Point set registration

In computer vision and pattern recognition, point set registration, also known as point matching, is the process of finding a spatial transformation that aligns two point sets. The purpose of finding such a transformation includes merging multiple data sets into a globally consistent model, and mapping a new measurement to a known data set to identify features or to estimate its pose. A point set may be raw data from 3D scanning or an array of rangefinders. For use in image processing and feature-based image registration, a point set may be a set of features obtained by feature extraction from an image, for example corner detection. Point set registration is used in optical character recognition and aligning data from magnetic resonance imaging with computer aided tomography scans.

The problem may be summarized as follows: Let ${\displaystyle \lbrace {\mathcal {M}},{\mathcal {S}}\rbrace }$ be two finite size point sets in a finite-dimensional real vector space ${\displaystyle \mathbb {R} ^{d}}$, which contain ${\displaystyle M}$ and ${\displaystyle N}$ points respectively. The problem is to find a transformation to be applied to the moving "model" point set ${\displaystyle {\mathcal {M}}}$ such that the difference between ${\displaystyle {\mathcal {M}}}$ and the static "scene" set ${\displaystyle {\mathcal {S}}}$ is minimized. In other words, a mapping from ${\displaystyle \mathbb {R} ^{d}}$ to ${\displaystyle \mathbb {R} ^{d}}$ is desired which yields the best alignment between the transformed "model" set and the "scene" set. The mapping may consist of a rigid or non-rigid transformation. The transformation model may be written as ${\displaystyle T}$ where the transformed, registered model point set is:

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