# A Point Cloud Registration Algorithm Based on Weighting Strategy for 3D Indoor Spaces
## 4 Experiment and Analysis
### Weighting Strategy Steps  
Based on Table 3, the decision matrix can be described as  
\[M=\left[ \begin{array}{cccccccccccccccc}0.200&0.200&0.200&0.600&0.600&0.400&1.000 &1.000&0.800&0.600&0.600&0.000\\ 0.977&0.979&0.870&0.918&0.685&0.887&0.896&0.832&0.922&0.836&0.841&0.888&0.804 \\ 0.989&0.987&0.957&0.932&0.735&0.902&0.919&0.890&0.946&0.844&0.909&0.930&0.918 \\ \end{array} \right] \tag{39}\]  
Equations (11) and (12) are applied to transform the decision matrix into fuzzy form, as shown in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Criteria** & & **IM** & **IoU** & **CPA** \\ \hline Ceiling & \({D^{\prime}_{ij}}^{L}\) & 0.0887 & 0.4437 & 0.4526 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.0987 & 0.4537 & 0.4626 \\ \hline Floor & \({D^{\prime}_{ij}}^{L}\) & 0.0874 & 0.4465 & 0.4511 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.0974 & 0.4565 & 0.4611 \\ \hline Wall & \({D^{\prime}_{ij}}^{L}\) & 0.0962 & 0.4214 & 0.4674 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.1062 & 0.4314 & 0.4774 \\ \hline Beam & \({D^{\prime}_{ij}}^{L}\) & 0.0997 & 0.4230 & 0.4623 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.1097 & 0.4330 & 0.4723 \\ \hline Column & \({D^{\prime}_{ij}}^{L}\) & 0.3477 & 0.2972 & 0.3401 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.3577 & 0.3072 & 0.3501 \\ \hline Window & \({D^{\prime}_{ij}}^{L}\) & 0.2538 & 0.3509 & 0.3802 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.2638 & 0.3609 & 0.3902 \\ \hline Door & \({D^{\prime}_{ij}}^{L}\) & 0.1968 & 0.3774 & 0.4107 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.2068 & 0.3874 & 0.4207 \\ \hline Table & \({D^{\prime}_{ij}}^{L}\) & 0.3726 & 0.2918 & 0.3205 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.3826 & 0.3018 & 0.3305 \\ \hline Chair & \({D^{\prime}_{ij}}^{L}\) & 0.3573 & 0.3062 & 0.3214 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.3673 & 0.3162 & 0.3314 \\ \hline Sofa & \({D^{\prime}_{ij}}^{L}\) & 0.3364 & 0.3181 & 0.3305 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.3464 & 0.3281 & 0.3405 \\ \hline Case & \({D^{\prime}_{ij}}^{L}\) & 0.2575 & 0.3445 & 0.3830 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.2675 & 0.3545 & 0.3930 \\ \hline Board & \({D^{\prime}_{ij}}^{L}\) & 0.2563 & 0.3469 & 0.3818 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.2663 & 0.3569 & 0.3918 \\ \hline Clutter & \({D^{\prime}_{ij}}^{L}\) & 0.0547 & 0.4294 & 0.5010 \\  & \({D^{\prime}_{ij}}^{H}\) & 0.0647 & 0.4394 & 0.5110 \\ \hline \end{tabular}
\end{table} 
Table 4: Decision matrix in fuzzy form. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & **Ceiling** & **Floor** & **Wall** & **Beam** & **Column Window** & **Door** & **Table** & **Chair** & **Sofa** & **Case** & **Board** & **Clutter** \\ \hline IM & 0.2 & 0.2 & 0.2 & 0.2 & 0.6 & 0.6 & 0.4 & 1 & 1 & 0.8 & 0.6 & 0.6 & 0.1 \\ IoU & 0.958 & 0.977 & 0.843 & 0.818 & 0.514 & 0.825 & 0.758 & 0.786 & 0.859 & 0.757 & 0.799 & 0.808 & 0.728 \\ CPA & 0.977 & 0.987 & 0.934 & 0.893 & 0.587 & 0.893 & 0.824 & 0.862 & 0.901 & 0.786 & 0.887 & 0.888 & 0.848 \\ \hline \end{tabular}
\end{table} 
Table 3: The average values of alternatives and criteria for six areas. The small positive number \(\Delta d\) in Equation (12) is set to be 0.005. The boundaries of each criterion and corresponding fuzzy RP points are calculated according to Equation (13), and the results are shown in Table 5.  
The decision matrix is normalized according to Equation (14), and the results are shown in Table 6.  
The distance correlations between different pairs of criteria are calculated according to Equation (7), and the results are shown as Table 7.  
The amounts of information and weights of criteria are calculated according to Equations (8)-(10) and corresponding ranks are assigned to the criteria based on the weights. The results are shown in Table 9.