The structure of shape and size variability is the primary concern in most applications (see Bookstein, 2016). However, the concept of a mean shape or mean size-and-shape also has an underpinning role to such analysis, for example when choosing a tangent space projection. We consider a situation where a population of shapes or size-and-shapes can be modelled by a probability distribution. Early definitions of a mean in a general manifold were given by Fréchet (1948) and Grove and Karcher (1973).

**Definition 6.1** *If d is a choice of distance in a manifold M then a* **population mean** *is given by:*

(6.1)

where *E _{X}*[ · ] is the expectation operator with respect to the random quantity

If μ_{M} is a global minimum then it has been called a ‘Fréchet mean’ in the literature, and the term ‘Karcher mean’ has been used when it is a local minimum, after the work of Grove and Karcher (1973) and Karcher (1977) on Riemannian centres of mass. Although the terminology is not universally applied, we shall use these terms in order to help distinguish global and local minimizers. A detailed history is given by Afsari (2011).

**Definition 6.2** *A* **population variance** *is:*

(6.2)

and if it is a global minimum then σ^{2}_{M} is called the Fréchet variance.

In the ...

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