*2.7. Reconstruction Process*

Assuming that the low-resolution image and the high-resolution image have the same sparse representation, the sparse representation of the low-resolution image can be obtained. The sparse representation α was estimated by solving the optimization problem of Equation (12).

$$\underset{\alpha}{\text{argmin}} \frac{1}{2} \|D^{low}\alpha - Y^{low}\|\_2^2 + \lambda \|\alpha\|\_1 \tag{12}$$

The reconstruction process is performed as follows. In this study, the resolution of RGB image was increased.


$$\begin{aligned} I\_{\rm sr}(j) &= D^{\rm high} \hat{\alpha}(j), \\ \hat{\alpha} &\in \mathbb{R}^{\rm Nd \times N\_{\rm patch}} \end{aligned} \tag{13}$$

$$
\hat{I}\_{\rm sr}(j) = I\_{\rm sr}(j) + \tilde{I}\_{\rm sr}(j) \tag{14}
$$

(5) Using the patches of the obtained high-resolution intensity ˆ*Isr*(*j*), the image is reconstructed. The mean value of the overlapped pixels is used as the value of the pixel in the adjacent overlapping patches.

### *2.8. Tradeo*ff *Process*

The intensity of SCMP is calculated using Equation (15) after obtaining the intensity *I low up* via Equation (6).

$$I\_{scmp}(k) = \frac{I\_{up}^{low}(k)}{PAN^{low}(k)} PAN^{high}(k)\_{\prime} \tag{15}$$

where *k* indicates the pixel position. The low-resolution PAN image, *PANlow*, is obtained using Equation (16).

$$PAR^{luv} = l\_{up}^{luv} + c\_1 MS\_{up, nir}^{luv} - c\_2 MS\_{up, llu}^{luv} - c\_3 MS\_{up, \xi rm}^{luv} - c\_4 MS\_{up, rad"}^{luv} \tag{16}$$

where *I low up* , *MSlow up*,*nir*, *MSlow up*,*blu*, *MSlow up*,*grn*, and *MSlow up*,*red* are the intensities of the low-resolution RGB image, NIR, blue, green, and red, respectively, and these are upsampled to the same sizes as those of the PAN image.

If the PAN image is corrected using the intensity *Iscmp* obtained via SCMP, there will be little loss of spatial information. Since the intensity correction is performed appropriately on the image, the spectral distortion when using SCMP is smaller than that of the other component substitution methods. Figure 2 shows the intensity of the original RGB image, the intensity image generated by SCMP, and the intensity image obtained by CS of Nihonmatsu and Yokohama images. From this figure, it can be seen that the intensity obtained by CS had less spatial information than SCMP. In order to increase the quality of the intensity reproduced by CS using SCMP, the intensities obtained by the CS and the SCMP were combined linearly using the tradeoff parameter τ. In the tradeoff process, the high-resolution intensity images were obtained by SCMP and CS, and these were linearly combined by Equation (17).

$$I^{high} = \tau I\_{sr} + (1 - \tau)I\_{scmp} \text{ s.t. } |\tau| \le 1 \tag{17}$$

**Figure 2.** Intensity of RGB images of (**a**) Nihonmatsu, (**b**) Yokohama. (i) Intensity of the original image, (ii) intensity of SCMP, (iii) intensity obtained via compressed sensing.
