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The Frenet-Serret framework for aligning geometric curves

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Abstract

Variations of the curves and trajectories in 1D can be analysed efficiently with functional data analysis tools. The main sources of variations in 1D curves have been identified as amplitude and phase variations. Dealing with the latter gives rise to the problem of curve alignment and registration problems. It has been recognised that it is important to incorporate geometric features of the curves in developing statistical approaches to address such problems. Extending these techniques to multidimensional curves is not obvious, as the notion of multidimensional amplitude can be dened in multiple ways. We propose a framework to deal with the curve alignment in multidimensional curves as 3D objects. In particular, we propose a new distance between the curves that utilises the geometric information of the curves through the Frenet-Serret representation of the curves. This can be viewed as a generalisation of the elastic shape analysis based on the square root velocity framework. We develop an efficient computational algorithm to find an optimal alignment based on the proposed distance using dynamic programming.