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Some remarks on the Frechet means and the shape of the means in shape spaces

Last modified: 2013-06-14

#### Abstract

Inference in shape analysis is related to problems where the invariance of rotations and translations (and possibly scaling) of data objects is required. Points which determine the shape of the object of interest (called landmarks) are usually observed in some arbitrary system of coordinates and possibly arbitrary scale. We need to make statistical inference which implicitly inherits the invariance of our data to the shape preserving transformations. Two different approaches have been used in shape analysis. In the first case a model for the landmarks is proposed and then one works with the induced distribution of the shapes, and a second approach includes working with models for shapes directly and a Frechet mean is of interest. In general, the shape of the means (from the first model) and the Frechet means (from the second model) are not the same. In this talk, we will identify the situations where these means coincide and in the situations where they differ we show that we can make some useful adaptations.

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