On analysis of discrete spatial fuzzy sets in 2 and 3 dimensions

The use of fuzzy set theoretical approaches for representing spatial relationships provides an intuitive way of expressing the diffuse localization and limits of image components. Fuzziness can be present in images as a consequence of noise introduced during the imaging process, in which case it should preferably be removed, and as imprecision inherent to the observed objects, in which case it provides important information that should be utilized during the image analysis process. A general goal for the research presented in this thesis has been to develop shape analysis methods that can be applied to fuzzy segmented images in 2D and 3D. A demand for the developed methods has been to respect the specific nature of a fuzzy representation of the studied shapes and, especially, the consequences of discretization. We have studied representation and reconstruction of a shape by using moments of both its crisp and fuzzy discretization. We show, both theoretically and statistically, that the precision of estimation of moments of a shape is increased if a fuzzy representation of a shape is used, instead of a crisp one. The signature of a shape based on the distance from the shape centroid is studied and two approaches for its calculation for fuzzy shapes are proposed. A comparison of the performance of fuzzy and crisp approaches is carried out through a statistical study, where a higher precision of shape signature estimation is observed for the fuzzy approaches. The measurements of area, perimeter, and compactness, as well as of volume, surface area, and sphericity, are considered, too. New methods are developed for the estimation of perimeter and surface area of a discrete fuzzy shape. It is shown through statistical studies that the precision of all the observed estimates increases if a fuzzy representation is used and that the improvement is more significant at low spatial resolutions. In addition, a defuzzification method based on feature invariance is designed, utilizing the improved estimates of shape characteristics from fuzzy sets to generate crisp discrete shapes with the most similar shape characteristics. This defuzzification method, performed on a fuzzy segmentation, can be seen as an alternative to a crisp segmentation of an image. The presented results can be applied wherever precise estimates of shape properties are required, especially in conditions of limited spatial resolution. We have showed, either theoretically, or empirically, that membership resolution available can be successfully utilized to overcome a lack of spatial resolution.