Context dependence in visual feature processing

We study the variant of the art gallery problem where we are given an orthogonal polygon P (possibly with holes) and we want to guard it with the minimum number of sliding cameras. A sliding camera travels back and forth along an orthogonal line segment s in P and a point p in P is said to be visible to the segment s if the perpendicular from p onto s lies in P. Our objective is to compute a set containing the minimum number of sliding cameras (orthogonal segments) such that every point in P is visible to some sliding camera. We study the following two variants of this problem: Minimum Sliding Cameras problem, where the cameras can slide along either horizontal or vertical segments in P, and Minimum Horizontal Sliding Cameras problem, where the cameras are restricted to slide along horizontal segments only. In this work, we design local search PTASes for these two problems improving over the existing constant factor approximation algorithms. We note that in the first problem, the polygons are not allowed to contain holes. In fact, there is a family of polygons with holes for which the performance of our local search algorithm is arbitrarily bad.

Given a collection of line segments in the plane we would like to connect the segments by their endpoints to construct a simple circuit. (A simple circuit is the boundary of a simple polygon). However, there are collections of line segments where this cannot be done. In this note it is proved that deciding whether a set of line segments admits a simple circuit is NP-complete. Deciding whether a set of horizontal line segments can be connected with horizontal and vertical line segments to construct an orthogonal simple circuit is also shown to be NP-complete.

Approximability of covering cells with line segments

We investigate here the intersection graphs of horizontal and vertical line segments in the plane, the so called B 0-VPG graphs. A forbidden induced subgraph characterization of B 0-VPG split graphs is given, and we present a linear time algorithm to recognize this class. Next, we characterize chordal bull-free B 0-VPG graphs and chordal claw-free B 0-VPG graphs.

Metamorphopsia, perceived as distortion of a shape, is experienced in age‐related macular degeneration (AMD): straight lines appear to be curved and wavy to AMD patients and some other retinal pathologies. Conventional clinical assessment largely relies on asking patients to identify irregularities in Amsler Grids ‐ a standardized set of equally spaced vertical and horizontal lines. Perceived distortions or gaps in the grid are a sign of macular pathology. Here, we developed an iterative Amsler Grid (IAG) procedure to obtain a quantifiable map of visual deformations. Horizontal and vertical line segments representing metamorphopsia are displayed on a computer screen. Line segments appearing distorted are adjusted by participants using the computer mouse to change their orientation in several iteratively such that they appear straight. Control participants are able to reliably correct deformations that simulate metamorphopsia while maintaining fixation in the center. In a pilot experiment, we attempted to obtain deformation maps from a small number of AMD patients. Whereas some patients with extensive scotomas found this procedure challenging, others were comfortable using the IAG and generating deformation maps corresponding to their subjective reports. This procedure may potentially be used to quantify local distortions and map them reliably in patients with early AMD.

Omni-directional sensors are useful in obtaining a 360/spl deg/ field of view with a single lens camera. Omni-directional stereo imaging systems were designed in our past research using a single camera and a mirror consisting of two concentric, radially symmetric lobes. If the central camera-mirror axis is vertical, a stereo image containing imagery from all azimuth directions from, two viewpoints can be captured within one image. A vertically posed catadioptric optical system that employs a mirror with a radial profile other than parabolic or hyperbolic cannot maintain the straightness of any nonvertical lines. However other mirror profiles are desirable for resolution distribution, sensor size and manufacturability. These other mirror shapes, including spherical mirrors, are said to be non-SVP (Single View-Point). Since they lack a virtual perspective point, they pose new feature extraction challenges. A method for processing the imagery from a panoramic non-SVP catadioptric stereo sensor to reconstruct a three-dimensional model of horizontal and vertical line features is introduced. Horizontal line segments are extracted using the panoramic Hough transform and vertical line segments are recognized as straight radial lines. These segments, and closed shapes they form are matched between the two lobe views to locate them in three-dimensions. The practical accuracy obtainable with such a system was explored, and some of the issues addressed to make this sensor work in real 3D reconstructions are described. Triangulating a 3D location requires better calibration information than is required for the robust feature extraction.

A riveted graph G(V,E) is a graph whose vertices are fixed on a plane. Its rectilinear drawing D(G) is a configuration of G on the plane in such a way that every edge is composed of horizontal and vertical line segments. The authors consider the problem of reducing the number of line segments and present an algorithm to obtain such a rectilinear drawing that the total number of line segments is no more than 3 mod E mod . It is noted that a drawing so obtained contains no edge composed of five line segments. The algorithm runs in O( mod E mod ) time and space. >

Improved visibility representation of plane graphs

A note on visibility graphs

We consider some problems on red and blue points in the plane lattice. An L-line segment in the plane lattice consists of a vertical line segment and a horizontal line segment having a common endpoint. There are some results on geometric graphs on a set of red and blue points in the plane. We show that some similar results also hold for a set of red and blue points in the plane lattice using L-line segments instead of line segments. For example, we show that if n red points and n blue points are given in the plane lattice in general position, then there exists a noncrossing geometric perfect matching covering them, each of whose edges is an L-line segment and connects a red point and a blue point.

This paper presents a simultaneous localization and mapping (SLAM) of an indoor hallway environment using Rao-Blackwellized particle filter (RBPF) along with a line segment as a landmark. Based on the fact that fluent line features can be extracted around the ceiling and side walls of hallway using vision sensor, a horizontal line segment is extracted from an edge image using Hough transform and is also tracked continuously by an optical flow method. A successive observation of a line segment gives initial state of the line in 3D space. For data association, registered feature and observed feature are matched in image space through a degree of overlap, an orientation of line, and a distance between two lines. Experiments show that a compact environmental map can be constructed with small number of horizontal line features in real-time.

Box-Rectangular Drawings of Plane Graphs

A sliding camera inside an orthogonal polygon P is a point guard that travels back and forth along an orthogonal line segment \(\gamma \) in P. The sliding camera g can see a point p in P if the perpendicular from p onto \(\gamma \) is inside P. In this paper, we give the first constant-factor approximation algorithm for the problem of guarding P with the minimum number of sliding cameras. Next, we show that the sliding guards problem is linear-time solvable if the (suitably defined) dual graph of the polygon has bounded treewidth. On the other hand, we show that the problem is NP-hard on orthogonal polygons with holes even if only horizontal cameras are allowed. Finally, we study art gallery theorems for sliding cameras, thus, give upper and lower bounds in terms of the number of sliding cameras needed relative to the number of vertices n.

Given an n-node plane graph G, the visibility representation of G is concerned with drawing each node of G using a horizontal line segment such that the line segments associated with any two adjacent nodes of G are vertically visible to each other. Finding most compact visibility representations of plane graphs is not only of theoretical importance but also of practical interest, and has received much attention in the community of algorithmic graph theory. In this paper, we give a linear-time algorithm to find a visibility representation of G no wider than ⌊4n/3⌋ - 2. Our result improves upon the previously known upper bound 4n/3 + 2⌈ √n⌉, providing a positive answer to a conjecture suggested in the literature about whether an upper bound 4n/3 + O(1) on the required width can be achieved for an arbitrary plane graph. In fact, our visibility representation achieves optimality in the upper bound of width because the bound differs from the previously known lower bound ⌊4n/3⌋ - 3 only by one unit.

In an orthogonal drawing of a planar graph $G$, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of $G$ is called an optimal orthogonal drawing if the number of bends is minimum among all orthogonal drawings of $G$. In this paper we give an algorithm to find an optimal orthogonal drawing of any given series-parallel graph of the maximum degree at most three. Our algorithm takes linear time, while the previously known best algorithm takes cubic time. Furthermore, our algorithm is much simpler than the previous one. We also obtain a best possible upper bound on the number of bends in an optimal drawing.