Define, Refine, Align: Correspondence-free 3D Line Alignment with Attentional, Equivariant and Rotational Layers

Lines can be extracted from sensors working beyond the visible spectrum, such as active depth cameras (e.g., infrared-based structured light or time-of-flight), LiDAR scanners, and thermal infrared imagers. Lines, however, are a far less popular representation of 3D scenes compared to point clouds. We wish to bridge this gap by introducing \textbf{Define, Refine, Align (DRA)}, a pipeline for aligning unordered 3D line sets without the need for prior correspondence evaluation. By operating in the Geometric Algebra $G(4,0,0)$, DRA leverages the expressive power of hypercomplex networks to represent both line bundles and their relative poses as multivector objects. Since a 3D scene can often be modeled as a bundle of lines, reconstructing it from multiple views necessitates the alignment of partially overlapping line sets. Conventional methods typically decompose this task into two sequential stages that are computationally intensive and prone to high latency, thus limiting real-time applications where acquisition is noisy and swift alignment is crucial. In contrast, DRA directly estimates the pose, offering a robust and efficient solution for real-time 3D line alignment. DRA outperforms all baselines that do not require the explicit evaluation of line correspondences on both indoor and outdoor scenes by up to $89.1\%$, as well as some correspondence-guided cases. We believe that the use of geometry-informed pipelines can constitute a promising research direction towards robust registration of partially overlapping 3D lines.