# A Point Cloud Registration Algorithm Based on Weighting Strategy for 3D Indoor Spaces
## 2 Weighting Strategy of Targets in Semantic Segmentation
### Steps of Calculation  
A decision matrix is defined as follows:  
\[\mathbf{M}=\begin{pmatrix}M_{11}&\ldots&M_{1m}\\ \vdots&\ddots&\vdots\\ M_{n1}&\ldots&M_{nm}\end{pmatrix} \tag{2}\]  
The variables \(n\) and \(m\) represent the quantity of alternatives and evaluation criteria, respectively. \(M_{ij}\) is the value of the \(j\)th criterion for the \(i\)th alternative, \(1\leq i\leq n\), \(1\leq j\leq m\). We specify \(n\) as 3 and \(m\) as 13 for our 3D indoor point cloud semantic segmentation task.  
Because each criterion has different dimensions, normalizing the elements in the decision matrix is necessary to unify the units. This paper's criteria anticipate higher values, leading to the adoption of a forward normalization approach:  
\[D_{ij}=\frac{M_{ij}-M_{j}^{-}}{M_{j}^{+}-M_{j}^{-}};i=1,\ldots,n;j=1,\ldots,m \tag{3}\]  
where \(M_{j}^{-}=\min(M_{1j},M_{2j},\ldots,M_{nj})\), \(M_{j}^{+}=\max(M_{1j},M_{2j},\ldots,M_{nj})\), and \(D_{i\bar{\jmath}}\) is the element \(M_{ij}\) of the decision matrix after normalization by dividing its distance from the maximum value by the length of the range.  
The Pearson correlation coefficient between two criteria \(j\) and \(k\) is defined as follows:  
\[r_{jk}=\frac{\sum\limits_{i=1}^{n}\left(D_{ij}-\overline{D_{j}}\right)\left(D_ {ik}-\overline{D_{k}}\right)}{\sqrt{\sum\limits_{i=1}^{n}\left(D_{ij}- \overline{D_{j}}\right)^{2}\sum\limits_{i=1}^{n}\left(D_{ik}-\overline{D_{k} }\right)^{2}}} \tag{4}\]  
where \(\overline{D_{j}}\) and \(\overline{D_{k}}\) are mean values of the \(j\)th and \(k\)th criteria after normalization by Equation (5):  
\[\overline{D_{j}}=\frac{1}{m}\sum\limits_{i=1}^{n}D_{ij};j=1,\ldots,m \tag{5}\]  
\[\overline{D_{k}}=\frac{1}{m}\sum\limits_{i=1}^{n}D_{ik};k=1,\ldots,m \tag{6}\]  
The standard deviation of each criterion is defined:  
\[\sigma_{j}=\sqrt{\frac{1}{m-1}\sum\limits_{i=1}^{n}\left(D_{ij}-\overline{D_ {j}}\right)^{2}};j=1,\ldots,m \tag{7}\]  
where \(\sigma\) is the standard deviation, which serves to contrast the intensity among each criterion of the same alternative.  
The measure of conflict among each criterion is defined:  
\[p_{j}=\sum\limits_{k=1}^{m}{(1-r_{jk});j=1,\ldots,m} \tag{8}\]  
where \(p_{j}\) is the measure of conflict between the \(j\)th criterion and the others, which indicates the correlation among the criteria. The more positive the correlation is, the smaller the value of the inconsistency will be.  
The amount of information of each criterion is defined:  
\[Q_{j}=\sigma_{j}p_{j} \tag{9}\]  
where \(Q_{j}\) is the amount of information of the \(j\)th criterion, which equals the product of standard deviation and measure of conflict. The greater \(Q_{j}\) is, the more information transmitted by the corresponding criterion, and the relatively higher importance it will have.  
Finally, the weights of the criteria are defined as follows:  
\[\omega_{j}=\frac{Q_{j}}{\sum\limits_{j=1}^{m}{Q_{j}}};j=1,\ldots,m \tag{10}\]  
where \(\omega_{j}\) indicates the weight of the \(j\)th criterion and will be applied in the subsequent ICP registration process.