Grey-scale morphology based on fuzzy logic

There exist several methods to extend binary morphology to grey-scale images. One of these methods is based on fuzzy logic and fuzzy set theory. Another approach starts from the complete lattice framework for morphology and the theory of adjunctions. In this paper, both approaches are combined. The basic idea is to use (fuzzy) conjunctions and implications which are adjoint in the definition of dilations and erosions, respectively. This gives rise to a large class of morphological operators for grey-scale images. It turns out that this class includes the often used grey-scale Minkowski addition and subtraction

A Unified Algebraic Framework for Fuzzy Image Compression and Mathematical Morphology

In this paper we show how certain techniques of image processing, having different scopes, can be joined together under a common "algebraic roof"

Quantale Modules and their Operators, with Applications

The central topic of this work is the categories of modules over unital quantales. The main categorical properties are established and a special class of operators, called Q-module transforms, is defined. Such operators - that turn out to be precisely the homomorphisms between free objects in those categories - find concrete applications in two different branches of image processing, namely fuzzy image compression and mathematical morphology

Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of L-fuzzy mathematical morphology

Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted. The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing

Fuzzy logic and mathematical morphology

In this report we investigate the general theory of grey-scale morphology within the framework of complete lattices and fuzzy logic. This includes grey-scale granulometries, hit-or-miss operators for grey-scale images, rank operators, and connected operators. We also show that the Matheron's representation theory does not hold for general grey-scale images and we present some results related to the representation theory. Besides these, in this report, we put forward a new approach to fuzzy morphology through the extension of infimum, supremum, and conjunction

( max , min )-convolution and Mathematical Morphology

International audienceA formal denition of morphological operators in (max, min)-algebra is introduced and their relevant properties from an algebraic viewpoint are stated. Some previous works in mathematical morphology have already encountered this type of operators but a systematic study of them has not yet been undertaken in the morphological literature. It is shown in particular that one of their fundamental property is the equivalence with level set processing using Minkowski addition and subtraction. Theory of viscosity solutions of the Hamilton-Jacobi equation with Hamiltonians containing u and Du is summarized, in particular, the corresponding Hopf-Lax-Oleinik formulas as (max, min)-operators. Links between (max, min)-convolutions and some previous approaches of unconventional morphology, in particular fuzzy morphology and viscous morphology, are reviewed

Fuzzy morphological operators in image processing

First of all, in this paper we propose a family of fuzzy implication operators, which the generalised Luckasiewicz's one, and to analyse the impacts of Smets and Magrez properties on these operators. The result of this approach will be a characterisation of a proposed family of inclusion grade operators (in Bandler and Kohout's manner) that satisfies the axioms of Divyendu and Dogherty. Second, we propose a method to define fuzzy morphological operators (erosions and dilations). A family of fuzzy implication operators and the inclusion grade are the basis for this method

Fuzzy techniques for noise removal in image sequences and interval-valued fuzzy mathematical morphology

Image sequences play an important role in today's world. They provide us a lot of information. Videos are for example used for traffic observations, surveillance systems, autonomous navigation and so on. Due to bad acquisition, transmission or recording, the sequences are however usually corrupted by noise, which hampers the functioning of many image processing techniques. A preprocessing module to filter the images often becomes necessary. After an introduction to fuzzy set theory and image processing, in the first main part of the thesis, several fuzzy logic based video filters are proposed: one filter for grayscale video sequences corrupted by additive Gaussian noise and two color extensions of it and two grayscale filters and one color filter for sequences affected by the random valued impulse noise type. In the second main part of the thesis, interval-valued fuzzy mathematical morphology is studied. Mathematical morphology is a theory intended for the analysis of spatial structures that has found application in e.g. edge detection, object recognition, pattern recognition, image segmentation, image magnification… In the thesis, an overview is given of the evolution from binary mathematical morphology over the different grayscale morphology theories to interval-valued fuzzy mathematical morphology and the interval-valued image model. Additionally, the basic properties of the interval-valued fuzzy morphological operators are investigated. Next, also the decomposition of the interval-valued fuzzy morphological operators is investigated. We investigate the relationship between the cut of the result of such operator applied on an interval-valued image and structuring element and the result of the corresponding binary operator applied on the cut of the image and structuring element. These results are first of all interesting because they provide a link between interval-valued fuzzy mathematical morphology and binary mathematical morphology, but such conversion into binary operators also reduces the computation. Finally, also the reverse problem is tackled, i.e., the construction of interval-valued morphological operators from the binary ones. Using the results from a more general study in which the construction of an interval-valued fuzzy set from a nested family of crisp sets is constructed, increasing binary operators (e.g. the binary dilation) are extended to interval-valued fuzzy operators