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$SE(3)$ Equivariant Convolution and Transformer in Ray Space

3D reconstruction and novel view rendering can greatly benefit from geometric priors when the input views are not sufficient in terms of coverage and inter-view baselines. Deep learning of geometric priors from 2D images requires each image to be represented in a $2D$ canonical frame and the prior to be learned in a given or learned $3D$ canonical frame. In this paper, given only the relative poses of the cameras, we show how to learn priors from multiple views equivariant to coordinate frame transformations by proposing an $SE(3)$-equivariant convolution and transformer in the space of rays in 3D. We model the ray space as a homogeneous space of $SE(3)$ and introduce the $SE(3)$-equivariant convolution in ray space. Depending on the output domain of the convolution, we present convolution-based $SE(3)$-equivariant maps from ray space to ray space and to $\mathbb{R}^3$. Our mathematical framework allows us to go beyond convolution to $SE(3)$-equivariant attention in the ray space. We showcase how to tailor and adapt the equivariant convolution and transformer in the tasks of equivariant $3D$ reconstruction and equivariant neural rendering from multiple views. We demonstrate $SE(3)$-equivariance by obtaining robust results in roto-translated datasets without performing transformation augmentation.

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