Registration

A surveyor returns from three days at a stone church with twelve terrestrial scans, each describing part of the building in its own scanner-local coordinate frame. Until the twelve frames agree on a common reference, the building does not exist in the dataset as one object, only as twelve disconnected fragments. And the agreement must be exact: sub-millimetre drift per station compounds, by the time the alignment reaches the apse on the far side of the nave, into doors that no longer close in the model and walls that diverge from the truth by several centimetres.

Registration is the hinge of the processing pipeline. It consumes the cleaned clouds of Chapter 6 and the normals of Chapter 7, on which its most accurate variants depend, and it leans on the spatial indexes of Chapter 4 for the millions of nearest-neighbour queries it issues. Everything downstream assumes its output: classification (Chapter 12), change detection (Chapter 14), and surface reconstruction (Chapter 15) all operate on the single unified cloud that registration delivers, and the SLAM systems met at the end of this chapter are registration run in real time. The chapter builds the family from the inside out: the fine alignment of ICP and its variants, the coarse methods that supply ICP's starting point, and the pose-graph machinery that scales pairwise alignment to a whole campaign.

We begin where every method must, with a precise statement of the problem.

The registration problem

Each scan in the surveyor's dataset captures part of the building from its own local coordinate system. To produce a complete 3D model, the scans must be stitched together by finding the spatial transformation that maps each scan into a common reference frame. The rigid transformation preserves all distances and angles within the cloud, leaving the object's shape and size unchanged. Formally, the pair $(\mathbf{R}, \mathbf{t})$ is chosen to minimise

where $d(\cdot, \mathcal{P}_T)$ is a distance metric from the transformed source point to the target cloud, and the $w_i$ are optional weights that downweight unreliable correspondences.