Compact Figures in Digital Pictures

摘要

The square of the perimeter of a figure divided by its area P2/A is a classical measure of the noncompactness of the figure. In Euclidean geometry, P2/A is always at least 4????????, 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 P2/A may be useful as a practical measure of noncompactness, it possesses certain pathological features of which users should be aware.