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# Deep Projective Rotation Estimation through Relative Supervision

*Proceedings of The 6th Conference on Robot Learning*, PMLR 205:1575-1585, 2023.

#### Abstract

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.