Soft Mathematical Morphology: Extensions, Algorithms and Implementations

Preface Linear filters have been used as the primary tools in signal and image processing. They exhibit many desirable properties, such as the superposition, which makes them easy to design and implement. Furthermore, they are the best solution in additive Gaussian noise suppression. However, when the noise is not additive or when there exist system non-linearities linear filters are not adequate. For example all image processing is, of necessity, non-linear. In such cases non-linear filters should be utilized. The advantage of non-linear filters is their ability to pass structural information while suppressing noise or removing clutter. Pattern and edge information are often crucial to image understanding, and in many circumstances it is possible to design non-linear filters that pass structural information in a manner superior to that of linear filters. Non-linear filtering has been developed along three lines: logical (set), geometrical (shape) and numerical (order based). These three approaches are deeply interrelated. Sh ape-based non-linear filtering is centered around mathematical morphology. Soft morphological filters are a relatively new subclass of non-linear filters. They combine morphological filters with weighted order statistics filters. Standard morphological opera tors are defined using local min and max operations. In soft mathematical morphology rather flexible definitions are provided, by substituting the min and max operators with general weighted order statistics. Soft morphological filters have been introduced to improve the results of standard morphological filters in noisy conditions. In this paper the trends in soft mathematical morphology are described. Fuzzy soft mathematical morphology applies the concepts of soft morphology to fuzzy sets. The definitions of the basic fuzzy soft morphological operations and their algebraic properties are provided. An algorithm for soft morphological structuring element decomposition is 3 also presented. Furthermore, techniques for software and hardware implementation of soft morphological operations are included.