An Application of Manifold Learning in Global Shape Descriptors

Fereshteh S Bashiri,Peggy Peissig,Roshan M D’Souza,Reihaneh Rostami,Zeyun Yu

doi:10.3390/a12080171

Fereshteh S Bashiri, Peggy Peissig + Show 3 more

Open Access

https://doi.org/10.3390/a12080171

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Abstract

With the rapid expansion of applied 3D computational vision, shape descriptors have become increasingly important for a wide variety of applications and objects from molecules to planets. Appropriate shape descriptors are critical for accurate (and efficient) shape retrieval and 3D model classification. Several spectral-based shape descriptors have been introduced by solving various physical equations over a 3D surface model. In this paper, for the first time, we incorporate a specific manifold learning technique, introduced in statistics and machine learning, to develop a global, spectral-based shape descriptor in the computer graphics domain. The proposed descriptor utilizes the Laplacian Eigenmap technique in which the Laplacian eigenvalue problem is discretized using an exponential weighting scheme. As a result, our descriptor eliminates the limitations tied to the existing spectral descriptors, namely dependency on triangular mesh representation and high intra-class quality of 3D models. We also present a straightforward normalization method to obtain a scale-invariant and noise-resistant descriptor. The extensive experiments performed in this study using two standard 3D shape benchmarks—high-resolution TOSCA and McGill datasets—demonstrate that the present contribution provides a highly discriminative and robust shape descriptor under the presence of a high level of noise, random scale variations, and low sampling rate, in addition to the known isometric-invariance property of the Laplace–Beltrami operator. The proposed method significantly outperforms state-of-the-art spectral descriptors in shape retrieval and classification. The proposed descriptor is limited to closed manifolds due to its inherited inability to accurately handle manifolds with boundaries.

Highlights

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
Some other manifold learning methods, e.g., Isomap, LLE, and Diffusion map, are based on spectral analysis of the high-dimensional manifold. In contrast to these methods that construct the orthogonal basis of their desired low-dimensional space using eigenfunctions of an LB operator, we develop our scale-invariant shape descriptor using the spectrum of the LB operator
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

Summary

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. 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. A local shape descriptor computes a feature vector for every point of a 3D model. 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. The LB operator is positive semi-definite, having non-negative eigenvalues λi that can be sorted as follows:

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