# A Survey on Deep Learning in 3D CAD Reconstruction
## 8 Metrics
### Reconstruction from a Point Cloud to a CAD Model  
Point2 Cyl, proposed by [PERSON] et al. [30], converts an original 3D point cloud into a set of CAD models of extruded cylinders. The following evaluation metrics are used to evaluate different aspects of the algorithm. The specific parameter values are shown in Table 3. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Publication**} & \multicolumn{2}{c}{**On the Dataset**} & \multirow{2}{*}{**Seg. \(\uparrow\)**} & \multirow{2}{*}{**Norm. \(\downarrow\)**} & \multirow{2}{*}{**B.B. \(\uparrow\)**} & \multirow{2}{*}{**E.A. \(\downarrow\)**} & \multirow{2}{*}{**E.C. \(\downarrow\)**} & \multirow{2}{*}{**Fit Cyl \(\downarrow\)**} & \multirow{2}{*}{**Fit Glob \(\downarrow\)**} \\ \hline Point2 Cyl [30] & CVPR’22 & Fusion Gallery & 0.736 & 8.547 & 0.911 & 8.137 & 0.0525 & 0.0704 & 0.0305 \\ H.V.+NJ [121] & - & Fusion Gallery & 0.409 & 12.264 & 0.595 & 58.868 & 0.1248 & 0.1492 & 0.0683 \\ \hline Point2 Cyl & CVPR’22 & DeepCAD & 0.833 & 8.563 & 0.919 & 7.923 & 0.0267 & 0.0758 & 0.0308 \\ H.V.+NJ & - & DeepCAD & 0.540 & 13.573 & 0.577 & 59.785 & 0.0435 & 0.1664 & 0.0459 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Point2 Cyl is compared with a strong geometric baseline jet fit (NJ), and H.V. is a simple hand-crafted baseline. \(\uparrow\): the higher the better, \(\downarrow\): the lower the better. **Segmentation IoU (Seg. )**: Uses the Hungarian algorithm to match the predicted extrusion segment with the actual extrusion segment. The segmentation result is compared with the true label. The total IoU loss (RloU loss) is defined as follows:  
\[\frac{1}{K}\sum_{k=1}^{K}\text{RoIU}(1(\widehat{W}_{:,k}),W_{:,k}) \tag{1}\]  
where \(1(\cdot)\) represents the one-hot encoding. **Normal angle error (Norm. )**: Measures the angular consistency between the predicted normal and the true normal. The average error is calculated using the following definition formula:  
\[\frac{1}{N}\sum_{i=1}^{N}\cos^{-1}\Bigl{(}\mathbf{1}(\widehat{\mathbf{N}}_{i,:}^{\text{T}}\mathbf{N}_{i,:})\Bigr{)} \tag{2}\]  
**Base/barrel classification accuracy (B.B. )**: Calculates the ratio of correctly predicted base and barrel labels. If the true label is consistent with the predicted label, it is counted. The formula is as follows:  
\[\frac{1}{N}\sum_{i=1}^{N}\mathbf{1}(\widehat{\mathbf{B}}_{i}==\mathbf{B}_{i}) \tag{3}\]  
**Extrusion axis angle error (E.A. )**: Measures the angular error between the GT and the predicted axis. The formula is as follows:  
\[\frac{1}{K}\sum_{k=1}^{K}\cos^{-1}\Bigl{(}\mathbf{1}(\mathbf{\hat{e}}_{k}^{ \text{T}}\mathbf{e}_{k})\Bigr{)}_{\circ} \tag{4}\]  
**Extrusion center error (E.C. )**: Measures the distance between the GT center and the predicted center:  
\[\frac{1}{K}\sum_{k=1}^{K}\bigl{\|}\mathbf{\hat{e}}_{k,:}-\mathbf{c}_{k,:} \bigr{\|}_{2} \tag{5}\]  
**Per-extrusion cylinder fitting loss (Fit Cyl. )**: A measure of how well each unrestricted extrusion cylinder parameter fits the GT extrusion cylinder segment:  
\[\frac{1}{K}\sum_{k=1}^{K}\left|\mathcal{F}=\sum_{s_{i}\in S_{k}}S(f_{\beta}( \hat{S}_{k}),s_{i})\right| \tag{6}\]  
where \(s_{k}\) is the segment point after projecting the extrusion parameters. **Global fitting loss (Fit Glob. )**: A measure of whether at least one predicted unbound extrusion cylinder can explain each barrel point:  
\[\frac{1}{\left|\mathbf{P}_{\text{barr}}\right|}\sum_{\mathbf{p}_{i}\in \mathbf{P}_{\text{barr}}}\min_{k=\{1,\dots,K\}}\left|\mathcal{S}(f_{\beta}( \hat{\mathbf{S}}_{k}),\mathbf{q}_{i})\right| \tag{7}\]  
where \(\mathbf{q}_{i}=\hat{s}_{k}\Pi(\mathbf{p}_{i},\hat{\mathbf{e}}_{k},\hat{\mathbf{ e}}_{k})\). The PS-CAD model proposed by [PERSON] et al. [49] reconstructs CAD modeling sequences from point cloud data. The evaluation indicators shown in Table 4 were used to evaluate different aspects of the algorithm.