On the perimeter and area of fuzzy sets

Fuzzy Distance Transform: Theory, Algorithms, and Applications

“…The following notions on fuzzy subsets are used in this paper; see [27,28] for details. A fuzzy subset S is a ring if µ S (x) =μ(r ), where r = x − x 0 for some x 0 ∈ n and µ : → [0, 1] is a membership function.…”

Perimeter and Area Estimations of Digitized Objects with Fuzzy Borders

“…We are particularly interested in the results of Rosenfeld [6] and Bogomolny [1]. Both papers are related to fuzzy subsets in the continuous domain, and utilize the gradient of a fuzzy membership function for the calculation of a perimeter of a set.…”

“…However, some simple interrelations, e.g., the isoperimetric inequality, that hold in the crisp case, do not hold if the perimeter and area are defined as in [6]. This fact initialized further research and resulted in the modified definition of the perimeter of a fuzzy subset [1].…”

“…We are interested in the results related to the fuzzy step subsets which were considered in both [6] and [1]. We utilize the following definitions, cited from [10] (Definition 1) and [1] (Definitions 2-5).…”

“…We utilize the following definitions, cited from [10] (Definition 1) and [1] (Definitions 2-5). Note: A fuzzy digital image is a fuzzy step subset, where sets S i are determined with respect to the connected sets of pixels having the same membership (greylevel) value.…”