# A Physicist's View on Partial 3D Shape Matching
## 5 Discussion  
### Full vs. Partial Shape Similarity  
The difference between the results for full shape comparison and partial shape comparison requires some clarifications. First, the LD-SIFT signatures outperform the WKS signatures for partial shape comparison, while we observe the opposite for full shape comparison. These different behaviors are attributed to the way those signatures are generated. We illustrate the differences in Figure 8. In this figure, we represent the LD-SIFT signatures as well as the WKS signatures at corresponding vertices at the tip on one ear of a cat and at the end of one paw, for two different poses, as well as for the vertices at the ear for a mesh that only includes the head of the cat. As illustrated on the right of the figure, the WKS signatures of the corresponding vertices are very similar for the two poses of the cats. In contrast, the LD-SIFT signatures exhibit more differences. Those signatures are computed from a rendering of the mesh in a plane that is local to the vertex of interest. The renderings for the vertex on the ear are similar, while the renderings for the paw are different. The differences in the images are mitigated by the fact that the SIFT signatures are computed only at the center of the image. It remains that the LD-SIFT signatures are computed from a rendering that relies on extrinsic geometry and as such are sensitive to nonrigid deformation. The WKS signatures differ, as they are computed from the Laplace-Beltrami operator which captures the intrinsic geometry of a mesh, and are therefore less sensitive to nonrigid deformations. Such differences between LD-SIFT and WKS signatures explain the differences observed in Figure 3. Those differences, however, are reverted when we consider partial meshes. The LD-SIFT signatures at the ear of the full cat and of the shape that only includes the head of the cat are very similar (left column, top and bottom row), while the corresponding WKS signatures are very different (right column). The WKS signature is based on the solution of the Schrodinger equation over the  
Figure 7: Distance matrices for shape similarity within the HEAD dataset using the optimal transport framework with LD-SIFT (**top**) and WKS signatures (**bottom**) to represent the vertices on the shapes, and with balanced, i.e., fixed masses (**left**), and unbalanced, i.e., variable masses (**right**), optimal transport optimization to find the correspondence between vertices of two different shapes. The color code is the same as in Figure 3.  
whole mesh describing the shape; as such, it remains a global descriptor. This explains the differences observed in Figure 7.  
Second, the balanced OT procedure finds a global correspondence, while the unbalanced OT procedure allows for partial matching. We illustrate the differences in Figure 1. This figure reports on an experiment in which the head of a cat is compared to its corresponding complete cat, using LD-SIFT signatures and either balanced or unbalanced optimal transport to compute the correspondence. At convergence, the transport plan defines matching between vertices of the cat head and of the complete cat. Pairs of vertices \((i,j)\) from the cat head and the complete cat, respectively, are considered as matches if the corresponding converged values \(G^{MF}(i,j)\) are larger than \(0.9\times m_{1}(i)\). All the matching pairs are then used to generate a rigid body transformation (scaling + rotation + translation) between the two meshes. The balanced OT framework impose a global registration of the two sets of points. As a consequence, the final correspondence matrix leads to the head being translated towards the center of mass of the complete cat, and rotated and scaled up such that it provides a maximum coverage of the vertices of the full cat.