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.

本文提出了一种基于学习的框架来从实际操作符编码的小型矩阵中重建三维形状，并引入了一种名为 OperatorNet 的新型神经网络，该网络以表示形状的一组线性操作符作为输入，并生成其 3D 嵌入。我们展示了这种方法在同一问题的纯几何方法中具有明显的优势，并演示了如何使用这些操作符合成全新的未见过的形状，并将其应用于形状插值和类比应用中。