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NearLinear Approximation Algorithms for Geometric Hitting Sets
 SCG'09
, 2009
"... Given a set system (X, R), the hitting set problem is to find a smallestcardinality subset H ⊆ X, with the property that each range R ∈ R has a nonempty intersection with H. We present nearlinear time approximation algorithms for the hitting set problem, under the following geometric settings: (i ..."
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Given a set system (X, R), the hitting set problem is to find a smallestcardinality subset H ⊆ X, with the property that each range R ∈ R has a nonempty intersection with H. We present nearlinear time approximation algorithms for the hitting set problem, under the following geometric settings: (i) R is a set of planar regions with small union complexity. (ii) R is a set of axisparallel drectangles in R d. In both cases X is either the entire ddimensional space or a finite set of points in dspace. The approximation factors yielded by the algorithm are small; they are either the same as or within an O(log n) factor of the best factors known to be computable in polynomial time.
D.: Fast and robust generation of cityscale seamless 3d urban models
 In: SIAM Conference on Geometric and Physical Modeling (GD/SPM). SIAM/ACM
, 2011
"... Since the introduction of the concept of “Digital Earth”, almost every major international city has been reconstructed in the virtual world. A large volume of geometric models describing urban objects has become freely available in the public domain via software like ArcGlobe and Google Earth. Alth ..."
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Since the introduction of the concept of “Digital Earth”, almost every major international city has been reconstructed in the virtual world. A large volume of geometric models describing urban objects has become freely available in the public domain via software like ArcGlobe and Google Earth. Although mostly created for visualization, these urban models can benefit many applications beyond visualization including city scale evacuation planning and earth phenomenon simulations. However, these models are mostly loosely structured and implicitly defined and require tedious manual preparation that usually takes weeks if not months before they can be used. Designing algorithms that can robustly and efficiently handle unstructured urban models at the city scale becomes a main technical challenge. In this paper, we present a framework that generates seamless 3D architectural models from 2D ground plans with elevation and height information. These overlapping ground plans are commonly used in the current GIS software such as ESRI ArcGIS and urban model synthesis methods to depict various components of buildings. Due to measurement and manual errors, these ground plans usually contain small, sharp, and various (nearly) degenerate artifacts. In this paper, we show both theoretically and empirically that our framework is efficient and numerically stable. Based on our review of the related work, we believe this is the first work that attempts to automatically create 3D architectural meshes for simulation at the city level. With the goal of providing greater benefit beyond visualization from this large volume of urban models, our initial results are encouraging.
Noname manuscript No. (will be inserted by the editor) Learning Optimal Metric for Image Alignment
"... Abstract Image alignment has been a long standing problem ..."
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Vis Comput DOI 10.1007/s0037101106450 ORIGINAL ARTICLE
, 2011
"... Creating building ground plans via robust Kway union A step toward largescale simulation in urban environment ..."
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Creating building ground plans via robust Kway union A step toward largescale simulation in urban environment
Discrete Comput Geom DOI 10.1007/s004540109312x On the Union of Cylinders in Three Dimensions
"... Abstract We show that the combinatorial complexity of the union of n infinite cylinders in R3, having arbitrary radii, is O(n2+ε), for any ε> 0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000), who establishe ..."
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Abstract We show that the combinatorial complexity of the union of n infinite cylinders in R3, having arbitrary radii, is O(n2+ε), for any ε> 0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000), who established a nearlyquadratic bound for the restricted case of nearly congruent cylinders. Our result extends, in a significant way, the result of Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000), in particular, a simple specialization of our analysis to the case of nearly congruent cylinders yields a nearlyquadratic bound on the complexity of the union in that case, thus significantly simplifying the analysis in Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000). Finally, we extend our technique to the case of “cigars ” of arbitrary radii (that is, Minkowski sums of linesegments and balls) and show that the combinatorial complexity of the union in this case is nearlyquadratic as well. This problem has been studied in Agarwal and Sharir (Discrete Comput. Geom. 24:645– 685, 2000) for the restricted case where all cigars have (nearly) equal radii. Based on our new approach, the proof follows almost verbatim from the analysis for infinite cylinders and is significantly simpler than the proof presented in Agarwal and Sharir
Fast and Robust Generation of City Scale Urban Ground Plan
"... Abstract Since the introduction of the concept of Digital Earth, almost every major international city has been reconstructed in the virtual world. A large volume of geometric models describing urban objects has become freely available in public domain via software like Google Earth. Although mos ..."
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Abstract Since the introduction of the concept of Digital Earth, almost every major international city has been reconstructed in the virtual world. A large volume of geometric models describing urban objects has become freely available in public domain via software like Google Earth. Although mostly created for visualization, these urban models can benefit many applications beyond visualization including video games, city scale evacuation plan, traffic simulation and earth phenomenon simulations. However, these urban models are mostly loosely structured and implicitly defined and require tedious manual preparation that usually take weeks if not months before they can be used. In this paper, we present a framework that produces welldefined ground plans from these urban models, an important step in the preparation process. Designing algorithms that can robustly and efficiently handle unstructured urban models at city scale is the main technical challenge. In this work, we show both theoretically and empirically that our method is resolution complete, efficient and numerically stable. Based on our review of the related work, we believe this is the first work that attempts to create urban ground plans automatically from 3D architectural meshes at city level. With the goal of providing greater benefit beyond visualization from this large volume of urban models, our initial results are encouraging.