On an Application of Lattice Integral Transforms in Image Processing

Abstract

The lattice integral transforms have been introduced to generalize lower and upper fuzzy transforms for lattice-valued functions that are used to approximate original functions from below and above. They are defined in complete analogy with classical integral transforms, particularly, the product of a lattice-valued function and a fuzzy relation called the integral kernel is integrated by a Sugeno-like fuzzy integral. In the article, we first investigate the conditions under which lattice integral transforms preserve (reverse) constant functions, which appears to be a fundamental presumption for a successful approximation of lattice-valued functions. Further, we show how the lattice integral transforms can be applied in image processing, more specifically, in non-linear filtering, compression/decompression, and opening/closing of images. We demonstrate that the filters based on integral transforms generalize the popular median filter as well as minimum and maximum filters, and also opening and closing defined using fuzzy morphological erosion and dilation. We illustrate the proposed methods in various selected images.