The focus of this research is the extension of topological operators with the addition of a inclusion measure. This extension is carried out in both crisp and fuzzy topological spaces. The mathematical properties of the new operators are discussed and compared with traditional operators. Ignoring small errors due to imperfections and noise in digital images is the main motivation in introducing the proposed operators. To show the effectiveness of the new operators, we demonstrate their utility in image database classification and shape classification. Each image (shape) category is modeled with a topological space and the interior of the query image is obtained with respect to different topologies. This novel way of looking at the image categories and classifying a query image shows some promising results. Moreover, the proposed interior and closure operators with inclusion degree is utilized in mathematical morphology area. The morphological operators with inclusion degree outperform traditional morphology in noise removal and edge detection in a noisy environment

The focus of this research is the extension of topological operators with the addition of a inclusion measure. This extension is carried out in both crisp and fuzzy topological spaces. The mathematical properties of the new operators are discussed and compared with traditional operators. Ignoring small errors due to imperfections and noise in digital images is the main motivation in introducing the proposed operators. To show the effectiveness of the new operators, we demonstrate their utility in image database classification and shape classification. Each image (shape) category is modeled with a topological space and the interior of the query image is obtained with respect to different topologies. This novel way of looking at the image categories and classifying a query image shows some promising results. Moreover, the proposed interior and closure operators with inclusion degree is utilized in mathematical morphology area. The morphological operators with inclusion degree outperform traditional morphology in noise removal and edge detection in a noisy environment