# An Application of Manifold Learning in Global Shape Descriptors
## 4 Experiments
### Retrieval Results  
In this section, we evaluate the general performance of our proposed shape descriptor and compare it with several state-of-the-art spectral-based global shape descriptors including Shape-DNA [20], cShape-DNA [30], and GPS [23] algorithms. We chose these methods because they are widely used by researchers (e.g., [51; 52; 53]) to develop new descriptors or applications, or to evaluate the performance of their proposed descriptors. Moreover, cShape-DNA represents the normalized version of the original Shape-DNA. Even though there are multiple ways to convert a local point descriptor to a global shape fingerprint, in this article we focus only on algorithms that have been originally introduced as global fingerprints. To this end, we take advantage of the source code made available on Dr. [PERSON]'s homepage ([[http://vision.mas.ecp.fr/Personnel/iasonas/descriptors.html](http://vision.mas.ecp.fr/Personnel/iasonas/descriptors.html)]([http://vision.mas.ecp.fr/Personnel/iasonas/descriptors.html](http://vision.mas.ecp.fr/Personnel/iasonas/descriptors.html))) [28], as well as the shape descriptor package provided by [PERSON] et al. [54] available on a GitHub repository ([[https://github.com/ChunyuanLI/spectral_descriptors](https://github.com/ChunyuanLI/spectral_descriptors)]([https://github.com/ChunyuanLI/spectral_descriptors](https://github.com/ChunyuanLI/spectral_descriptors))) to generate the Shape-DNA and GPS descriptors, respectively. We also compare the performance of shape retrieval using the code provided for evaluation by SHREC'11 [55]. The shape descriptors are compared using the TOSCA dataset to discriminate between different classes of 3D objects. In this experiment, we used the first 33 non-zero eigenvalues \((d=33)\). Then, to visualize the locations of objects in the shape space, we projected them onto a 2D plane using Principle Component Analysis (PCA). Figure 3 displays the effectiveness of our method compared with the fingerprints of interest. Figure 3 reveals that LESI can differentiate models of various classes significantly better than the other methods for a refined and normalized dataset. Despite the large isometric deformations in each class, the proposed LESI method clusters all models of the same class together very tightly. Even though all human models ([PERSON], [PERSON], and [PERSON]) are very similar, it can distinguish the woman from the men. However, it fails to discriminate models of [PERSON] from [PERSON]. Looking more closely at Figure 3 and comparing the performance of Shape-DNA and cShape-DNA in the separation of [PERSON] and [PERSON], we conjecture whether objects' scale is one of the most distinctive features of [PERSON] and [PERSON]'s differentiation. LESI descriptor is invariant to isometry, object's location, and rotation. LESI learns the overall structure of each model, which it is similar for  
Figure 3: 2D PCA projection of shape descriptors computed from: (**a**) original Shape-DNA; (**b**) cShape-DNA; (**c**) GPS; and (**d**) LESI algorithms on TOSCA dataset. men models. We suspect that using the parameter \(t\) as defined in Equation (7) may cause LESI to fail to discriminate models of different men. Therefore, we computed LESI descriptor for [PERSON] and [PERSON], using \(t=5\) and \(t=15\). Figure 4 shows two samples of [PERSON] and [PERSON], which demonstrates how much they look alike. In addition, PCA projection of LESI descriptors for all models of [PERSON] and [PERSON] using three different values for \(t\)--two constant and one variable in proportion to the size of each model--is shown. From the results in Figure 4, we conclude that LESI algorithm can differentiate different models of men, if we use constant \(t\) value. However, for the sake of simplicity and consistency, we continue the rest of this section using \(t\) as defined in Equation (7). To demonstrate the power of our method in classifying objects with low intra-class similarity compared with other shape descriptors, the same experiment was carried out on the McGill dataset.