Computational geometry in digital space

Date of Completion

January 2006


Artificial Intelligence|Computer Science




This work investigates digital space in terms of geometrical and topological properties. It also examines the applicability of digital space on different problem domains. The first problem area relates to the topology of regions on 2D space, proposing algorithms to infer topological relationships between regions in planar space. Geometrical relationships that required boundary definition were handled with extra overhead (as cell complexes) by other researchers. ^ The second problem area concerns geometrical problems defined for isothetic rectangles. Isothetic rectangles have been a graphical representation for concurrency control since 1980's. However, current systems lack a visual view of the system with a tool to verify the solution given. I present in this research a new graphical technique based on Yannakakis progress graphs that provide a safe and deadlock free solution. This new framework will allow dynamic mode of operation for calculating or verifying a safe schedule rather than restricting the solution to static systems where all required resources must be reserved in advance. ^ The third area relates to 3D rendering. In 3D visualization, the visual faces of the objects in the scene have to be defined, as some objects overlay others depending on their depth. There exists a technique called z-buffering that keeps track of the depth of each pixel in the screen buffer. This technique is simple but has some limitations such as it consumes the bandwidth. We enhanced that technique using IDR technology to reduce the redundant computations that can reduce the bandwidth problem. ^ We defined a diagrammatic reasoning framework to solve geometric problems in discrete space. This framework is applicable as a diagrammatic reasoning mechanism. In addition IDR has a parallel nature. The inter-diagrammatic reasoning (IDR) operators compute on the pixel level independently, which speeds up the complexity of the IDR algorithms. IDR works on the diagram as a whole allowing for a compact specification as a set of diagrams and can be considered in an IDR equation to compute the required solution. This technique is application independent as it is general enough to be defined as a "framework for diagrammatic reasoning". ^ For all of these problems, we shall focus on tractable algorithms using IDR.^