## 3 Simulation  
The CAVE multispectral image dataset [28] can be used for simulation. It consists of 31 images from 400 to 700 nm, with 10 nm intervals. First, calculate the ideal detected images using Equation (1). Then, compare the recovered multispectral images with the reference images. In this study, we compared the traditional point-by-point optimization method and our improved method with different thresholds. Figure 3 shows some reference multispectral images (Figure 3b) and recovered images at 450, 500, 550, 600, 650, and 700 nm. The reconstructed images have some errors with reference images at 500 and 700 nm, but there are no obvious differences between our method (Figure 3d,e) and the point-by-point optimization method (Figure 3c). To see the difference more clearly, we selected one pixel of the 'F' region in Figure 3a and compared the recovered spectra. In the recovery process, we used the mean value for optimization initially and adjusted the spectra by segment linear calibration. In order to examine the results of the linear calibration, three conditions were compared: (1) point-by-point optimization directly, (2) optimization using the mean value, and (3) optimization using the mean value and adding the linear calibration. In our  
Figure 3: (**a**) Initial image for simulation. (**b**–**e**) Parts of spectral images at 450, 500, 550, 600, and 650 nm. (**b**) Reference spectral images. (**c**) Recovered images using point-by-point optimization. (**d**) Recovered images using our method, _th_ = 0.1. (**e**) Recovered images using our method, _th_ = 0.5.  
relevant for threshold values, which will affect the calculation speed and accuracy. Figure 4 displays the spectra differences, comparing the reference under different thresholds. A smaller cosine value means more similar spectra. That is, the reconstructed spectra is better with a small cosine value. Traditional point-by-point optimization has no issues with different thresholds--it has fixed values for different thresholds. When the threshold value was small, the spectra reconstructed using mean values were close to those of the point-by-point optimization method. The spectra were reconstructed best when using mean values that added calibration. As the threshold increased, the cosine values increased for those that used mean value reconstruction only. When the threshold was 0.8, the difference was much greater. Fortunately, by adding the linear calibration, the differences or biases were compensated for. Therefore, linear calibration is necessary for maintaining a high recovery quality. Figure 5 shows the recovered spectra for the different regions ('A-F') in Figure 3a. Although there were some biases at peak regions, especially at the narrow peak for the smooth constraint, the recovered spectra generally displayed similar trends to the reference spectra. Besides comparing single pixels, we also compared the total images. The simulated image had 512 \(\times\) 512 pixels, and a Dell notebook with an Intel Core i5-6200 CPU at 2.3 GHz was used for computation. Table 1 shows the computation time, PSNR (peak signal-to-noise  
Figure 4: Spectral similarity between reference spectra and recovered spectra using different thresholds during reconstruction. Figure 5: Spectra comparison for different spatial regions and colors (reference spectra, point-by-point optimization directly, our method with \(th\) = 0.8). The (**a–f**) curves correspond to the spatial regions ‘A–F’ in Figure 3a. ratio), SAM (spectral angle mapping), and SMAPE (symmetric mean absolute percentage error) [20]. When the threshold increased, the computation time greatly decreased. Further, the computation time was only 5.5% of that of the point-by-point optimization method when the threshold was 0.5. Here, MATLAB R2014a (32 bit) was used for computation. If the C/C++ codes are programmed, the computation time may be reduced further. PSNR indicates the image spatial quality of recovered images.