## 2 Methods
### Monochromatic Value Space Transformation  
RGB color space sometimes does not show part of information we need, and some important information cannot be distinguished from the color space. Therefore, transforming a color value space to a monochromatic value space is more conducive to the extraction of laser stripe. In this paper, a linear transformation is used because it does not cause large jumps, singular value points, or discontinuities, and is faster than nonlinear transformation [15]. A gray image is the most typical monochromatic value image generated by linear transformation. In a discrete color snow sculpture image \(f_{ij}\) (with a size of \(M\times N\)), the tristimulus values of the pixels are \(R_{ij}\), \(G_{ij}\) and \(B_{ij}\). The linear transformation is defined as Equation (1). \[F_{ij}=w_{r}R_{ij}+w_{g}G_{ij}+w_{b}B_{ij} \tag{1}\]  
where \(F_{ij}\) is the monochromatic value image after transformation, and \(i\) and \(j\) represent the row and column indexes of pixelsrespectively, with \(I=1\), \(2\)..., \(M\), \(j=1\), \(2\)..., \(N\), \(w_{r},w_{g},w_{b}\in R\). From Equation (1), the monochromatic value image is determined by  
Figure 3: (**a**) A snow sculpture (penguin) image scanned by structure light in sunlight; (**b–d**) Color histograms of (**a**) in R, G and B space. the transformation coefficients. So, we need to define anobjective function to search for theoptimal \(w_{r},w_{g}\) and \(w_{b}\) to highlight the laser characteristics. The projected stripe of the laser is relatively concentrated and presents a concise structure, and its profiles showing all the luminance row vectors are shown in Figure 4. There is a high contrast between the laser and background, implying a highSignal to Noise Ratio (SNR). The objective function should make full use of this characteristic so that the laser stripe can be extracted easily after transformation. Therefore, the objective function is taken as a measure of the contrast. The pixel values in the laser stripe are close to their averages in Figure 4. In these areas, the contrast increases with energy concentration, and the waveform is steep, suggesting that kurtosis is high. This can be seen as a contrast between the laser stripe and the background. Thus, it is reasonable to choose the kurtosis to define the contrast. The kurtosis \(K\) is defined as Equation (2). \[K=\frac{\kappa_{4}}{\kappa_{2}^{2}}=\frac{\mu_{4}}{\sigma^{4}}-3, \tag{2}\]  
where \(\kappa_{4}\) is the fourth cumulant, \(\kappa_{2}\) is the second cumulant, and \(\mu_{4}\) and \(\sigma^{4}\) are the mean of fourth moment and the square of the variance t of the probability distribution respectively. In particular, the kurtosis is zero in the case of normal distribution. We expect that the laser stripe has the maximum kurtosis after transformation, while the noise signals from background have very low kurtosis, suggesting they are very disordered. In signal systems, disorder is synonymous with entropy. Entropy is affected by the amount of information and its random properties. The more information there is, the more disordered the signalsare, and the greater the entropy is. [PERSON] [16] first used the minimum entropy deconvolution technique and called it the minimum entropy method. Therefore, maximizing kurtosis is equivalent to minimizing entropy, and the transformation model can be called the minimum entropy model. When \(K\) is less than 0 or if the signals come from a pulse source, minimizing is better than maximizing for the purpose of this study [17,18]. Therefore, the objective function can be further defined as Equation (3). \[abs(K)=\Big{|}\frac{\mu_{4}}{\sigma^{4}}-3\Big{|}, \tag{3}\]  
Equation (3) is fully differentiable, thus the maximizing function can be defined as Equation (4). \[K^{2}=\Big{(}\frac{\mu_{4}}{\sigma^{4}}-3\Big{)}^{2}, \tag{4}\]  
Figure 4: Profiles of the laser. (**a**) Images captured including laser stripe on snow sculpture surface; (**b**) Luminance values row by row.