# A Physicist's View on Partial 3D Shape Matching
## 4 Experimental Results
### Full Shape Similarity
#### 4.1.3 Comparisons with Other Shape Matching Tools: SHREC19  
We consider shape correspondence, namely, the identifications of corresponding points between two (or more) 3D shapes. Those correspondences are embedded in the optimal transport plan. We used the SHREC19 benchmark [53] to gauge how well our algorithms can retrieve correspondences. Each 3D shape in this benchmark is given as a triangular mesh of its surface. These shapes were created using 3D scans of actual objects. Each object was captured in several poses that each corresponded to a different type of deformation. The deformations are divided into four categories, or \"groups\", for classification purposes. These four categories are articulated (group 0), isometric (group 1), nonisometric (group 2), and topologic/geometric (group 3) deformations, respectively. Example of shapes for each test set are provided in Figure 5. We used the low-resolution version of this benchmark, with each shape represented with a mesh with approximately 10,000 vertices and 20,000 triangles. SHREC19 then consists of 76 pairs of shapes regrouped in four test sets (see [53] for details). For each pair of shapes \((X,Y)\) in the benchmark, the ground truth correspondence between the vertices of \(X\) and the vertices of \(Y\) is known. To evaluate the quality of correspondence recovery with our algorithms, we considered normalized geodesics between the ground truth and the predicted correspondence. Briefly, for a pair of shapes \((X,Y)\), if \(p_{i}\) is a point on a shape \(X\), \(y_{i}\) is its predicted correspondence \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Dataset** & \multicolumn{5}{c}{**Molars**} \\ \hline
**Classification** & **N a** & **cP b** & **cWhb** & **LD-SIFT c** & **WKS c** \\ \hline Genera & 24 & 91 & 68 & 24 & 80 \\ Families & 17 & 92 & 75 & 28 & 85 \\ Superfamilies & 5 & 95 & 83 & 69 & 94 \\ \hline \hline \end{tabular}  
* Number of groups at the taxonomic level considered. b Results from [14]. c Our results, using either LD-SIFT or WKS as signatures, and unbalanced OT for computing the correspondence.  
\end{table} 
Table 4: Success rates (percentage) of leave-one-out classification experiments for the Molar anatomical dataset. on shape \(Y\), and \(z_{i}\) the ground truth position of \(p_{i}\) on \(Y\). The normalized geodesic error \(\epsilon(p_{i})\) is computed as follows:  
\[\epsilon(p_{i})=\frac{d_{Y}(y_{i},z_{i})}{area(Y)^{1/2}},\]  
where \(d_{Y}(y_{i},z_{i})\) is the geodesic distance between \(y_{i}\) and \(z_{i}\) on the surface of \(Y\). We use the algorithm from [67] to compute this geodesic distance. We compared the LD-SIFT and WKS vertex signatures combined with either balanced or partial OT in their abilities to define those correspondences. In each experiment, a pair of shapes \(i\) and \(j\) is represented with their complete sets of vertices, \(S_{i}\) and \(S_{j}\). For each vertex \(x\) on \(i\), its correspondence \(y\) on \(j\) is set to the index of the maximum value on the row corresponding to \(x\) in the OT transport plan. Figure 6 shows the corresponding cumulative geodesic errors for the four algorithms for the four test sets in SHREC19. As observed with the TOSCA8 dataset as well as with the anatomical dataset, the combination of WKS signatures with balanced OT leads to the best results for all four SHREC19 test sets. Figure 5: A few shapes for each group of the SHREC19 benchmark. Figure 6: Cumulative distribution functions for the geodesic errors for all four methods (see text for details). Results are shown for all four test sets in SHREC19, which consider articulated deformations (test set 0), isometric deformations (test set 1), nonisometric deformations (test set 2), and topological or geometric deformations (test set 3). The Workshop on 3D Object Retrieval, which took place in Genova, Italy, in May 2019, included a competition on comparing 3D shape registration. The SHREC19 dataset was initially utilized as a benchmark for that competition [53].