A fuzzy morphological hit-or-miss transform for grey-level images: A new approach

The translation of the hit-or-miss transform (HMT) for grey-level images from the HMT for binary images is not simple. Initially established as a powerful tool for morphological binary image processing, some generalizations for grey-level images have been proposed in the literature. The difficulty lies in the definition of the complement of a given grey-level image. In this paper, after performing a detailed review of the different approaches proposed in the literature, including those based on fuzzy logic, we propose the definition of the hit-or-miss transform for grey-level images in the framework of the fuzzy mathematical morphology based on fuzzy conjunctions and implications under the duality with respect to a fuzzy negation approach. Some theoretical properties of this operator are studied when a general fuzzy conjunction is considered. In particular, we prove that this fuzzy morphological HMT reduces to the binary one when it is applied to binary images. After that, we focus on the fuzzy morphological HMT derived from t-norms and we introduce the concept of “part of an image” which will guide the study of the desirable properties of our operator and the interpretability of the obtained results. Some preliminary experimental results provide evidence of the potential of this tool to be feasible in design algorithms for detecting patterns. Moreover, some comparisons with other hit-or-miss transforms proposed in the literature are performed.