Minimum point correspondences between two point clouds to solve for transformation between them - for Wahba and Orthogonal Procrustes algorithms

Task at hand

We investigate what the minimum number of corresponding points needed is to register two point sets. This problem seems deceptively simple but I couldn't find a clear answer in standard computer vision texts. Therefore, this post offers a concise answer from multiple perspectives: theoretical, algorithmic and from geometric intuition by going through the original papers from the 1960s.

What you are reading is the main blog post providing theoretical/geometric justification.

This repo supplements the blog post by providing the code in the form of ipython notebook:

GitHub - Shubodh/3D_point-set-registration: A tutorial on 3D point set registration using SVD based approaches. We also investigate further objectives such as min. no. of point correspondences.

Resources

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Citation

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Question

Given P1 and P2 as a set of two-point clouds in frames 1 and 2, how many point correspondences are needed between the two frames to solve for the transformation between them? Why?

Given two corresponding point sets:

$$ \begin{array}{l} Q=\left\{q_{1}, \ldots, q_{N}\right\} \\ P=\left\{p_{1}, \ldots, p_{N}\right\} \end{array} $$

with correspondences $C=\{(i, j)\}$.