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.

遵循形式遵循功能的假设，诸如生长和萎缩之类的过程会导致时间变化，在一系列医学图像中可以看到这些变化。正如100多年前达西·汤普森（D'Arcy Thompson）所假设的那样，这些动作所固有的变化模型可以帮助理解工作流程。我们考虑如何在此过程中使用使用有限维平面Lie组的图像配准（与一般衍射同构）。可以将识别出的变形描述为李代数中的点，因此可以在线性空间中描述诸如进化变化，生长和疾病变形等过程。适当的李群的选择成为建模选择，可以使用模型选择进行选择。奥卡姆（Occam）的剃刀建议使用参数数量最少的组（汤普森称为“简单转换”）。我们以汤普森的三个有蹄哺乳动物的大炮骨为例以及一组人类头骨发育的轮廓曲线展示了我们的方法，并取得了可喜的结果。