Fig. 1. Low discrimination power of the definition of degree of intersection between two fuzzy sets using the maximum of intersection: μ_int(μ, ν′)=μ_int(μ, ν), although μ and ν′ strongly overlap and should be considered as more intersecting than μ and ν.

Fig. 2. Illustration of Eq. (34) when using different definitions for the degree of intersection. Using the maximum of the intersection we obtain μ_adj(μ, ν)=α₁ (=0.36) and μ_adj(μ, ν′)=α₃ (=0.35), and using the fuzzy hypervolume μ_adj(μ, ν)=α₂ (=0.67) and μ_adj(μ, ν′)=α₄ (=0.34).

Fig. 3. Magnetic resonance image of a brain (one slice of a 3D volume).

Fig. 4. Top: 5 fuzzy objects representing a few internal brain structures of the image shown in Fig. 3 (membership values rank between 0 and 1, from white to black). From left to right: right caudate nucleus (cn2), right lateral ventricle (v2), left lateral ventricle (v1), left caudate nucleus (cn1), left putamen (p1). Bottom: superimposition of these fuzzy objects (the maximum membership value is displayed at each point).

Fig. 5. Left: a fuzzy reference object. Right: fuzzy landscape representing the relationship ‘to the left of’ obtained by a fuzzy dilation by a directional structuring element.

Fig. 6. Two examples where the relative position of objects with respect to the reference object is difficult to define in a ‘all-or-nothing’ manner.

Translation of crisp concepts in their fuzzy equivalents

Results obtained for fuzzy adjacency

Labels of structures are given in Fig. 4. High degrees are obtained between structures where adjacency is expected, while very low degrees are obtained in the opposite case.

Distances between fuzzy sets of Fig. 4 using the definitions involving comparison of membership functions only

Distances between fuzzy sets of Fig. 4 using the geometrical approach: weighted average distance using 3 different t-norms, fuzzification (using integral over α-cuts) of mean, min and Hausdorff distances

Distances between fuzzy sets of Fig. 4 using the morphological approach, for the nearest point distance and the Hausdorff distance, using two different t-norms

Relative position of object A (rectangle) with respect to object R (square) of Fig. 6, using centroid, aggregation, compatibility methods, and using the morphological approach (the [N, Π] intervals are given, as well as the average value)

Angle or force histograms are computed using r=0, r=2 and r=5 (the angle histogram method corresponds to r=0).

Relative position of object B with respect to object R of Fig. 6, using centroid, aggregation, compatibility methods, and using the morphological approach (the [N, Π] intervals are given, as well as the average value)

Angle or force histograms are computed using r=0, r=2 and r=5.