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.

Abstract: We study the problem of computing the visibility graph defined by a set P of n points inside a polygon Q: two points p,q e P are joined by an edge if the segment ‾pq ⊂ Q. Efficient output-sensitive algorithms are known for the case in which P is the set of all vertices of Q. We examine the general case in which P is an arbitrary set of points, interior or on the boundary of Q and study a variety of algorithmic questions. We give an output-sensitive algorithm, which is nearly optimal, when Q is a simple polygon. We introduce a notion of "fat" or "robust" visibility, and give a nearly optimal algorithm for computing visibility graphs according to it, in polygons Q that may have holes. Other results include an algorithm to detect if there are any visible pairs among P, and algorithms for output-sensitive computation of visibility graphs with distance restrictions, invisibility graphs, and rectangle visibility graphs.

Abstract: In this paper, we study the Obstructed Nearest Neighbor (ONN) problem: given a set of points and a set of polygonal obstacles in two dimensions, find the k nearest neighbors to a query point according to the length of the shortest obstacle-avoiding path between two points ONN query is useful both as a stand-alone tool in geographical information systems and as a primitive for spatial data analysis such as clustering and classification in the presence of obstacles We propose an efficient ONN algorithm that processes only the data points and obstacles relevant to the query in an incremental way and thus filters out a large number of points and obstacles Experiments on spatial data sets show the algorithm scales well with respect to the input data size and the number of nearest neighbors requested

Abstract: An evolutionary technique with a Fuzzy Inference System (FIS) is offered for planning time-optimal trajectories on a predefined Visibility Graph Method Dijkstra (VGM-D) path of a Nomad 200 mobile robot (MR). First of all, the segmented trajectory is generated by the VGM-D algorithm. Line and curve segments are the components of the trajectory. The number of intersections of the segmented VGM-D path determines the curve segments number. It is assumed that, at each curve segment, translation velocity v t is taken as constant. The Differential Evolution (DE) algorithm finds v t values of all the curve segments, which minimize the trajectory tracking time. Line segments lengths are used to calculate the constraints of the problem according to the Nomad 200's limitations on the translation velocity and acceleration/deceleration. The structures of the curve segments are modeled by FIS to decrease the DE's execution time. Another FIS model is used to define the upper bound of the translation velocities on the cu...

Abstract: Methods based on configurational theory were proved to be highly effective in urban space analysis, in order to highlight the distribution of the levels of attractiveness towards activities. Nevertheless, both Axial Analysis and Visibility Graph Analysis (the most prominent analytic techniques) appear affected by some evident faults, that somehow do limit their actual use: in particular, those faults affect the definition of the axial system, due to the arbitrariness in drawing the lines that cover the urban grid and the lacking of correspondence between them and the streets of the settlement. In VGA, that studies the configurational features of the internal points of the grid, the problem concerns the heaviness of the data (thousands of numerical values) and the difficulty in referring them to a single urban space (either a street or a square).

Abstract: A triangulated polygon is a 2-connected maximal outerplanar graph A unit bar-visibility graph (UBVG for short) is a graph whose vertices can be represented by disjoint, horizontal, unit-length bars in the plane so that two vertices are adjacent if and only if there is a non-degenerate, unobstructed, vertical band of visibility between the corresponding bars We give combinatorial and geometric characterizations of the triangulated polygons that are UBVGs To each triangulated polygon G we assign a character string with the property that G is a UBVG if and only if the string satisfies a certain regular expression Given a string that satisfies this condition, we describe a linear-time algorithm that uses it to produce a UBV layout of G

Abstract: An important geometric structure used in robotic path planning and computer graphics is the visibility graph. In this paper, we present a new parallel algorithm to construct the reduced visibility graph that is appropriate for finding shortest paths in a convex polygonal environment. A key feature of the algorithm is that it supports easy mapping to hardware. The computational complexity is O(p/sup 2/+log((n/p)/sup 2/)) where p is the number of objects and n is the total number of vertices. An efficient FPGA implementation of the algorithm is presented. The design operates at approximately 48 Mhz. Further, the implementation for an environment with roughly 60 vertices requires 90% of an XCV3200E.

Abstract: This paper introduces new graph geometric measures and models for architectural planning. Based upon geometrical distance among nodes, the ′minimum-path graph′ is proposed to analyse all shortest-p...

Abstract: Visibility graphs constitute a useful data structure for environment representation in the context of robot path planning. A central element in the construction of the basic visibility graph and its variants is tangent determination. This letter presents new schemes and hardware designs for key elements in tangent construction and identification of obstructed tangents. The designs have been synthesized using Synopsys Design Compiler 2001.08-SP1, and results show they are appropriate for development of a cost-effective and efficient visibility graph generation system.

Abstract: A new time series clustering algorithm named morphing cluster dynamics is put forward. This algorithm can capture two levels of inter-time-series interaction behavior among a large set of time series: (i) group level, which includes the merges and splits of clusters, and (ii) individual-to-group level, such as a time series joins or leaves a cluster. In order to do so, it first reduces the TS set into a system morphing graph, a multipartite graph is employed to represent the latent interactions, and then extract strong interaction patterns from this graph. A system morphing graph is built by three steps: firstly, the input time series set is divided into sets of segments along the time line, and each segment set serves as a partite of the graph; then, each segment set is clustered and the resulting clusters serves as the vertices in the graph; third, the edges are built according to member-sharing relationships between clusters. System morphing graphs can model the captured TS interactions. Because this algorithm both cuts and concatenates time series, it does not fit into either whole clustering or subsequence clustering.

Abstract: A triangulation is said to be Hamiltonian if its dual graph contains a Hamiltonian path. A sequential triangulation is a Hamiltonian triangulation having the additional property that the “turns” in the Hamiltonian path alternate left/right. Such triangulations are useful in computer graphics rendering and are related to a new type of two-guard walk. In this paper we present a simple algorithm that determines all sequential triangulations (or equivalently all sequential two-guard walks) of a simple vertex input polygon. The previous best algorithm uses the polygon’ s visibility graph and hence runs in worse case time [1].

Abstract: The visibility polygon of a simple polygon P is the set of points which are visible from a visibility source in P such as a point or an edge Since a visibility polygon is the set of points, the set operations such as intersection, union, or difference can be executed on them The intersection (resp union) of two visibility polygons is the set of points which are visible from both (resp either) of the corresponding two visibility sources The difference of two visibility polygons is the set of points which are visible from only a visibility source Previously, the best known algorithm for the set operations of two polygons with total n vertices takes O(nlogn ＋ k) time, where k is the output size In this paper, we present O(n) time algorithms for computing the intersection, the union, and the difference of given two visibility polygons, which are optimal

Abstract: Numerous methods have been developed to solve the motion planning problem, among which the Voronoi diagram, visibility graph, and potential fields are well-known techniques. In this paper, a new path planning algorithm is presented where these three methods are integrated for the first time in a single architecture. After constructing the generalized Voronoi diagram of C-space, we introduce a novel procedure for its abstraction, producing a pruned generalized Voronoi diagram. A broad freeway net is then developed through a new a-MID (maximal inscribed discs) concept. A potential function is assigned to the net to form an obstacle-free network of valleys. Afterwards we take advantage of a bidirectional search, where the visibility graph and potential field modules execute alternately from both start and goal configurations. A steepest descent mildest ascent search technique is used for local planning and avoiding local minima. The algorithm provides a parametric tradeoff between safest and shortest paths and generally yields shorter paths than the Voronoi and potential field methods, and faster than the visibility graph. It also performs well in complicated environments. © 2004 Wiley Periodicals, Inc.