Research On The Theory Of Grayscale Morphological Connectivity In Terms Of Lattice Homomorphism

Connectivity theory originates from the topology and graph-theoretic frame, and is widely used in image processing and analysis, particularly in image segmentation, image filter, object detection, motion analysis and pattern recognition. Because of the limitation of the early con-nectivity analysis in topological space graph-theoretic framework, classical analysis concepts are not suitable for digital image processing and analysis in practise. J. Serra proposed the ax-iomatization of connectivity for crisp sets, which encapsulates the characteristic of topological and graph-theoretic analysis. However, in order to meet the actual demand of image processing and analysis, the connectivity theory applicable to images in different types must be established. Motivated by it, Serra brought up the idea of connectivity analysis in complete lattices. From then, the morphological connectivity analysis has a unified theoretical basis and framework.This dissertation is dedicated to the theory of connectivity analysis, formulated by means of lattice homomorphism. Firstly, generally introduce the essential mathematical operators, and then discuss the definition and property of fuzzy logic based grayscale morphological opera-tors. In addition, the present important connectivity theory is summarized. On that basis, main work in the dissertation focus on the problem of connectivity preserving between distinct image spaces in a new manners. The major contributions in this dissertation are as follows:(1) To aim at grayscale images, a kind of grayscale morphological operators based on fuzzy sets and fuzzy logic is proposed. The operators are constructed by extending crisp in-clusion and intersection to the fuzzy case, which measure the degree of fuzzy inclusion and intersection employing with fuzzy implication and conjunction. Furthermore, we have proved the decomposition theorem of fuzzy logic based grayscale morphology operators, and estab-lished the relation between the proposed grayscale operators and the binary ones.(2) In the framework of complete lattices, connectivity analysis theory based on lattice homomorphism mappings is proposed. In addition, it has been proved that a homomorphism mapping satisfying certain conditions can preserve connectivity. Theoretical investigation on relationship of connectivity classes and connected operators between different lattices based on homomorphism mappings is explored.(3) For grayscale images, an image segmentation method based on Serra homomorphism and flattening operator is given under the framework of lattice homomorphism connectivity analysis. In the proposed method, image can be segmented perfectly in terms of connected operator, flattening, which can both smooth image and preserve the contour of objects. Conse-quently, over segmentation problem has been solved.(4) Combined with multiscale analysis, a novel edge detection means is put forward re-curring to lattice connectivity. Some experiments are carried out to show effects and efficiency of lattice homomorphism theory.