Fuzzy Geometry

The digital geometry plays a key role in many image processing applications. Generally, the gray-level images will be thresholded to calculate geometrical measures such as area, perimeter, diameter, compactenss etc. of an object. Since the images or their segments have ill-defined or non-crisp boundaries, it is sometimes appropriate to consider them as fuzzy sets. Azriel Rosenfeld has introduced the concept of fuzzy digital geometry:

"The standard approach to image analysis and recognition begins by segmenting the image into regions and computing various properties of and relationships among these regions. However, the regions are not always 'crisply' defined; it is sometimes more appropriate to regard them as fuzzy subsets of the image... It is not always obvious how to measure geometrical properties of fuzzy sets, but definitions have been given and basic properties established for a variety of such properties and relationships, including connectedness and surroundedness, convexity, are, perimeter and compactness, extent and diameter"
(A. Rosendfeld, The Fuzzy Geometry of Image Subsets).

For instance, if we have to calculate the elongatedness of an object, the thresholding step is often not optimal (loss of information, see Fig.1). Therefore, it is more suitable to operate on the original gray-level image and not commit ourselves to such crisp classifications.

If we define an appropriate membership function m for the object, the area a and the perimeter p of the object can be calculated as follows:

Area of a fuzzy sets:

Perimeter of a fuzzy set:

Based on area and perimeter, the compactness of a fuzzy set can be determined as follows:

Other measures and relationships (such as length, width, medial axis, density etc., see Table 1) are defined in the literature (see the papers of Azriel Rosenfeld, Sankar K. Pal, Ashish Ghosh). The application areas of fuzzy geometry are image enhancement, segmentation and representation.

Table 1. Different topics of fuzzy geometry in image processing
(From: Tizhoosh, H.R., On a systematic Introduction into Fuzzy Image Processing (in German). In Proc. AFN'97 Annual Meeting, Magdeburg, Germany, pp. 39-45)