*2.5. Estimation of Coe*ffi*cients for Intensity Correction*

The coefficient estimated by SCMP is used for the intensity correction in the proposed method. It is possible to obtain an intensity correction value with little spectral distortion with SCMP. This coefficient is calculated by

$$\underset{c}{\text{argmin}} \|Ac - d\|\_{2}^{2} \text{ s.t. } c \ge 0,\tag{5}$$

where

$$A = \begin{bmatrix} -MS\_{\text{surr}}^{\text{lurr}}(1) & MS\_{\text{blu}}^{\text{lurr}}(1) & MS\_{\text{blu}}^{\text{lurr}}(1) & MS\_{\text{pru}}^{\text{lurr}}(1) & MS\_{\text{pru}}^{\text{lurr}}(1) \\ \vdots & \vdots & \vdots & \vdots \\ -MS\_{\text{nir}}^{\text{lurr}}(k) & MS\_{\text{blu}}^{\text{lurr}}(k) & MS\_{\text{pru}}^{\text{lurr}}(k) & MS\_{\text{lu}}^{\text{lurr}}(k) \\ \vdots & \vdots & \vdots & \vdots \\ -MS\_{\text{nir}}^{\text{lurr}}(N) & MS\_{\text{blu}}^{\text{lurr}}(N) & MS\_{\text{pru}}^{\text{lurr}}(N) & MS\_{\text{ru}}^{\text{lurr}}(N) \end{bmatrix} \in \mathbb{R}^{N \times 4},$$

$$c = \begin{bmatrix} c\_1 \\ c\_2 \\ c\_3 \\ c\_4 \end{bmatrix} \in \mathbb{R}^{4 \times 1},$$

$$d = \begin{bmatrix} I^{\text{lur}}(1) - PAR^{\text{lvl}}(1) \\ \vdots \\ I^{\text{lurr}}(k) - PAR^{\text{lvl}}(k) \\ \vdots \\ I^{\text{lurr}}(N) - RAM^{\text{lvl}}(k) \end{bmatrix} \in \mathbb{R}^{N \times 1},$$

where *<sup>k</sup>* indicates the pixel position. *<sup>N</sup>* is the number of pixels and *PANhigh* <sup>∈</sup> <sup>R</sup>*<sup>N</sup>* is the downsampled PAN test image, of which the size is the same as that of *I low* obtained via the bicubic interpolation. The suffixes *nir*, *blu*, *grn*, and *red* represent the NIR, blue, green, and red color components of the MS image, respectively. For *MSlow nir* , *MSlow blu* , *MSlow grn*, and *MSlow red* , test MS images are used. *I low* is calculated by

$$I^{low} = \frac{MS\_{rad}^{low} + MS\_{\mathcal{S}^{rm}}^{low} + MS\_{blu}^{low}}{3} \tag{6}$$

Note that Equation (6) does not include NIR because it is the intensity of the RGB image.
