Compact Figures in Digital Pictures

Abstract

The square of the perimeter of a figure divided by its area P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /A is a classical measure of the noncompactness of the figure. In Euclidean geometry, P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /A is always at least 4π, and takes on this minimum value only if the figure is a circle; larger values correspond to figures that are less compact than circles. In digital pictures, however, certain squares or octagons-depending on how P is measured-yield smaller values than do digitized circles. Thus for digital pictures, though P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /A may be useful as a practical measure of noncompactness, it possesses certain pathological features of which users should be aware.