Fuzzy adjacency between image objects

The notion of adjacency has a strong interest for image processing and pattern recognition, since it denotes an important relationship between objects or regions in an image, widely used as a feature in model-based pattern recognition. A crisp definition of adjacency often leads to low robustness in the presence of noise, imprecision, or segmentation errors. We propose two approaches to cope with spatial imprecision in image processing applications, both based on the framework of fuzzy sets. These approaches lead to two completely different classes of definitions of a degree of adjacency. In the first approach, we introduce imprecision as a property of the adjacency relation, and consider adjacency between two (crisp) objects to be a matter of degree. We represent adjacency by a fuzzy relation whose value depends on the distance between the objects. In the second ap approach, we introduce imprecision (in particular spatial imprecision) as a property of the objects, and consider objects to be fuzzy subsets of the image space. We then represent adjacency by a relation between fuzzy sets. This approach is, in our opinion, more powerful and general. We propose several ways for extending adjacency to fuzzy sets, either by using alpha-cuts, or by using a formal translation of binary equations into fuzzy ones. Since set equations are more easily translated into fuzzy terms, we shall privilege set representations of adjacency, particularly in the framework of fuzzy mathematical morphology. Finally, we give some hints on how to compare degrees of adjacency, typically for applications in model-based pattern recognition.