Descriptors of Two-Dimensional Fuzzy Geometric Objects

This paper studies basic concepts of fuzzy planimetry and their representation in digital images. Several shape descriptors and their associated measures—area, perimeter, diameter, height, and width—are introduced and calculated for fuzzy figures such as the fuzzy square, fuzzy rectangle, fuzzy disc, and fuzzy ellipse. A new fuzzy shape descriptor, elongation, is proposed, defined through the widths of the fuzzy set in two orthogonal directions; it takes values in the interval [1,∞), is theoretically well grounded, and behaves in a way that matches human perception and can be predicted in advance. This shape-based measure is invariant to rotation and translation of the underlying fuzzy shape. Elongation is further adapted to digital images, i.e., to digital fuzzy shapes, for which its definition is suitably adjusted. Experiments on synthetically generated digital fuzzy images illustrate the behaviour of the new measure and confirm the theoretical results obtained for it.