Fuzzy Shape Analysis

Foreword The intention of this report is to list and analyze some of the published results related to different approaches of applying fuzzy set theory to fuzzy shape analysis. These results provide a good background for further development of fuzzy shape analysis methods, which is our main goal. It should be noted that only fuzzy shape analysis techniques are considered, and not other approaches to use fuzzy set theory in (crisp) shape analysis (like, e.g., fuzzy reasoning). Even the methods referring to grey-level images are studied only if their adjustment to fuzzy sets is straightforward, by simple normalization of grey levels to the interval [0, 1]. The focus is, thus, on the shapes obtained by segmentation techniques which assign to the image pixels application-dependent membership values to the fuzzy object (shape). The overall organization of the paper is as follows: At the beginning (Section 1) a brief introduction is given. It refers to the classification and evaluation of existing crisp shape analysis methods, as well as to the general approaches when introducing fuzziness into the binary concepts. Section 2 introduces some basic fuzzy shape definitions. In Sections 3 and 4, we report on a class of shape analysis methods which produce a numerical shape descriptor, such as extent, diameter, area, perimeter, shape signature, Fourier transform based motion descriptor, and moments. Sections 5 is related to shape descrip-tors which produce an image (non-numerical result) as an output; we report on convexity, symmetry, distance transform, medial axis transform, and mathematical morphology, in fuzzy settings. Section 6 contains some comments on the reported results. The reporting style balances between " easy to follow " and " get the information " concept; we would be very glad if both, rather than none, is achieved.