Geodesic convolutional neural networks on Riemannian manifolds

TL;DR

This paper introduces Geodesic Convolutional Neural Networks (GCNN), extending CNNs to non-Euclidean manifolds for improved shape analysis tasks like retrieval and correspondence.

Contribution

It presents a novel GCNN framework that operates on Riemannian manifolds using geodesic polar coordinates and learned filters for shape analysis.

Findings

Achieves state-of-the-art results in shape retrieval.

Demonstrates effective shape correspondence and description.

Provides a general approach for non-Euclidean deep learning.

Abstract

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 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 GCNN to learn invariant shape features, allowing to achieve state-of-the-art performance in problems such as shape description, retrieval, and correspondence.