Processes such as growth and atrophy cause changes through time that can be visible in a series of medical images, following the hypothesis that form follows function. As was hypothesised by D’Arcy Thompson more than 100 years ago, models of the changes inherent in these actions can aid understanding of the processes at work. We consider how image registration using finite-dimensional planar Lie groups (in contrast to general diffeomorphisms) can be used in this process. The deformations identified can be described as points in the Lie algebra, thus enabling processes such as evolutionary change, growth, and deformation from disease, to be described in a linear space. The choice of appropriate Lie group becomes a modelling choice and can be selected using model selection; Occam’s razor suggests that groups with the smallest number of parameters (which Thompson referred to as ‘simple transformations’) are to be preferred. We demonstrate our method on an example from Thompson of the cannon-bones of three hoofed mammals and a set of outline curves of the development of the human skull, with promising results.
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This research was supported by the Royal Society of New Zealand Marsden Fund (Grant No. MAU0908).
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Marsland, S., McLachlan, R.I. & Zarre, R. Analysing ‘Simple’ Image Registrations. J Math Imaging Vis (2021). https://doi.org/10.1007/s10851-021-01018-2
- Growth and form
- Image registration
- Lie groups