# A Physicist's View on Partial 3D Shape Matching
## 4 Experimental Results
### Full Shape Similarity  
#### 4.1.1 Synthetic Data: TOSCA  
We tested our different procedures first on the synthetic TOSCA dataset [65; 66], available at [[http://tosca.cs.technion.ac.il/book/resources_data.html](http://tosca.cs.technion.ac.il/book/resources_data.html)]([http://tosca.cs.technion.ac.il/book/resources_data.html](http://tosca.cs.technion.ac.il/book/resources_data.html)), accessed on 1 March 2019. We used eight classes of objects from this dataset (see Figure 2: cat, centaur, horse, dog, seahorse, two male figures ([PERSON] and [PERSON]), and one female figure ([PERSON])). Each class consists of the same shapes under different poses and/ or different triangulations. The differences between the poses echo nonrigid transformations within objects (see [65; 66]for details). In addition, the poses within a class are represented with different topologies (i.e., different triangulations), and may even have different genera. The corresponding dataset, TOSCA8, includes six instances in each class, i.e., a total of 48 classes. Each pose is represented with a triangulated surface mesh with approximately 3400 vertices and 6600 faces, with the exception of the seahorse meshes that include approximately 2100 vertices and 4200 faces. In the first set of experiments, we compared four different approaches for shape similarity, with two different signatures for representing the vertices of the shape, LD-SIFT and WKS, combined with two different methods to compute the correspondences between those vertices, namely, balanced (with fixed masses) optimal transport, which we will refer to as simply OT, and unbalanced (with variable masses) optimal transport, which we will name Partial OT. In each experiment, a pair of shapes is represented with their sets of vertices, \(S_{1}\) and \(S_{2}\), and the cost matrix \(C\) between those vertices, such that \(C(k,l)\) between a vertex \(k\) on shape 1 and a vertex \(l\) on shape 2 is equal to the distance between their signature vectors (see previous section for the choice of distance for the two types of signatures considered). For balanced OT, the masses of the vertices are set as uniform, while for unbalanced OT, we allow for mass creation and deletion. We computed a set of matrices \(D(\beta)\) for \(\beta\) ranging between 1000 and \(10^{11}\), such that \(D(\beta)(k,l)\) is the optimized transport energy \(U_{\beta}^{MF}(S_{k},S_{l})\) (balanced OT), or the regularized optimized transport energy \(SU_{\beta}^{MF}(S_{k},S_{l})\) (unbalanced OT), i.e., the temperature-based distance between the sets of vertices \(S_{k}\) and \(S_{l}\) of the shapes \(k\) and \(l\). Note that for the balanced OT, \(U_{\beta}^{MF}(S_{k},S_{l})\) is a true distance, while for the unbalanced OT, \(SU_{\beta}^{MF}(S_{k},S_{l})\) satisfies the first two properties of a metric (identity and symmetry), but not the triangle inequality. We also computed \(D(\infty)\), the matrix of distances at convergence in \(\beta\). See Figure 3 for graphical representations of \(D(\infty)\) for the four sets of experiments. As observed in Figure 3, the four distance matrices are visually different. To provide a more quantitative assessment of these differences, we performed a set of classification experiments. We started with randomly selecting one shape from each of the eight classes to build a training set. We then performed 1-nearest neighbor classification experiment over the remaining shapes, where \"nearest\" is defined based on the distance value in one of the distance matrices. The difference between the predicted class of a shape with the actual class of that shape allows us to estimate the probability of correct classification \(P(\infty)\), given a distance matrix \(D(\infty)\). Results for the four sets of experiments are given in Table 1.  
Figure 2: A few representative shapes in our TOSCA8 dataset that includes 8 classes (from **left** to **right** and **top** to **bottom**): cat, centaur, [PERSON], dog, horse, [PERSON], Victoria, and seahorse.