Geometric Deep Learning: Going beyond Euclidean data

We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.

Section: Discussionmentioning

confidence: 99%

Functional Maps Representation On Product Manifolds

Rodolà

Lähner²,

Bronstein

et al. 2019

“…For example, the number of 2D views rendered from a 3D model must be large enough so that the entire shape surface is captured as accurate as possible [KAMC17]. Usually, these approaches (and the volumetric representation which will come later) miss some levels of details during the process of transforming 3D shapes into 2D views [BBL*17]. Besides, different parts of a shape may be appeared in more than one view, therefore, an effective technique needs to be taken to unify the parts specifically [KAMC17].…”

Section: Discussionmentioning

confidence: 99%

“…For more information on applying CNN to spectral domain, we refer the readers to a recent article by Bronstein et al . [BBL*17]. They present an in‐depth discussion on the considerations for CNN applications in shape analysis tasks for spatial and spectral domains.…”

Section: Data‐driven 3d Shape Descriptorsmentioning

confidence: 99%

A Survey on Data‐Driven 3D Shape Descriptors

Rostami

Bashiri

Rostami

et al. 2018

Recent advances in scanning device technologies and improvements in techniques that generate and synthesize 3D shapes have made 3D models widespread in various fields including medical research, biology, engineering, etc. 3D shape descriptors play a fundamental role in many 3D shape analysis tasks such as point matching, establishing point‐to‐point correspondence, shape segmentation and labelling, and shape retrieval to name a few. Various methods have been proposed to calculate succinct and informative descriptors for 3D models. Emerging data‐driven techniques use machine learning algorithms to construct accurate and reliable shape descriptors. This survey provides a thorough review of the data‐driven 3D shape descriptors from the machine learning point of view and compares them in different criteria. Also, a comprehensive taxonomy of the existing descriptors is proposed based on the exploited machine learning algorithms. Advantages and disadvantages of each category have been discussed in detail. Besides, two alternative categorizations from the data type and the application perspectives are presented. Finally, some directions for possible future research are also suggested.

“…In the recent years, several methods have been proposed for analyzing and processing 3D shapes by building on the success of machine learning methods, and especially those based on deep learning (see, for example, recent overviews in [MRB*16, BBL*17]). These methods are based on the notion that the solutions to many problems in geometric data analysis can benefit from large data repositories.…”

Section: Related Workmentioning

confidence: 99%

PCPNet Learning Local Shape Properties from Raw Point Clouds

Guerrero

Kleiman

Ovsjanikov

et al. 2018

275

251

In this paper, we propose PCPNET, a deep‐learning based approach for estimating local 3D shape properties in point clouds. In contrast to the majority of prior techniques that concentrate on global or mid‐level attributes, e.g., for shape classification or semantic labeling, we suggest a patch‐based learning method, in which a series of local patches at multiple scales around each point is encoded in a structured manner. Our approach is especially well‐adapted for estimating local shape properties such as normals (both unoriented and oriented) and curvature from raw point clouds in the presence of strong noise and multi‐scale features. Our main contributions include both a novel multi‐scale variant of the recently proposed PointNet architecture with emphasis on local shape information, and a series of novel applications in which we demonstrate how learning from training data arising from well‐structured triangle meshes, and applying the trained model to noisy point clouds can produce superior results compared to specialized state‐of‐the‐art techniques. Finally, we demonstrate the utility of our approach in the context of shape reconstruction, by showing how it can be used to extract normal orientation information from point clouds.