Safe Landing Site Selection Based on Computational Geometry and Genetic Algorithm

<p indent=0mm>To address potential safety threats posed by various terrain conditions on the planetary surface, the recent development of autonomous planetary landing site assessment and selection methods is systematically reviewed in this paper. First, the mission characteristics and main technical challenges in landing site selection are summarized. Critical factors to be considered in landing site selection are introduced together with the selection process and applications of the Safety Index method. Then, an overview of the research progress of autonomous landing site assessment and selection methods is provided. Past missions and theoretical developments are analyzed from the aspects of hazard detection, terrain safety assessment, and comprehensive landing site selection. Lastly, based on the growing requirements of mission intelligence in future planetary exploration, several suggestions and recommendations with regard to landing site selection technology development trends are proposed.

This paper has two goals. First, we point out that most problems in computational geometry in fact have fast parallel algorithms (that is, in NC*) by reduction to the cell decomposition result of Kozen and Yap. We illustrate this using a new notion of generalized Voronoi diagrams that subsumes all known instances. While the existence of NC* algorithms for computational geometry is theoretically significant, it leaves much to be desired for specific problems. Therefore, the second part of the paper surveys some recent results in a fast growing list of parallel algorithms for computational geometry.

In school development problem, determining location is one of important things to consider. In this research, the purpose is to determine the location of SMAN 9 Denpasar if it will be built. One of algorithms in computational geometry that can be used to find solution of facility location problem is multiplicatively weighted Voronoi diagram in two dimensions. The result of weighted Voronoi diagram shows the influence of each site to the surrounding area. Then, the location of SMAN 9 Denpasar is obtained by determining the center of the largest empty circle of the weighted Voronoi diagram.

We consider a collectionH ofn hyperplanes in Ed (where the dimensiond is fixed). An ?-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all Ed and such that the interior of any simplex is intersected by at most?n hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices (which is asymptotically optimal). Forr≤n1??, where ?>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn)O(1)rd?1). In the plane we achieve a time boundO(nr) forr≤n1??, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm. For ann point setX⊆Ed and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ? R; R is a simplex}), In timeO(n(logn)O(1)rd?1+rO(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ?-nets for other range spaces usually encountered in computational geometry. These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.

Solid modelling and computational geometry are based on classical topology and geometry in which the basic predicates and operations, such as membership, subset inclusion, union and intersection, are not continuous and therefore not computable. But a sound computational framework for solids and geometry can only be built in a framework with computable predicates and operations. In practice, correctness of algorithms in computational geometry is usually proved using the unrealistic Real RAM machine model of computation, which allows comparison of real numbers, with the undesirable result that correct algorithms, when implemented, turn into unreliable programs. Here, we use a domain-theoretic approach to recursive analysis to develop the basis of an e3ective and realistic framework for solid modelling. This framework is equipped with a well de5ned and realistic notion of computability which re6ects the observable properties of real solids. The basic predicates and operations on solids are computable in this model which admits regular and non-regular sets and supports a design methodology for actual robust algorithms. Moreover, the model is able to capture the uncertainties of input data in actual CAD situations. c 2002 Elsevier Science B.V. All rights reserved.

Algorithms in computational geometry often use the real-RAM model of computation. This model assumes that exact real numbers can be stored in memory and retreived in constant time, and that field operations (+, –, *, /) and certain other operations, square root, sine, and cosine for instance, are exact, and can be applied in constant time. These assumptions are often difficult to discharge at implementation time. Even well-understood algorithms, like line-sweep for polygon union [PS85], present much trouble. Why? Such algorithms obtain good combinatorial complexity bounds by exploiting geometric orders on the input. These relations are often implicit, so an algorithm can only probe them pointwise with a collection of predicate functions answering, e.g., “Is point p left of edge e?”. In implementations, the reply must depend on arithmetic with finitely represented numbers. The trouble originates here.

Introduced by Agarwal, Har-Peled, and Varadarajan [J. ACM, 2004], an epsilon-kernel of a point set is a coreset that can be used to approximate the width, minimum enclosing cylinder, minimum bounding box, and solve various related geometric optimization problems. Such coresets form one of the most important tools in the design of linear-time approximation algorithms in computational geometry, as well as efficient insertion-only streaming algorithms and dynamic (non-streaming) data structures. In this paper, we continue the theme and explore dynamic streaming algorithms (in the so-called turnstile model). Andoni and Nguyen [SODA 2012] described a dynamic streaming algorithm for maintaining a (1+epsilon)-approximation of the width using O(polylog U) space and update time for a point set in [U]^d for any constant dimension d and any constant epsilon>0. Their sketch, based on a polynomial method, does not explicitly maintain an epsilon-kernel. We extend their method to maintain an epsilon-kernel, and at the same time reduce some of logarithmic factors. As an application, we obtain the first randomized dynamic streaming algorithm for the width problem (and related geometric optimization problems) that supports k outliers, using poly(k, log U) space and time.

Algorithms in computational geometry are of increasing importance in computer-aided design, for example, in the layout of integrated circuits. The efficient computation of the intersection of several superimposed figures is a basic problem. Plane figures defined by points connected by straight line segments are considered, for example, polygons (not necessarily simple) and maps (embedded planar graphs). The regions into which the plane is partitioned by these intersecting figures are to be processed in various ways such as listing the boundary of each region in cyclic order or sweeping the interior of each region. Let m be the total number of points of all the figures involved and s be the total number of intersections of all line segments. A two plane-sweep algorithm that solves the problems above is presented; in the general case (non convexity) in time O((n+s)log-n) and space O(n+s); when the regions of each given figure are convex, the same can be achieved in time O(n log n +s) and space O(n)

This work presents an adapted version of the Computational Geometry Algorithm (CGA) used for the development of audio-based applications and services. The CGA algorithm analyses an audio stream and produces a unique set of points that can be considered to be the audio data “fingerprint”. It is shown that this fingerprint is coding-independent, a fact that can render the proposed algorithm suitable for multiple purposes, including the categorisation of content identity and the identification of audio clips, hence providing support for the realisation of audio sorting/searching tasks and services. Additionally, based on specific novel applications and services, the overall algorithmic performance and efficiency characteristics of the CGA algorithm are discussed and analysed.

Enhancements to array processors in the form of broadcast buses have been proposed for improving the speeds of algorithms in linear algebra, image processing and computational geometry. In this paper, we consider certain practical issues in such arrays which include fewer processors connected to the broadcast buses and finite time, namely log N time for broadcasting data to N processors. We propose a modified broadcast bus VLSI architecture which consists of a √N × √N mesh connected structure and a hierarchy of broadcast buses in each row and in each column such that every bus has k PE's. That is, in any row or column, in the first level there are \(\frac{{\sqrt N }}{k}\)buses with groups of k PE's connected to each bus. One PE from each of this group is connected to second level of buses in a similar manner. This is recursively done until there are only k PE's left which are connected by a broadcast bus. With this architecture, maximum, minimum, or sum of N values can be found in O(log N), median row of a binary picture in O(log N), shortest distance between two points in O(log N). This architecture is well suited for parallel processing of applications in Linear Algebra, Image Processing, Computational Geometry and Numerical Computations. These restricted connections to buses reduces the I/O ports significantly, and therefore, is more suitable for VLSI implementation.

Abstract. The European Space Agency (ESA) is planning a Lunar Lander mission in the 2018 timeframe that will demonstrate precise soft landing at the polar regions of the Moon. To ensure a safe and successful landing a careful risk analysis has to be carried out. This is comprised of identifying favorable target areas and evaluating the surface conditions in these areas. Features like craters, boulders, steep slopes, rough surfaces and shadow areas have to be identified in order to assess the risk associated to a landing site in terms of a successful touchdown and subsequent surface operation of the lander. In addition, global illumination conditions at the landing site have to be simulated and analyzed. The Landing Site Risk Analysis software framework (LandSAfe) is a system for the analysis, selection and certification of safe landing sites on the lunar surface. LandSAfe generates several data products including high resolution digital terrain models (DTMs), hazard maps, illumination maps, temperature maps and surface reflectance maps which assist the user in evaluating potential landing site candidates. This paper presents the LandSAfe system and describes the methods and products of the different modules. For one candidate landing site on the rim of Shackleton crater at the south pole of the Moon a high resolution DTM is showcased.

The aim of this study was to compare three methods for computing a target point for use in autonomous or semi-autonomous aspirations and biopsies. Given a 3D binary image of the object of interest, the procedures computed the target point. The following approaches were tested: the method #1 - center of mass, the method #2 - largest projection area + largest empty circle and the method #3 - largest empty circle + largest empty circle. Each procedure was tested on four cases obtained from Magnetic Resonance Imaging scans used to diagnose Baker’s cysts. The methods were analyzed and compared in terms of their safety and computation time. In terms of safety, the best results were obtained with the third procedure, which used the largest empty circle + largest empty circle combination. The second method - the largest projection area + largest empty circle - offered good compromise between safety and computation time. It can be used to estimate target points for medical tool path planning in aspiration or biopsy.

COMBINATORIAL AND DISCRETE GEOMETRY Finite Point Configurations, J. Pach Packing and Covering, G. Fejes Toth Tilings, D. Schattschneider and M. Senechal Helly-Type Theorems and Geometric Transversals, R. Wenger Pseudoline Arrangements, J.E. Goodman Oriented Matroids, J. Richter-Gebert and G.M. Ziegler Lattice Points and Lattice Polytopes, A. Barvinok New! Low-Distortion Embeddings of Finite Metric Spaces, P. Indyk and J. Matousek New! Geometry and Topology of Polygonal Linkages, R. Connelly and E.D. Demaine New! Geometric Graph Theory, J. Pach Euclidean Ramsey Theory, R.L. Graham Discrete Aspects of Stochastic Geometry, R. Schneider Geometric Discrepancy Theory and Uniform Distribution, J.R. Alexander, J. Beck, and W.W.L. Chen Topological Methods, R.T. Zivaljevic Polyominoes, S.W. Golomb and D.A. Klarner POLYTOPES AND POLYHEDRA Basic Properties of Convex Polytopes, M. Henk, J. Richter-Gebert, and G.M. Ziegler Subdivisions and Triangulations of Polytopes, C.W. Lee Face Numbers of Polytopes and Complexes, L.J. Billera and A. Bjoerner Symmetry of Polytopes and Polyhedra, E. Schulte Polytope Skeletons and Paths, G. Kalai Polyhedral Maps, U. Brehm and E. Schulte ALGORITHMS AND COMPLEXITY OF FUNDAMENTAL GEOMETRIC OBJECTS Convex Hull Computations, R. Seidel Voronoi Diagrams and Delaunay Triangulations, S. Fortune Arrangements, D. Halperin Triangulations and Mesh Generation, M. Bern Polygons, J. O'Rourke and S. Suri Shortest Paths and Networks, J.S.B. Mitchell Visibility, J. O'Rourke Geometric Reconstruction Problems, S.S. Skiena New! Curve and Surface Reconstruction, T.K. Dey Computational Convexity, P. Gritzmann and V. Klee Computational Topology, G. Vegter Computational Real Algebraic Geometry, B. Mishra GEOMETRIC DATA STRUCTURES AND SEARCHING Point Location, J. Snoeyink New! Collision and Proximity Queries, M.C. Lin and D. Manocha Range Searching, P.K. Agarwal Ray Shooting and Lines in Space, M. Pellegrini Geometric Intersection, D.M. Mount New! Nearest Neighbors in High-Dimensional Spaces, P. Indyk COMPUTATIONAL TECHNIQUES Randomization and Derandomization, O. Cheong, K. Mulmuley, and E. Ramos Robust Geometric Computation, C.K. Yap Parallel Algorithms in Geometry, M.T. Goodrich Parametric Search, J.S. Salowe New! The Discrepancy Method in Computational Geometry, B. Chazelle APPLICATIONS OF DISCRETE AND COMPUTATIONAL GEOMETRY Linear Programming, M. Dyer, N. Megiddo, and E. Welzl Mathematical Programming, M.H. Todd Algorithmic Motion Planning, M. Sharir Robotics, D. Halperin, L.E. Kavraki, and J.-C. Latombe Computer Graphics, D. Dobkin and S. Teller New! Modeling Motion, L.J. Guibas Pattern Recognition, J. O'Rourke and G.T. Toussaint Graph Drawing, R. Tamassia and G. Liotta Splines and Geometric Modeling, C.L. Bajaj New! Surface Simplification and 3D Geometry Compression, J. Rossignac Manufacturing Processes, R. Janardan and T.C. Woo Solid Modeling, C.M. Hoffmann New! Computation of Robust Statistics: Depth, Median, and Related Measures, P.J. Rousseeuw and A. Struyf New! Geographic Information Systems, M. van Kreveld Geometric Application of the Grassmann-Cayley Algebra, N.L. White Rigidity and Scene Analysis, W. Whiteley Sphere Packing and Coding Theory, G.A. Kabatiansky and J.A. Rush Crystals and Quasicrystals, M. Senechal New! Biological Applications of Computational Topology, H. Edelsbrunner New! GEOMETRIC SOFTWARE Software, J. Joswig Two Computation Geometry Libraries: LEDA and CGAL, L. Kettner and S. Naher Index of Defined Terms New! Index of Cited Authors

We present parallel computational geometry algorithms that are scalable, architecture independent, easy to implement, and have, with high probability, an optimal time complexity for uniformly distributed random input data. Our methods apply to multicomputers with arbitrary interconnection network or bus system. The following problems are studied in this paper: (1) lower envelope of line segments, (2) visibility of parallelepipeds, (3) convex hull, (4) maximal elements, (5) Voronoi diagram, (6) all-nearest neighbors, (7) largest empty circle, and (8) largest empty hyperrectangle. Problems 2-8 are studied for d-dimensional space, d=O(1). We implemented and tested the lower envelope algorithm and convex hull algorithm (for d=3 and d=4) on a CM5. The results indicate that our methods are of considerable practical relevance. >

The Mars Exploration Rover (MER) mission was intended to land two identical rovers at different sites showing evidence of past water activity. This description of the landing site selection process is based on the extensive public documentation at the Mars Landing Site website at NASA Ames Research Center (marsoweb.nas.nasa.gov/landingsites/index.html) and a summary by Golombek et al. (2003). Maps and tables released at the time contain minor discrepancies, which are corrected here as noted.