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12
Delaunay Triangulations of Imprecise Points in Linear Time after Preprocessing
, 2008
"... An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one ..."
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Cited by 24 (6 self)
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An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.
Largest and Smallest Convex Hulls for Imprecise Points
 ALGORITHMICA
, 2008
"... Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we d ..."
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Cited by 18 (4 self)
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Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n log n) to O(n^13), and prove NPhardness for some other variants.
Preprocessing imprecise points and splitting triangulations
 UTRECHT UNIVERSITY
, 2009
"... Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a ..."
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Cited by 15 (4 self)
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Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity n in O(n log n) time so that if one point per set is specified with precise coordinates, a triangulation of the points can be computed in linear time. In our solution, we solve another problem which we believe to be of independent interest. Given a triangulation with red and blue vertices, we show how to compute a triangulation of only the blue vertices in linear time.
Approximating Largest Convex Hulls for Imprecise Points
, 2007
"... Assume that a set of imprecise points is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NPhardness when the imp ..."
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Cited by 7 (1 self)
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Assume that a set of imprecise points is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NPhardness when the imprecise points are modelled as line segments, and give linear time approximation schemes for a variety of models, based on the coreset paradigm.
Basic Algorithms of Computational Geometry with Imprecise Input
, 2005
"... The domaintheoretic model of computational geometry provides us with continuous and computable predicates and binary operations. It can also be used to generalise the theory of computability for real numbers and real functions into geometric objects and geometric operations. A geometric object is c ..."
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Cited by 5 (0 self)
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The domaintheoretic model of computational geometry provides us with continuous and computable predicates and binary operations. It can also be used to generalise the theory of computability for real numbers and real functions into geometric objects and geometric operations. A geometric object is computable if it is the effective limit of a sequence of finitary partial objects of the same type as the original object. We are also provided with two different quantitative measures for approximation using the Hausdorff metric and the Lebesgue measure. In this thesis, we introduce a new data type to capture imprecise data or approximate points on the plane, given in the shape of compact convex polygons. This data type in particular includes rectangular approximation and is invariant under linear transformations of coordinate system. Based on the new data type, we define the notion of a number of partial geometric operations, including partial perpendicular bisector and partial disc and we show that these operations and the convex hull, Delaunay triangulation and Voronoi diagram are Hausdorff and Scott continuous and nestedly Hausdorff and Lebesgue computable. We develop algorithms to obtain the partial convex hull, partial Delaunay triangulation and partial Voronoi diagram. We prove that the complexity of the partial convex hull is N log N in 2D and 3D, whereas the partial Delaunay triangulation and partial Voronoi diagram algorithms for nondegenerate data have the same complexity as their classical counterparts. 2
Computing the discrete Fréchet distance with imprecise input
, 2010
"... We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algori ..."
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We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2 O(d2) m 2 n 2 log 2 (mn) the Fréchet distance lower bound between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 2 (mn)+(m 2 +n 2) log(mn)). In the ddimensional orthogonal case, where points are modelled as axisparallel boxes, and we use the L∞ distance, we give an O(dmn log(dmn))time algorithm. We also give efficient O(dmn)time algorithms to approximate the Fréchet distance upper bound, as well as the smallest possible Fréchet distance lower/upper bound that can be achieved between two imprecise point sequences when one is allowed to translate them. These algorithms achieve constant factor approximation ratios in “realistic ” settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).
Part I Case for Support Computing with arbitrary precision curves
, 2004
"... Previous research track record and plans for the future The proposer has a longstanding interest in the theory and applications of computing with arbitrary precision (AP), which began with his PhD training (1996–2000) at the University of Birmingham under the supervision of Prof. Jung. PhD thesis. ..."
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Previous research track record and plans for the future The proposer has a longstanding interest in the theory and applications of computing with arbitrary precision (AP), which began with his PhD training (1996–2000) at the University of Birmingham under the supervision of Prof. Jung. PhD thesis. The thesis (Konečn´y 2000) built on the work of Wiedmer, Weihrauch, Edalat, Ko and others on representing real numbers as infinite streams of digits in an arbitrary precision computation. The thesis addressed, in this context, one of the first fundamental questions in computational complexity theory: Which realnumber functions can be computed to an arbitrary precision without an ever growing need for more memory? This question was answered for many different ways of representing the real numbers as infinite streams of symbols and for all reasonably wellbehaved functions (in some precise sense). This result is also described and proved in two journal articles (Konečn´y 2004, Konečn´y 2002), each for different types of realnumber representations. The thesis extends the articles in terms of the scope of representations and also generalises the theorem in another direction. It covers not only functions but also onetomany mappings. Such mappings arise naturally from the fact that each real number can be represented in many ways: for different representations of the same arguments different, but all correct results may be computed. More importantly, some frequently occurring practical problems, such as finding a zero of a polynomial, cannot be computed as a function but rather as a onetomany mapping. Work in Edinburgh. More recently, the proposer worked as a research fellow for the EPSRC funded project “Type
Visual Hull from Imprecise Polyhedral Scene
"... Abstract—We present a framework to compute the visual hull of a polyhedral scene, in which the vertices of the polyhedra are given with some imprecision. Two kinds of visual event surfaces, namely VE and EEE surfaces are modelled under the geometric framework to derive their counterpart object, name ..."
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Abstract—We present a framework to compute the visual hull of a polyhedral scene, in which the vertices of the polyhedra are given with some imprecision. Two kinds of visual event surfaces, namely VE and EEE surfaces are modelled under the geometric framework to derive their counterpart object, namely partial VE and partial EEE surfaces, which contain the exact information of all possible visual event surfaces given the imprecision in the input. Correspondingly, a new definition of visual number is proposed to label the cells of Euclidean space partitioned by partial VE and partial EEE surfaces. The overall algorithm maintains the same computational complexity as the classical method and generates a partial visual hull which converges to the classical visual hull as the input converges to an exact value. Keywordsvisual hull; shape from silhouettes; imprecise input; solid domain; quadratic surface I.
Basic Measures for . . .
, 2008
"... Most algorithms in computational geometry tend to assume that all input is exact, with no imprecision or error. Most realworld data however, has some imprecision (for example due to measurement error). Thus, there exists a need for algorithms that can produce meaningful output for imprecise input d ..."
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Most algorithms in computational geometry tend to assume that all input is exact, with no imprecision or error. Most realworld data however, has some imprecision (for example due to measurement error). Thus, there exists a need for algorithms that can produce meaningful output for imprecise input data. In this thesis, I present results on the computation of upper and lower bounds on various basic measures (such as diameter, width, closest pair, volume of smallest enclosing ball and volume of minimum axis aligned bounding box) for imprecise point sets in R d. I model the imprecision by representing an imprecise point set as a set of regions (balls or polytopes), such that each point may lie anywhere within one of the regions. This work is an extension of previous research by Löffler and van Kreveld on imprecise point sets in R², to higher dimensions.