infocus #52 December 2018 The Problem of Perimeter

In Euclidean geometry, perimeter is a defined and readily measured property of shapes. In the real world, and especially in digitised images of real world objects, it is less well defined and much less readily measured, in spite of the fact that practically all image analysis programmes report numeric values, and use them in shape characterisation.

DOI: 10.22443/rms.inf.1.167

Few real world objects have ideal Euclidean boundaries; most reveal more and more irregularities as magnification increases. Whether this increase corresponds to exactly fractal scaling or not, it still implies that measured perimeter is an artefact of the resolution used to examine the object.

When an image of the object is digitised, it becomes necessary to decide how to define the boundary and the pixels that determine it, and how to measure the boundary length. Several different definitions and algorithms, with very different levels of computational requirements, give rise to quite different measurement results.

These are illustrated and compared, including ones not known to be previously published. Difficulties in extending the methods to 3D surface area measurements are
shown.

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