An efficient algorithm for touring a sequence of given convex polygons in the plane

The problem of cutting a convex polygon P out of a piece of paper Q with minimum total cutting length is a well studied problem in computational geometry. Researchers studied several variations of the problem, such as P and Q are convex or non-convex polygons and the cuts are line cuts or rays cuts. In this paper we consider yet another variation of the problem where Q is a circle and P is convex polygon such that P is bounded by a half circle of Q and all the cuts are line cuts. We give a simple linear time O(log n)-approximation algorithm for this problem where n is the number of vertices of P.

Algorithms for line clipping against convex polygon have been studied for a long time and many research papers have been published so far. In spite of the latest graphical hardware development and significant increase of performance the clipping is still a bottleneck of the graphical pipeline. This paper presents a new robust and fast algorithm for line clipping by a convex polygon. The algorithm uses a small preprocessing in order to obtain significant speed up. The proposed algorithm is especially convenient for applications where points or lines are represented in homogeneous coordinates. The algorithm does not use division in floating point representation as the resulting points are in homogeneous coordinates. The algorithms benefit if vector-vector hardware supported operations can be used

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.

一様ランダム分布乱数点の凸型領域内への散布法

In conducting their work in the industry quickly, precisely, and safely, mobile robots must be able to determine the position and direction of movement in their work environment. Several algorithms have been developed to solve maze rooms, however, when the room is huge with several obstacles which could be re-placed in other parts of the room, determining the path for a mobile robot will be difficult. This can be done by mapping the work environment and determining the position of the robot so that the robot has good path planning to get the optimal path. In this research, a Genetic Algorithm (GA) will be used to determine the fastest route that a robot may take when moving from one location to another. The method used is to design a mobile robot work environment, design genetic algorithm steps, create software for simulation, test the algorithm in 6 variations of the work environment, and analyze the test results. The genetic algorithm can determine the shortest path with 93% completeness among the 6 possible combinations of the start point, target point, and position of obstacles. The proposed GA, it can be argued, can be used to locate the shortest path in a warehouse with different start and end points.

The concept of convex polygon-offset distance function was introduced in 2001 by Barequet, Dickerson, and Goodrich. Using this notion of point-to-point distance, they showed how to compute the corresponding nearest- and farthest-site Voronoi diagram for a set of points. In this paper we generalize the polygon-offset distance function to be from a point to any convex object with respect to an m-sided convex polygon, and study the nearest- and farthest-site Voronoi diagrams for sets of line segments and convex polygons. We show that the combinatorial complexity of the nearest-site Voronoi diagram of n disjoint line segments is O(nm), which is asymptotically equal to that of the Voronoi diagram of n point sites with respect to the same distance function. In addition, we generalize this result to the Voronoi diagram of disjoint convex polygonal sites. We show that the combinatorial complexity of the nearest-site Voronoi diagram of n convex polygonal sites, each having at most k sides, is \(O(n(m+k))\). Finally, we show that the corresponding farthest-site Voronoi diagram is a tree-like structure with the same combinatorial complexity.

Extraction of filled-in information from document images in the presence of template poses challenges due to geometrical distortion. Filled-in document image consists of null background, general information foreground and vital information imposed layer. Template document image consists of null background and general information foreground layer. In this paper a novel document image registration technique has been proposed to extract imposed layer from input document image. A convex polygon is constructed around the content of the input and the template image using convex hull. The vertices of the convex polygons of input and template are paired based on minimum Euclidean distance. Each vertex of the input convex polygon is subjected to transformation for the permutable combinations of rotation and scaling. Translation is handled by tight crop. For every transformation of the input vertices, Minimum Hausdorff distance (MHD) is computed. Minimum Hausdorff distance identifies the rotation and scaling values by which the input image should be transformed to align it to the template. Since transformation is an estimation process, the components in the input image do not overlay exactly on the components in the template, therefore connected component technique is applied to extract contour boxes at word level to identify partially overlapping components. Geometrical features such as density, area and degree of overlapping are extracted and compared between partially overlapping components to identify and eliminate components common to input image and template image. The residue constitutes imposed layer. Experimental results indicate the efficacy of the proposed model with computational complexity. Experiment has been conducted on variety of filled-in forms, applications and bank cheques. Data sets have been generated as test sets for comparative analysis.

3-D analysis of blood vessels from volumetric CT and MR datasets has many applications ranging from examination of pathologies such as aneurysm and calcification to measurement of cross-sections for therapy planning. Segmentation of the vascular structures followed by tracking is an important processing step towards automating the 3-D vessel analysis workflow. This paper demonstrates a fast and automated algorithm for tracking the major arterial structures that have been previously segmented. Our algorithm uses anatomical knowledge to identify the start and end points in the vessel structure that allows automation. Voxel coding scheme is used to code every voxel in the vessel based on its geodesic distance from the start point. A shortest path based iterative region growing is used to extract the vessel tracks that are subsequently smoothed using an active contour method. The algorithm also has the ability to automatically detect bifurcation points of major arteries. Results are shown for tracking the major arteries such as the common carotid, internal carotid, vertebrals, and arteries coming off the Circle of Willis across multiple cases with various data related and pathological challenges from 7 CTA cases and 2 MR Time of Flight (TOF) cases.

A 2-approximation algorithm for the zookeeper's problem

This paper proposes an improved method of exact cellular decomposition to plan the coverage path in polygon area. Firstly, the problem of Coverage Path Planning (CPP) in the convex polygon area is transformed to the width calculation of the convex polygon, and the strategy of flying along the vertical direction of width is presented to reach the least number of turns. Secondly, a convex decomposition algorithm of minimum width sum based on the greedy recursive method is presented to decompose the concave area into some convex subregions. Finally, a subregion connection algorithm based on minimum traversal of undirected graph is proposed to connect the coverage paths of the subregions. The simulation shows that the proposed method is feasible and effective.

Finding minimal convex nested polygons

Understanding the movement of visitors contributes to appropriate management plans for visitor destinations. Determination of this movement is composed of not only the identification of the starting and end points together with the sequence of stops but the characterisation of the route pattern as well. The type of this pattern depends on the human apart from outer physical parameters and the available time thus it has a relationship with the behaviour and type of the tourist. Understanding and making use of these are important measures for organisations managing natural areas. Observation of visitors in natural areas is not possible in every environment while reducing the tread of certain popular sites can be made only with the help of such knowledge. This explains that literature describing different patterns analyse inter- destination relationships. In this paper the identification of the movement pattern of hikers in a mountainous study area is presented. Visitor flow analysis of a study area in Hungary (180 km2 area of Central Börzsöny in Duna-Ipoly National Park) was performed on the basis of a questionnaire survey carried out with the help of volunteers. Based on the analysis of the obtained data the accurate description of the movement of both individual hikers and groups became possible. Mathematics based definition of hiking routes completed in mountainous areas is new to the available data in the literature. Patterns of one day routes were classified into 6 types and two types of hiking routes requiring several days were also identified. Different route types were observed at the different entrances of the destination. Such information helps the identification of visitor types at the starting points and the design of appropriate visitor management measures.

Summary form only given. Tubular and tree structures appear very commonly in biomedical images like vessels, microtubules or neuron cells. Minimal paths have been used for long as an interactive tool to segment these structures as cost minimizing curves. The user usually provides start and end points on the image and gets the minimal path as output. These minimal paths correspond to minimal geodesics according to some adapted metric. They are a way to find a (set of) curve(s) globally minimizing the geodesic active contours energy. Finding a geodesic distance can be solved by the Eikonal equation using the fast and efficient Fast Marching method. In the past years we have introduced different extensions of these minimal paths that improve either the interactive aspects or the results. For example, the metric can take into account both scale and orientation of the path. This leads to solving an anisotropic minimal path in a 2D or 3D+radius space. On a different level, the user interaction can be minimized by adding iteratively what we called the keypoints, for example to obtain a closed curve from a single initial point. The result is then a set of minimal paths between pairs of keypoints. This can also be applied to branching structures in both 2D and 3D images. We also proposed different criteria to obtain automatically a set of end points of a tree structure by giving only one starting point. More recently, we introduced a new general idea that we called Geodesic Voting or Geodesic Density. The approach consists in computing geodesics between a given source point and a set of points scattered in the image. The geodesic density is defined at each pixel of the image as the number of geodesics that pass over this pixel. The target structure corresponds to image points with a high geodesic density. We will illustrate different possible applications of this approach. The work we will present involved as well F. Benmansour, Y. Rouchdy and J. Mille at CEREMADE.

The ocelot Leopardus pardalis is a medium-sized Neotropical spotted cat with a wide geographic range. The present study was conducted in a reserve in an Atlantic Forest area in southern Brazil, and provided information on ocelot ecology through photographic records. Density estimated by the half of the mean maximum distance moved method resulted in 0.04 ocelots per km2. The minimum home range sizes were obtained by the minimum convex polygon method and were in agreement with other studies for the males recorded. Our female’s home range size was smaller than ranges reported elsewhere. The nocturnal pattern recorded is probably related with prey activity, since it is suggested that ocelots adjust their movements to the probabilities of encountering local prey, or with an evolutionary factor, aiming to avoid competition/predation with larger cats. The low density observed could be a consequence of the conditions of the study area, small and isolated, which indicates the necessity for larger reserves and mechanisms to connect isolated populations.

A new scheme of backward Raman scattering, in which coupling between the forward and backward Stokes processes is caused by injecting an additional backward pump field, is demonstrated both experimentally and theoretically. Four wave mixing among the pump and Stokes fields leads to gain enhancement of backward Raman scattering. A factor of gain enhancement of 2 was obtained with injection of a 0.4 mJ seed Stokes and a 5 mJ backward pump wave.