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51
Dynamic planar convex hull
 Proc. 43rd IEEE Sympos. Found. Comput. Sci
, 2002
"... In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage o ..."
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Cited by 63 (1 self)
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In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.
Practical shadow mapping
 Journal of Graphics Tools
, 2000
"... In this paper we propose several methods that can greatly improve image quality when using the shadow mapping algorithm. Shadow artifacts introduced by shadow mapping are mainly due to low resolution shadow maps and/or the limited numerical precision used when performing the shadow test. These probl ..."
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Cited by 58 (9 self)
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In this paper we propose several methods that can greatly improve image quality when using the shadow mapping algorithm. Shadow artifacts introduced by shadow mapping are mainly due to low resolution shadow maps and/or the limited numerical precision used when performing the shadow test. These problems especially arise when the light source’s viewing frustum, from which the shadow map is generated, is not adjusted to the actual camera view. We show how a tight fitting frustum can be computed such that the shadow mapping algorithm concentrates on the visible parts of the scene and takes advantage of nearly the full available precision. Furthermore, we recommend uniformly spaced depth values in contrast to perspectively spaced depths in order to equally sample the scene seen from the light source. 1.
Classroom examples of robustness problems in geometric computations
 In Proc. 12th European Symposium on Algorithms, volume 3221 of Lecture Notes Comput. Sci
, 2004
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Efficient Reverse kNearest Neighbor Search in Arbitrary Metric Spaces
, 2006
"... The reverse knearest neighbor (RkNN) problem, i.e. finding all objects in a data set the knearest neighbors of which include a specified query object, is a generalization of the reverse 1nearest neighbor problem which has received increasing attention recently. Many industrial and scientific appl ..."
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Cited by 32 (9 self)
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The reverse knearest neighbor (RkNN) problem, i.e. finding all objects in a data set the knearest neighbors of which include a specified query object, is a generalization of the reverse 1nearest neighbor problem which has received increasing attention recently. Many industrial and scientific applications call for solutions of the RkNN problem in arbitrary metric spaces where the data objects are not Euclidean and only a metric distance function is given for specifying object similarity. Usually, these applications need a solution for the generalized problem where the value of k is not known in advance and may change from query to query. However, existing approaches, except one, are designed for the specific R1NN problem. In addition — to the best of our knowledge — all previously proposed methods, especially the one for generalized RkNN search, are only applicable to Euclidean vector data but not for general metric objects. In this paper, we propose the first approach for efficient RkNN search in arbitrary metric spaces where the value of k is specified at query time. Our approach uses the advantages of existing metric index structures but proposes to use conservative and progressive distance approximations in order to filter out true drops and true hits. In particular, we approximate the knearest neighbor distance for each data object by upper and lower bounds using two functions of only two parameters each. Thus, our method does not generate any considerable storage overhead. We show in a broad experimental evaluation on realworld data the scalability and the usability of our novel approach.
An adaptable and extensible geometry kernel
 In Proc. Workshop on Algorithm Engineering
, 2001
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Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 23 (4 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
KNearest Neighbor Search for Fuzzy Objects
"... The KNearest Neighbor search (kNN) problem has been investigated extensively in the past due to its broad range of applications. In this paper we study this problem in the context of fuzzy objects that have indeterministic boundaries. Fuzzy objects play an important role in many areas, such as biom ..."
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Cited by 8 (1 self)
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The KNearest Neighbor search (kNN) problem has been investigated extensively in the past due to its broad range of applications. In this paper we study this problem in the context of fuzzy objects that have indeterministic boundaries. Fuzzy objects play an important role in many areas, such as biomedical image databases and GIS. Existing research on fuzzy objects mainly focuses on modelling basic fuzzy object types and operations, leaving the processing of more advanced queries such as kNN query untouched. In this paper, we propose two new kinds of kNN queries for fuzzy objects, Adhoc kNN query (AKNN) and Range kNN query (RKNN), to find the k nearest objects qualifying at a probability threshold or within a probability range. For efficient AKNN query processing, we optimize the basic bestfirst search algorithm by deriving more accurate approximations for the distance function between fuzzy objects and the query object. To improve the performance of RKNN search, effective pruning rules are developed to significantly reduce the search space and further speed up the candidate refinement process. The efficiency of our proposed algorithms as well as the optimization techniques are verified with an extensive set of experiments using both synthetic and real datasets.