Quad-edge

Abstract: A winged edge polyhedron representation is stated and a set of primitives that preserve Euler''s F-E+V = 2 equation are explained. Present use of this representation in artificial intelligence for computer graphics and world modeling is illustrated and its intended future application to computer vision is described.

Abstract: Mobile robot localization and navigation requires a map - the robot's internal representation of the environment. A common problem is that path planning becomes very inefficient for large maps. In this paper we address the problem of segmenting a base-level map in order to construct a higher-level representation of the space which can be used for more efficient planning. We represent the base-level map as a graph for both geometric and appearance based space representations. Then we use a graph partitioning method to cluster nodes of the base-level map and in this way construct a high-level map, which is also a graph. We apply a hierarchical path planning method for stochastic tasks based on Markov decision processes (MDPs) and investigate the effect of choosing different numbers of clusters

Abstract: A method and apparatus are provided for calculating potential paths between source and destination locations. First and second map databases are provided that are indicative of roadway networks for geographic regions bounded by region edges and containing source and destination locations. The first and second map databases, are non-adjacent, non-contiguous such that the region edges of the first map database are separate and distinct from region edges of the second map database. Potential paths are calculated through the roadway network of the first map database up to a node or segment at which each potential path intersects a region edge of the first map database, thereby defining a node/edge coordinate. A transition location is obtained in the second map database that geographically corresponds to the node/edge coordinate at which a given potential path intersected the region edge of the first map database. The calculation continues from the transition location through the roadway network of the second map database. The method and apparatus may include organizing the map databases into a map hierarchy to define tiers for the map databases. The calculation process searches potential paths utilizing the tier-one map databases until each potential path intersects a map edge of the tier-one map, databases. Thereafter, the search through potential paths continues automatically based on the lower tier map databases.

Abstract: Space-variant sampling of visual input is ubiquitous in the higher vertebrate brain, because a large input space may be processed with high peak precision without requiring an unacceptably large brain mass. Space-variant sampling has been studied in computer vision for decades. A major obstacle to exploiting this architecture in machines, and understanding its role in biology, is the lack of algorithms that generalize beyond regular samplings. Most image processing algorithms implicitly assume a Cartesian grid underlying the sensor. This thesis generalizes image processing to a sensor architecture described by an arbitrary graph. This data structure separates the sensor topology, expressed by the graph edge structure, from its geometry, represented by coordinates of the vertex set. The combinatorial Laplacian of the sensor graph is a key operator underlying a series of novel image processing algorithms. First, a new graph partitioning algorithm for segmentation is presented that heuristically minimizes the ratio of the perimeter of the partition border and the area of the partitions, under a suitable definition of graph-theoretic area. This approach produces high quality image segmentations. Interpolation of missing data on graphs is developed, using a combinatorial version of the Dirichlet Problem, i.e., minimizing the average gradients of the interpolated values while maintaining fixed boundary conditions. This leads to the solution of the Laplace Equation, which represents the steady-state of the diffusion process for stated boundary conditions. Results compare favorably to both isotropic and anisotropic diffusion for filling-in of missing data. A pyramid graph is defined by connecting vertical and horizontal levels of the Laplacian pyramid data structure. The isoperimetric algorithm, run on the graph pyramid, yields an improved segmentation at little extra computational cost. Finally, a small-world graph topology is employed by randomly introducing a few new edges to the image graph. This results in a large speed-up in computation time, with identical final results. The algorithms developed in this thesis do not require that the data associated with the graph are embedded in two-dimensions or even have a metric structure. Therefore, this approach to generalized image processing may find wider application in other areas of discrete data processing.