Internal Report. Tutorial: A Critical Assessment of the Use of Riemann Manifolds for Shape Analysis.

The book “The Statistical Theory of Shape ” [?] describes the basic approaches now taken by the Kendall School of shape analysis and covers a subset of the material also to be found in Kendall et. al.’s“Shape and Shape Theory ” [?]. It explains the motivation for, and the approaches to computing statistical tests on measured landmarks. It discusses the fundamental problems associated with this and also solutions described in terms of Riemann manifolds. These manifolds are used in general relativity as a way of encoding the change in oriented space distances (expresses as a ‘metric’) as a function of the local gravitational field. The key point is that the mathematical machinery exists to apply non-linear transformations to these spaces while maintaining meaningful definitions of physical distance. This review compares these approaches with conventional methods for statistical analysis. As a consequence, though the page by page mathematical descriptions of the approach are of course beyond reproach (though perhaps overly “set theoretic ” for a general audience), I have some (I would say substantial) misgivings regarding the practical use of these techniques in scientific studies. Despite the heavy presentation style, this book provides a good opportunity to summarise the appropriate criticisms with respect to what many people would regard to be a fair introduction to the topic. This work has been done for the purposes of documenting my opinions for collaborators and PhD students