Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

Image Processing of Surface in Cad/Cam Applications

Because of difficulty of Reconstructing 3D surface from the 2D captured images in terms of getting a higher resolution compared with the original surface so it has been adopted treatment of surface images after reconstructing the surface by of assembling all 2D im- ages and apply various filters to get the surface closest to the original surface of the sample under test. Various ways have been applied such as Laplacian and Median. It has been found that Median operator gave the best treatment to the surface where it was adopted to find the toolpath through the surface points for the purpose of machining using a CNC milling machine.