OperatorNet: Recovering 3D Shapes From Difference Operators

TL;DR

OperatorNet is a novel neural framework that reconstructs 3D shapes from functional operators, outperforming geometric methods and enabling shape synthesis through operator algebra.

Contribution

The paper introduces OperatorNet, a neural architecture for 3D shape reconstruction from functional operators, and a new extrinsic operator for improved accuracy and shape synthesis.

Findings

Outperforms previous geometric methods in shape reconstruction

Achieves high accuracy even with incomplete shape information

Enables shape interpolation and analogy using operator algebra

Abstract

This paper proposes a learning-based framework for reconstructing 3D shapes from functional operators, compactly encoded as small-sized matrices. To this end we introduce a novel neural architecture, called OperatorNet, which takes as input a set of linear operators representing a shape and produces its 3D embedding. We demonstrate that this approach significantly outperforms previous purely geometric methods for the same problem. Furthermore, we introduce a novel functional operator, which encodes the extrinsic or pose-dependent shape information, and thus complements purely intrinsic pose-oblivious operators, such as the classical Laplacian. Coupled with this novel operator, our reconstruction network achieves very high reconstruction accuracy, even in the presence of incomplete information about a shape, given a soft or functional map expressed in a reduced basis. Finally, we demonstrate that the multiplicative functional algebra enjoyed by these operators can be used to synthesize entirely new unseen shapes, in the context of shape interpolation and shape analogy applications.