{0}/2\) is a constant that does not affect the waveform of the spectral curve. It is the integrated response of the feature to the reflection of electromagnetic waves and contains the spatial information of the image. Replacing the high-spatial-resolution image with \(A_{0}/2\) for harmonic reconstruction inversion can realize spatial-spectral fusion. However, there is a large difference between the high-spatial-resolution image and the gray value of the \(\varphi_{h}\) image element.  
Therefore, the fusion of \(A_{0}/2\) and high-spatial-resolution images is performed using the GS transform to improve the spatial resolution of \(A_{0}/2\) while rendering the gray value of image elements of the GS-transform-fused images closer to \(A_{0}/2\). Then, the GS-transform-fused image, \(C_{h}\), and \(\varphi_{h}\) are harmonic-reconstructed and inverted to complete the fusion of hyperspectral and high-spatial-resolution images that are closer to the real feature spectral reflectance.