  Orientation estimation is the core to a variety of vision and robotics tasks
such as camera and object pose estimation. Deep learning has offered a way to
develop image-based orientation estimators; however, such estimators often
require training on a large labeled dataset, which can be time-intensive to
collect. In this work, we explore whether self-supervised learning from
unlabeled data can be used to alleviate this issue. Specifically, we assume
access to estimates of the relative orientation between neighboring poses, such
that can be obtained via a local alignment method. While self-supervised
learning has been used successfully for translational object keypoints, in this
work, we show that naively applying relative supervision to the rotational
group $SO(3)$ will often fail to converge due to the non-convexity of the
rotational space. To tackle this challenge, we propose a new algorithm for
self-supervised orientation estimation which utilizes Modified Rodrigues
Parameters to stereographically project the closed manifold of $SO(3)$ to the
open manifold of $\mathbb{R}^{3}$, allowing the optimization to be done in an
open Euclidean space. We empirically validate the benefits of the proposed
algorithm for rotational averaging problem in two settings: (1) direct
optimization on rotation parameters, and (2) optimization of parameters of a
convolutional neural network that predicts object orientations from images. In
both settings, we demonstrate that our proposed algorithm is able to converge
to a consistent relative orientation frame much faster than algorithms that
purely operate in the $SO(3)$ space. Additional information can be found at
https://sites.google.com/view/deep-projective-rotation/home .
