Spherical convolutional neural networks have been introduced recently as a tool to learn powerful feature representations of 3D shapes. Since spherical convolutions are equivariant to 3D rotations, the latent space of a SphericalCNN provides a natural representation for applications where 3D data may be observed in arbitrary orientations.
In this paper we explore if it is possible to learn 2D image embeddings with a similar equivariant structure: embedding the image of a 3D object should commute with rotations of the object. Our proposal is to bootstrap our model with supervision from a Spherical CNN pretrained with 3D shapes. Given an equivariant latent representation for 3D shapes, we introduce a novel supervised cross-domain embedding architecture that learns to map 2D images into the Spherical CNN's latent space. Our model is only optimized to produce the embeddings from an image's corresponding 3D shape. The trained model learns to encode images with 3D shape properties and is equivariant to 3D rotations of the observed object.
We show that learning only a rich embedding for images with appropriate geometric structure is in and of itself sufficient for tackling numerous applications. We show evidence from two different applications, relative pose estimation and novel view synthesis. In both settings we demonstrate that equivariant embeddings are sufficient for the application without requiring any task-specific supervised training.