# An Application of Manifold Learning in Global Shape Descriptors
## 1 Introduction  
Three-dimensional models are ubiquitous data in the form of 3D surface meshes, point clouds, volumetric data, etc. in a wide variety of domains such as material and mechanical engineering [1], genetics [2], molecular biology [3], entomology [4], and dentistry [5, 6], to name a few. Processing such large datasets (e.g., shape retrieval, matching, or recognition) is computationally expensive and memory intensive. For example, to query against a large database of 3D models to find the closest match for a 3D model of interest, one needs to develop an appropriate similarity measure as well as an efficient algorithm for search and retrieval. Shape descriptors assist with the example problem by providing discriminating feature vectors for shape retrieval [7, 8] and play a fundamental role when dealing with shape analysis problems such as shape matching [9, 10] and classification [11]. In general, there are two types of shape descriptors: _local descriptors_, also called _point signatures_, and _global descriptors_, referred to as _shape fingerprints_. A local shape descriptor computes a feature vector for every point of a 3D model. On the other hand, a global shape descriptor represents the whole 3D shape model in the form of a low-dimension vector. A descriptor that is informative and concise captures as much information as possible from the 3D shape including the geometric and topological features. Such a vector drastically lowers the shape analysis burdens in terms of both computational intensity and memory. While many successful non-spectral shape descriptors have been proposed in the literature, spectral descriptors have proved to be beneficial in many applications [12; 13]. The spectral methods take advantage of eigenvalues and eigenvectors computed from the eigen-decomposition of the Laplace-Beltrami (LB) operator applied on the surface of 3D shapes. These methods have found successful applications in graph processing [14], computational biology [15], and point-to-point correspondence [16]. Several comprehensive surveys [17; 18; 19] have been conducted on studying and classifying data-driven 3D shape descriptors to which we refer interested readers for more information. Our objective in this paper is to develop a concise and informative global, spectral shape descriptor. One of the first spectral descriptors introduced to the computer graphics community is Shape-DNA, developed by [PERSON] et al. in 2006 [20]. Shape-DNA attracted a great deal of attention for its unique isometric and rotation invariant features [20]. Since then, several local as well as global shape descriptors have been introduced in accordance with Shape-DNA such as Heat Kernel Signature (HKS) [21], Wave Kernel Signature (WKS) [22], and Global Point Signature (GPS) [23]. The common ground between these methods is the discretization approach used to solve the Laplacian eigenvalue problem, which uses a _cotangent weighting scheme_ along with area normalization. Although there are many advantages of using variations of the cotangent scheme, there are several limitations. First, by their nature, they are limited to triangulated meshes. Second, they do not perform well when dealing with degenerate and non-uniform sampled meshes [24; 25]. In addition, their convergence error depends on factors such as the linearity of the function on the surface [25]. One possible approach to address these limitations is through the use of manifold learning, which is investigated in the current contribution. Nonlinear dimensionality reduction techniques, known as _manifold learning_, assume the existence of a low-dimensional space, in which, a high-dimensional manifold can be represented without much loss of information [26]. Similar to global descriptors, manifold learning methods attempt to learn the geometry of a manifold in order to extract a low dimensional vector of features that is informative and discriminative. However, unlike shape descriptors, the number of dimensions of a space does not confine manifold learning methods.