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15
Nice Point Sets Can Have Nasty Delaunay Triangulations
 In Proc. 17th Annu. ACM Sympos. Comput. Geom
, 2001
"... We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in ..."
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Cited by 47 (5 self)
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We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in IR 3 with spread A has complexity il(min{A 3 , uA, u2}) and O (min{A 4, u2}). For the case A = D(v/), our lower bound construction consists of a gridlike sample of a right circular cylinder with constant height and radius. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has nearquadratic complexity.
Dense Point Sets Have Sparse Delaunay Triangulations
"... Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms ..."
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Cited by 31 (2 self)
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Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since threedimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worstcase running time \Omega (n2). However, this behavior is almost never observed in practice except for highlycontrived inputs. For all practical purposes, threedimensional Delaunay triangulations appear to have linear complexity. This frustrating
Combinatorial and experimental methods for approximate point pattern matching
 Algorithmica
, 2003
"... Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary ..."
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Cited by 6 (0 self)
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Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high degree polynomials, and require robust calculations of intersection points of high degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average case analysis and a detailed empirical study of our methods.
An image retrieval system using FPGAs
 in Proceedings of the Asia and South Pacific Design Automation Conference (ASPDAC ’03
, 2003
"... Abstract — The main contribution of this paper is to present an image retrieval system using FPGAs. Given a template image and a database of a number of images, our system lists all images that contain a subimage similar to. More specifically, a hardware generator in our system creates the Verilog H ..."
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Cited by 6 (2 self)
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Abstract — The main contribution of this paper is to present an image retrieval system using FPGAs. Given a template image and a database of a number of images, our system lists all images that contain a subimage similar to. More specifically, a hardware generator in our system creates the Verilog HDL source of a hardware that determines whether has a similar subimage to for any image and a particular template. The created Verilog HDL source is embed in an FPGA using the design tool provided by the FPGA vendor. Since the hardware embedded in the FPGA is designed for a particular template, it is an instancespecific hardware that allows us to achieve extreme acceleration. We evaluate the performance of our image matching hardware using a PCIconnected Xilinx FPGA and a timing analyzer. Since the generated hardware attains up to 3000 speedup factor over the software solution, our approach is promising.
Smoothed analysis of probabilistic roadmaps
 In Fourth SIAM Conference of Analytic Algorithms and Computational Geometry
, 2007
"... The probabilistic roadmap algorithm is a leading heuristic for robot motion planning. It is extremely efficient in practice, yet its worst case convergence time is unbounded as a function of the input’s combinatorial complexity. We prove a smoothed polynomial upper bound on the number of samples req ..."
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Cited by 5 (0 self)
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The probabilistic roadmap algorithm is a leading heuristic for robot motion planning. It is extremely efficient in practice, yet its worst case convergence time is unbounded as a function of the input’s combinatorial complexity. We prove a smoothed polynomial upper bound on the number of samples required to produce an accurate probabilistic roadmap, and thus on the running time of the algorithm, in an environment of simplices. This sheds light on its widespread empirical success. 1
Improved Approximation Bounds for Planar Point Pattern Matching
"... We consider the well known problem of matching two planar point sets under rigid transformations so as to minimize the directed Hausdorff distance. This is a well studied problem in computational geometry. Goodrich, Mitchell, and Orletsky [GMO94] presented a very simple approximation algorithm for t ..."
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Cited by 2 (1 self)
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We consider the well known problem of matching two planar point sets under rigid transformations so as to minimize the directed Hausdorff distance. This is a well studied problem in computational geometry. Goodrich, Mitchell, and Orletsky [GMO94] presented a very simple approximation algorithm for this problem, which computes transformations based on aligning pairs of points. They showed that their algorithm achieves an approximation ratio of 4. We consider a minor modification to their algorithm, which is based on aligning midpoints rather than endpoints. We show that this algorithm achieves an approximation ratio not greater than 3.13. Our analysis is sensitive to the ratio between the diameter of the pattern set and the Hausdorff distance, and we show that the approximation ratio approaches 3 in the limit. Finally, we provide lower bounds that show that our approximation bounds are nearly tight.
A Sweep Line Algorithm for Nearest Neighbour Queries
"... We introduce a novel algorithm for solving the nearest neighbour problem when the query points are known in advance, which is based on Fortune's plane sweep algorithm. The crucial idea is to use the wavefront for solving the nearest neighbour queries as the Voronoi diagram is being computed, in ..."
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Cited by 1 (0 self)
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We introduce a novel algorithm for solving the nearest neighbour problem when the query points are known in advance, which is based on Fortune's plane sweep algorithm. The crucial idea is to use the wavefront for solving the nearest neighbour queries as the Voronoi diagram is being computed, instead of storing it in an auxiliary data structure, as the algorithm presented by Lee and Yang [9] does, and then querying that data structure. Although
Title of dissertation: A FEATUREBASED SHAPE SIMILARITY ASSESSMENT FRAMEWORK
"... The popularity of 3D CAD systems is resulting in a large number of CAD models being generated. Availability of these CAD models is opening up new ways in which information can be archived, analyzed, and reused. 3D geometric information is one of the main components of CAD models. Therefore shape sim ..."
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The popularity of 3D CAD systems is resulting in a large number of CAD models being generated. Availability of these CAD models is opening up new ways in which information can be archived, analyzed, and reused. 3D geometric information is one of the main components of CAD models. Therefore shape similarity assessment is a fundamental geometric reasoning problem that finds several different applications. In many design and manufacturing applications, the gross shape of the 3D parts does not play an important role in the similarity assessment. Instead certain attributes of part features play a dominant role in determining the similarity between two parts. Different featurebased models are usually created using their own coordinate systems. Therefore, featurebased shape similarity assessment involves finding the optimal alignment transformations for two sets of feature vectors. The optimal alignment corresponds to the minimum value of a distance function that is computed between the two sets of feature vectors being aligned. In order to compute the distance function the closest neighbor to each feature vector needs to be identified. We have developed optimal feature alignment algorithms based on the partitioning of the transformation space into