In the AI framework, edge detection is an important task especially when images are affected by uncertainties and/or inaccuracies. Thus, usual edge detectors are unsuitable, so it is necessary to exploit fuzzy tools as Versaci-Morabito edge detector proposing a procedure to adaptively construct fuzzy membership functions. In this chapter, the authors reformulate this approach exploiting a new formulation for adaptively fuzzy membership functions but characterized by a more reduced computational load making the approach more attractive for any real-time applications. Furthermore, the chapter provides new mathematical results not yet proven in previous works.
TopIntroduction
Edge detection marks points in a digital image where light intensity changes abruptly. Abrupt changes in the properties of an image are symptom of events or changes in the physical world of which the images are the representation. These changes can be: discontinuity in depth and orientation of surfaces, modification of material properties and variations in lighting from the surrounding environment. Outline recognition is a research field of image processing, in particular of the branch of feature extraction.
The contour recognition generates images containing much less information than the original ones, since it eliminates most of the details not relevant to the identification of the contours, while preserving the essential information to describe the shape and structural and geometric characteristics of the objects represented. As first step it is possible to use the gradient of the image which will have a low value in correspondence of areas with few changes in intensity. By applying a threshold to the latter, one is able to generate a new image containing only the outlines of the object depicted, ignoring any disturbances.
There are many other methods for recognizing contours groupable into two categories: search-based and zero-crossing methods. Search-based methods recognize boundaries by looking for the maxima/minima of the I- derivative of the image, looking for the direction where the maximum local gradient occurs. Zero-crossing methods look for points where the II- derivative passes through zero, usually the Laplacian function or a differential expression of a non-linear function. More systematically, the standard edge detection procedures can be classified into I and II order procedures. For example, the gradient operator technique, together the procedure elaborated by Roberts Prewitt and Sobel belong to I order procedures; while the Laplacian of Gaussian technique and the Zero Cross Operator are II order procedures. In 1986 the American John F. Canny designed an algorithm for the recognition of the contours that is now defined as the gold standard in this field.
However, the images can be affected by uncertainties and/or inaccuracies so the usual edge detection techniques fail or, at best, do not provide optimal results. Therefore, it is necessary to use fuzzy techniques which, notoriously, manage the problem with high levels of performances: on the one hand, makes the approach more “readable” ever by non-experts and, on the other hand, favors its updating through the expert knowledge. Many researchers are actively engaged in research activities concerning fuzzy edge detection with highly competitive results. Among these, the work published in (Versaci & Morabito, 2021) based on fuzzy entropy computations, presented results almost comparable to the results achieved by the Canny procedure. However, in (Versaci & Morabito, 2021) many mathematical statements have only been intuitively verified without rigorous mathematical proof. Furthermore, the adaptive construction procedure of the fuzzy membership functions used there required a considerable computational effort that hardly conciles with any real-time applications. So, the goal of this chapter is twofold: to complete what is presented in (Versaci & Morabito, 2021) by a set of Propositions with rigorous proofs that fill in the aforementioned gaps; to propose a new procedure for the adaptive construction of fuzzy membership functions with reduced computational complexity (replacing the one proposed in (Versaci & Morabito, 2021)) without experiencing appreciable drops in performance (testing the procedure on a larger database of images than stated in (Versaci & Morabito, 2021)). Finally, for the sake of completeness, the chapter also offers a broader description of the more well-known edge detection techniques currently in use than those reported in (Versaci & Morabito, 2021).