# An Application of Manifold Learning in Global Shape Descriptors
## 5 Discussion  
In this study, motivated by the unique properties of Laplacian Eigenmap (i.e., locality preservation, structural equivalence, and dimensionality reduction) and inspired by the existing spectral-based shape descriptors, we investigated the application of manifold learning in deriving a shape fingerprint in order to address the limitations tied to popular cotangent-based shape descriptors. We proposed a global descriptor (LESI) with an easy-to-compute and efficient normalization technique that facilitates applications such as shape classification and retrieval. Our method applies fewer restrictions on the class of meshes as well as improves the quality of tessellations. Analogous to other spectral descriptors, LESI uses the spectrum of the LB operator, which is independent of the shape location, informative (contains a considerable amount of geometrical and topological information), and above all isometric invariant. We compared the discriminating power of LESI with three prominent descriptors from the literature, namely Shape-DNA, cShape-DNA, and GPS, and found it to be superior. In the first set of experiments illustrated in Figures 3 and 5, our method substantially outperforms the others. The superiority of LESI is more significant when the McGill dataset is used (Table 2 and Figure 6). This dataset includes wide variations in mesh structure and scales, causing the failure of the other methods to generate acceptable results. However, LESI, due to utilizing a different method of discretization to form the LB operator, focuses on the vicinity rather than the quality of the triangulation. Therefore, our technique, unlike other methods, is not affected by the low quality of polygon meshes. The second set of experiments evaluated the reliability of our method in the presence of noise, scale variations, as well as different sampling rates. LESI shows impressive robustness against the first two sets of perturbation. Despite the negative impact of down sampling in LESI descriptor, it continues to show better performance when compared to cShape-DNA and GPS. It should be noted that the result could also be improved by increasing the size of the output vector. In addition to the discriminating power of the descriptor, degenerate and non-uniform meshes may also cause failure of an algorithm to converge. The cotangent weight-based algorithms were not able to compute the descriptors for two shapes from the McGill dataset. GPS also failed to compute descriptors for six models of the down sampled TOSCA dataset. However, our technique converges at all times despite the quality of the polygon mesh structure. Figure 11: 2D PCA projection of shape descriptors computed by: (**a**) original Shape-DNA; (**b**) cShape-DNA; (**c**) GPS; and (**d**) LESI algorithms from down sampled TOSCA dataset by rate of 20%. Figure 12: The Euclidean pairwise distance matrix of shape descriptors computed by (from left to right) Shape-DNA, cShape-DNA, GPS, and LESI algorithms from from down sampled TOSCA dataset by rate of 20%. Moreover, LESI, unlike cotangent weight-based techniques, is not confined to the triangulated meshes as it disregards the mesh geometry [58]. LESI inherits this property from the capability of manifold learning techniques in coping with high dimensional data. The discretization of the LB operator using cotangent weights on the quadrilateral meshes is not as straightforward as on triangular meshes. To compute the LB operator on a quadrilateral mesh, all rectangles need to be divided into triangles. It could be done easily; however, as for each quad there are two possible triangulations, thus the result is not unique. The time needed to compute a descriptor, for all spectral-based descriptors discussed in this research, can be divided into two parts: (1) constructing the \(\mathbf{L}\) and \(\mathbf{D}\) matrices; and (2) solving the generalized eigenvalue problem. The computation time depends on \(n\) (graph size) and \(d\) (descriptor length).