Geodesic Convolutional Neural Networks on Riemannian Manifolds

Feature descriptors play a crucial role in a wide range of geometry analysis and processing applications, including shape correspondence, retrieval, and segmentation. In this paper, we introduce Geodesic Convolutional Neural Networks (GCNN), a generalization of the convolutional neural networks (CNN) paradigm to non-Euclidean manifolds. Our construction is based on a local geodesic system of polar coordinates to extract "patches", which are then passed through a cascade of filters and linear and non-linear operators. The coefficients of the filters and linear combination weights are optimization variables that are learned to minimize a task-specific cost function. We use ShapeNet to learn invariant shape features, allowing to achieve state-of-the-art performance in problems such as shape description, retrieval, and correspondence.

1. Introduction

Feature descriptors are ubiquitous tools in shape analysis. Broadly speaking, a local feature descriptor assigns to each point on the shape a vector in some multi -dimensional descriptor space representing the local structure of the shape around that point. A global descriptor describes the whole shape. Local feature descriptors are used in higher-level tasks such as establishing correspondence between shapes [35], shape retrieval [8], or segmentation [43]. Global descriptors are often produced by aggregating local descriptors e.g. using the bag-of-features paradigm. Descriptor construction is largely application dependent, and one typically tries to make the descriptor discriminative (capture the structures that are important for a particular application, e.g. telling apart two classes of shapes), robust (invariant to some class of transformations or noise), compact (low dimensional), and computationally-efficient.