## 2 Materials and Methods
### PSCDR: A Deep Contrastive Regression Method for Point Cloud Data
#### 2.2.3 Supervised Contrastive Loss for Point Cloud and Deep Regression
This formula combines two aspects of similarity: the first term, involving cosine similarity, measures how aligned the feature vectors \(\mathrm{f_{j}}\) and \(\mathrm{f_{q}}\) are (higher cosine similarity means more aligned). The second term, \(\mathrm{d_{CD}}\), measures the geometric dissimilarity between the original point clouds \(\mathrm{\widetilde{S_{j}}}\) and \(\mathrm{\widetilde{S_{q}}}\) using Chamfer distance (lower Chamfer distance means more similar shapes). The coefficients \(\mu\) and \(\lambda\) balance the contribution of these two similarity aspects. The calculation formula for \(\mathrm{Sim_{pf}}\) is as follows:  
\[\mathrm{Sim_{pf}}\bigg{(}\mathrm{f_{j}},\mathrm{f_{q_{r}}},\mathrm{\widetilde{S _{j}}},\mathrm{\widetilde{S_{q}}}\bigg{)}=\mu\cdot\frac{1}{\mathrm{Cossim}( \mathrm{f_{j}},\mathrm{f_{q}})+2}+\lambda\cdot\mathrm{d_{CD}}(\mathrm{ \widetilde{S_{j}}},\mathrm{\widetilde{S_{q}}}) \tag{3}\]  
\[\mathrm{d_{CD}}(\mathrm{\widetilde{S_{j}}},\mathrm{\widetilde{S_{q}}})=\frac{ 1}{\left|\mathrm{\widetilde{S_{j}}}\right|}\sum_{\mathrm{P_{x}}\in\mathrm{ \widetilde{S_{j}}}}\min_{\mathrm{P_{y}}\in\mathrm{\widetilde{S_{q}}}}\left\| \mathrm{P_{x}}-\mathrm{P_{y}}\right\|_{2}^{2}+\frac{1}{\left|\mathrm{ \widetilde{S_{q}}}\right|}\sum_{\mathrm{P_{y}}\in\mathrm{\widetilde{S_{q}}}} \min_{\mathrm{P_{y}}\in\mathrm{\widetilde{S_{q}}}}\left\|\mathrm{P_{y}}- \mathrm{P_{x}}\right\|_{2}^{2} \tag{4}\]  
where \(\mathrm{Cossim}(\cdot,\cdot)\) is the cosine similarity calculation function and \(\mathrm{d_{CD}}(\cdot,\cdot)\) is the function used to compute the Chamfer distance value. For two point cloud samples, a larger Chamfer distance indicates greater dissimilarity between the two sets of point clouds. According to Equation (3), the more similar the two point cloud samples, the smaller the sum of the first and second components; conversely, the less similar the two samples, the larger the sum of the first and second components. \(\mathrm{\widetilde{S_{j}}}\) and \(\mathrm{\widetilde{S_{q}}}\) are two sets of augmented point clouds, and \(\left|\mathrm{\widetilde{S_{j}}}\right|\) and \(\left|\mathrm{\widetilde{S_{q}}}\right|\) represent the number of points contained in each set. In Equation (4), the first term represents the average of the sum of the minimum distances from any point \(\mathrm{P_{x}}\) in \(\mathrm{\widetilde{S_{j}}}\) to \(\mathrm{\widetilde{S_{q}}}\), while the second term represents the average of the sum of the minimum distances from any point \(\mathrm{P_{y}}\) in \(\mathrm{\widetilde{S_{q}}}\) to \(\mathrm{\widetilde{S_{j}}}\). The loss function \(\Lambda_{\mathrm{PSCDR}}\) is defined as the average value of the sum of the losses for all augmented samples, expressed as:  
\[\Lambda_{\mathrm{PSCDR}}=\frac{1}{2\mathrm{N}}\sum_{\mathrm{j}=0}^{2\mathrm{N }}\Lambda_{\mathrm{PSCDR_{j}}} \tag{5}\]  
The total loss function is the weighted sum of the \(\Lambda_{\mathrm{PSCDR}}\) and the regression loss, expressed as:  
\[\Lambda_{\mathrm{sum}}^{\mathrm{PSCDR}}=\alpha_{2}\Lambda_{\mathrm{P}}+\beta_ {2}\Lambda_{\mathrm{PSCDR}} \tag{6}\]