# A Point Cloud Registration Algorithm Based on Weighting Strategy for 3D Indoor Spaces
## 2 Weighting Strategy of Targets in Semantic Segmentation
### Improvement of Steps  
The criteria for labels in the semantic segmentation application background do not strictly adhere to a linear correlation. For instance, there is no inherent positive or negative correlation between the importance of a particular label and its CPA. The classical CRITIC method, unfortunately, fails to adequately consider this kind of fuzzy nonlinearity [36]. Therefore, we make some improvements to CRITIC based on the methodology proposed by [PERSON] and [PERSON] [37]. In the normalization procedure, we include fuzzy set theory and Hamming distance as similarity measures. Additionally, we use the distance correlation instead of the Pearson correlation to describe both the linear and nonlinear relationships between the criteria.  
The following procedures allow for improvements: First, the elements of decision matrix D are converted into percentages.  
\[M^{\prime}_{ij}=\frac{M_{ij}}{\sum\limits_{i=1}^{n}{M_{ij}}} \tag{11}\]  
Then, the fuzzy form of elements of decision matrix \(D\) is defined as follows:  
\[\begin{cases}D^{\prime\,L}_{ij}=M^{\prime}_{ij}-\Delta d\\ D^{\prime\,H}_{ij}=M^{\prime}_{ij}+\Delta d\end{cases} \tag{12}\]  
where \(\Delta d\) is a very small positive number for all \(i\) and \(j\) values: \(0<D^{\prime}_{ij}+\Delta d<1\) and \(0<D^{\prime}_{ij}-\Delta d<1\). The boundaries of fuzzy RP points are defined as:  
\[\begin{cases}L_{j}=\sum\limits_{i=1}^{n}{\frac{D^{\prime\,L}_{ij}}{n}}\\ H_{j}=\sum\limits_{i=1}^{n}{\frac{D^{\prime\,H}_{ij}}{n}}\end{cases} \tag{13}\]where \(L_{j}\) is the lower bound of the fuzzy RP points and \(H_{j}\) is the upper bound of the fuzzy RP points. The normalization procedure can be described as:  
\[D^{\prime}_{ij}=\begin{cases}1,&\left[D^{\prime\;L}_{ij},D^{\prime\;H}_{ij} \right]\subseteq\left[L_{j},H_{j}\right]\\ 1-\frac{\left|D^{\prime\;L}_{ij}-L_{j}\right|+\left|D^{\prime\;H}_{ij}-H_{j} \right|}{2\left(D^{\prime\;H}_{ij}-D^{\prime\;L}_{ij}\right)},&\left[D^{\prime \;L}_{ij},D^{\prime\;H}_{ij}\right]\
ot\subset\left[L_{j},H_{j}\right]\end{cases} \tag{14}\]  
where \(D^{\prime}_{ij}\) is the element after normalization, which belongs to fuzzy set [\(D^{\prime}_{ij}\),\(D^{\prime}_{ij}\),\(D^{\prime}_{ij}\)] and \(D^{\prime}_{j}\),\(D^{\prime}_{j}\),\(D^{\prime}_{j}\),\(D^{\prime}_{j}\),\(D^{\prime}_{j}\),\(H^{+}=\text{max}(D^{\prime}_{j})^{H}\),\(D^{\prime}_{j}\),\(\ldots\),\(D^{\prime}_{nj}\),\(H\)).  
The distance correlation is defined as:  
\[\rho_{jk}=\frac{\frac{\sum\limits_{i=1}^{n}\left(D^{\prime}_{ij}- \overline{D^{\prime}_{j}}\right)\left(D^{\prime}_{ik}-\overline{D^{\prime}_{k }}\right)}{\sqrt{\frac{\sum\limits_{i=1}^{n}\left(D^{\prime}_{ij}- \overline{D^{\prime}_{j}}\right)^{2}}{n}*\frac{\sum\limits_{i=1}^{n} \left(D^{\prime}_{ik}-\overline{D^{\prime}_{k}}\right)^{2}}{n}}}} \tag{15}\]  
where \(\overline{D^{\prime}_{j}}\) and \(\overline{D^{\prime}_{k}}\) are mean values of the \(j\)th and \(k\)th criteria after normalization by Equation (16), the numerator refers to the covariance between the \(j\)th and \(k\)th criteria, \(cov(j,k)\), and the denominator refers to the square root of the product of the variances of the two criteria, \(var(j)\) and \(var(k)\).  
Finally, Equation (14) is used for normalization instead of Equation (3), and Equation (15) is used for the distance correlation coefficient \(\rho_{jk}\) instead of the Pearson correlation coefficient \(r_{jk}\) from Equation (6). In Equation (10), \(r_{jk}\) is replaced with \(\rho_{jk}\) to calculate the measure of conflict.