16 Curves, surfaces and volumes

16.1 Shape factors and random sets

Shape analysis is a much wider topic than just landmark analysis. We have concentrated most of our work on point set configurations, but the extension to more general objects is widespread.

Some simple measures for describing the shapes of sets include shape factors and other moment-based measures. These shape factors are dimensionless and are often used in microscopy (Exner and Hougardy 1988) and image analysis (Glasbey and Horgan 1995). Commonly used measures include the area–perimeter ratio

(16.1) numbered Display Equation

which has maximum value 1 for a circle and minimum value 0, with A the area and P the perimeter. The elongation shape factor is:

numbered Display Equation

where B is the breadth of the object measured orthogonally to the maximum length L between the two most extreme points. See Stoyan and Stoyan (1994, Chapter 8) for further examples of simple shape measures.

Stoyan and Molchanov (1995) have developed theory for the shapes of set-valued data, and include methodology for the calculation of set-valued means. Distance measures between sets, such as the Hausdorff distance, can be used to develop estimated mean sets in a manner analogous to Fréchet means for shapes discussed in Chapter 6. The distance between a point x and a set C is:

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