P. Agarwal and S. Suri. Surface approximation and geometric partitions. Proc. 5th Annu. ACM Sympos. Discrete Algorithms, 1994. pp 24-33.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....problem for surfaces The support problem is a simpler optimization problem, using only one sided constraints. Remarkably, if we were to attempt to solve SUPPORT over all nonsmooth functions with arbitrary break points, rather than for splines with fixed break points, the problem would be NP hard [1]. Fig. 16. A sharp 2D channel. left) The control polygon of the solution lies well outside the channel. right) The control polygon of the solution after subdivision still violates the channel boundaries illustrating the need for the tight slefe bounds. Fig. 17. Bilinear barrier surface (left) ....

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. SIAM Journal on Computing, 27(4):1016--1035, August 1998.

.... Given some input mesh = p i j ] the task is to find another mesh # = p # i # j ] which has a prescribed number of triangles and minimizes the approximation error #M M (cf. 7, 10] For most applications, the computation of the exact global optimum is far too complex [1]. Hence, one usually tries to find solutions # with approximate optimality where the computation costs can be traded for geometric suboptimality. Over the last years the greedy optimization paradigm has established the de facto standard for mesh decimation algorithms. In the greedy approach, ....

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proceedings of 5th ACM-SIAM Symposium on Discrete Algorithms, pages 24--33, 1994.

....the decimation problem can be understood as an instance of the knapsack problem with the number of decimation operations being the objective function and the geometric approximation error being the capacity function. Obviously, finding the optimal decimation sequence is a very complex problem [1] and consequently one has to find solutions with approximate optimality. The above best first strategy is, in fact, a greedy strategy to find a decimation sequence that is close to optimal. In this paper, we are using a different optimization strategy to address the mesh decimation problem. ....

P. Agarwal, S. Suri, "Surface Approximation and Geometric Partitions", Proceedings of 5th ACM-SIAM Symposium on Discrete Algorithms, pp.24-33, 1994.

....few methods able to perform adaptive surface simplification of a massive mesh. It may be the only method able to do so efficiently. 2 PREVIOUS WORK Polygonal simplification [7, 4] has been an area of active research for close to a decade. Optimal approximation of a surface is known to be NP Hard [1], and hence most research has focused on developing heuristic methods. The fundamental principle underlying most of these methods is that of partitioning the vertex set of a mesh into disjoint clusters and unifying the vertices within each cluster. This process of vertex unification will cause ....

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 24--33,

....Summary We began this research project in the summer of 1995. At that time, there were relatively few publications on the subject of general polygonal mesh simplification; much of the earlier work focused on the simplification of convex polyhedra [Das and Joseph 1990] and polyhedral terrains [Agarwal and Suri 1994]. There were several publications on the more general problem, though. The most well known of these were [Rossignac and Borrel 1992] Schroeder et al. 1992] Turk 1992] and [Hoppe et al. 1993] During the course of this research, the field of automatic simplification has become much more ....

....Given some target complexity, n, and an input model, I, compute the approximation, A, with the minimum error, e, described above. In computational geometry, it has been shown that computing the min # problem is NPhard for both convex polytopes [Das and Joseph 1990] and polyhedral terrains [Agarwal and Suri 1994]. Thus, algorithms to solve these problems have evolved around finding polynomial time approximations that are close to the optimal. 18 Let k 0 be the size of a min # approximation. An algorithm has been given in [Mitchell and Suri 1992] for computing an e approximation of size O(k 0 log n) for ....

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Agarwal, Pankaj K. and Subhash Suri. Surface Approximation and Geometric Partitions. Proceedings of 5th ACM-SIAM Symposium on Discrete Algorithms. 1994. pp. 24-33.

....(DP) algorithm; and (3) showing that the DP solution can be transformed into a solution of the original problem, with a small factor increase in total length. This method is not new; e.g. it has been used by Gonzalez et al. 16] in optimal partitioning problems, and by Agarwal and Suri [1] and Mitchell [22] in other geometric separation and approximation problems. 2 Geometric Preliminaries We restrict our attention to problems in two dimensions. We will work with planar polygonal subdivisions and triangulations, always within some bounded polygonal region, R. The length of a ....

.... points of only one color (from R or from B, but not from both) If, instead of minimizing the Euclidean length, we want to minimize the combinatorial size of a separating simple polygon (i.e. the number of vertices or links ) then an O(logk ) approximation bound has recently been established ([1, 22]) where k is the size of an optimal separator; the problem is known to be NP hard in this case [14] Here, we give an approximation result for the problem of minimizing the Euclidean length of the separator, a problem shown to be NP hard in [3, 12] Consider a minimum length separating simple ....

P. K. Agarwal and S. Suri. Surface approximation and geometric partitions. In Proc. Fifth Annual ACM-SIAM Sympos. on Discrete Algorithms, 1994, pages 34--43.

....generate a resolution of the original space. The BSP trees have a wide usage in many areas of computer science. They are used, for example, in hidden surface removal using painters algorithm [11] visibility solution [19] shadow generation [7] objects modeling [14, 20] surface approximation [3], or robot motion planning [5] When we split the space by a hyperplane then some objects can be unwillingly divided into two or more parts. In such way, the original scene will be divided into a lot of fragments. However, the eOEciency of algorithms beneting from BSP depends on the size of ....

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 2433, 1994.

....generate a resolution of the original space. The BSP trees have a wide usage in many areas of computer science. They are used, for example, in hidden surface removal using painters algorithm [11] visibility solution [19] shadow generation [7] objects modeling [14, 20] surface approximation [3], or robot motion planning [5] When we split the space by a hyperplane then some objects can be unwillingly divided into two or more parts. In such way, the original scene will be divided into a lot of fragments. However, the e#ciency of algorithms benefiting from BSP depends on the size of ....

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 24--33, 1994.

.... to be versatile, with applications in many problems apart from hiddensurface removal [8, 101] global illumination [23] shadow generation [29, 30] solid modelling [75, 78, 102] ray tracing [74] robotics [12] and approximation algorithms for network design [66] and surface simplification [7]. The efficiency of our model repair and hidden surface removal algorithms algorithms as well as the applications mentioned above inherently depends on the size and height of the BSP. In Chapters 4 6, we describe our algorithms for constructing BSPs of small size. We show that BSPs of ....

.... that use the BSP in a variety of different applications: hidden surface removal itself [8, 101] global illumination [23] shadow generation [29, 30] solid modelling [75, 78, 102] ray tracing [74] robotics [12] and approximation algorithms for network design [66] and surface simplification [7]. As a result, there has been a lot of effort to construct BSPs of small size. While several simple heuristics have been developed for constructing BSPs of reasonable sizes [8, 24, 47, 73, 101, 102] provable bounds were first obtained by Paterson and Yao. They show that a BSP of size O(n log n) ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, SIAM Journal on Computing, 27 (1998), 1016--1035. 152 153

....the original space. The BSP trees have a wide usage in many areas of computer science. They are used, for example, in hidden surface removal using painters algorithm [Fuchs80] visibility solution [Telle92] shadow generation [Chin89] objects modelling [Naylo90, Thiba87] surface approximation [Agarw94], or robot motion planing [Balli93] When we split the space by some hyperplane then some objects can be unwillingly divided into two or more parts. In such way, the original scene can be divided into a lot of fragments. An example of two alternative BSPs for a set of segments is displayed on ....

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 2433, 1994.

....problem can be solved in polynomial time [HS91a] Notice, however, that a slightly different two dimensional point set separation problem, which asks for a general separating polygon, is NP hard; this is exactly the red blue separation problem mentioned in Section 8.2.3. Agarwal and Suri [AS94] showed that the three dimensional surface fitting problem is NP hard using a reduction from planar 3 SAT. It is not clear, however, whether the problem is NP complete, as good bounds are not known on the numerical precision needed to represent the optimal function. Agarwal and Suri also devised ....

P.K. Agarwal and S. Suri. Surface approximation and geometric partitions. In Proc. 5th ACM/SIAM Symp. Discrete Algorithms, pages 24--33, 1994.

....the original space. The BSP trees have a wide usage in many areas of computer science. They are used, for example, in hidden surface removal using painters algorithm [Fuchs80] visibility solution [Telle92] shadow generation [Chin89] objects modelling [Naylo90, Thiba87] surface approximation [Agarw94], or robot motion planing [Balli93] When we split the space by some hyperplane then some objects can be unwillingly divided into two or more parts. In such way, the original scene can be divided into a lot of fragments. An example of two alternative BSPs for a set of segments is displayed on ....

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 24--33, 1994.

....complex monotone chains, our algorithm can obtain approximate ratios between 22 and 32 . 3.3 Some comments on approximating polyhedral terrains Approximating (simplifying) polyhedral terrains is a very important problem in GIS and spatial databases. Theoretically this problem is NP hard [AS94] and some provably good approximations are known [AS94; AD97] Most previous work are based on heuristics (see [AD97] for a complete list of references) We believe our algorithm, combined with the recent practical convex decomposition algorithm of Chazelle et al. CDST97] will produce good ....

....number of convex pieces. Therefore, this idea should work well and besides that, both the heuristic algorithm of Chazelle et al. and ours runs in O(n log n) time so the running time should not be causing any problem even if the data has a huge size. We note that the approximation algorithm of [AS94] has a time complexity of O(n 8 ) and the one proposed in [AD97] runs in O(n 2 ffi ) time. 4 Concluding Remarks In this paper we present a very simple probablistic algorithm to approximate 2D and 3D geometric objects. This algorithm is very easy to implement and the empirical results are ....

P. Agarwal, and S. Suri. Surface approximation and geometric partitions. In Proc. 5th Sympos. Discrete Algorithms, pages 24--33, 1994.

....time algorithms are known, however, for sub optimal solutions with provable size and quality bounds. If the optimal L# solution for a given error tolerance has m o vertices, there is an O(n 7 ) algorithm 15 to find an approximation with the same error using m= O(m o log m o ) vertices [87, 1], but this is far too slow to be practical for large problems. 3.2 Manifold Surfaces We now turn our attention from height fields and parametric surfaces to manifolds and manifolds with boundary. In general, the manifold can have arbitrary genus and be nonorientable 7 unless stated otherwise. ....

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 24--33, 1994. (Also available as Duke U. CS tech report, ftp://ftp.cs.duke.edu/ dist/techreport/1994/1994-21.ps.Z).

....conveying the shape of an object within some error bound of the original mesh. Unfortunately, the space of possible meshes is so large that searching for the globally best mesh is impractical. In fact, the simpler problem of computing a minimal facet polyhedral terrain model is an NP hard problem[1]. Therefore, most simplification algorithms to date are greedy, iterative algorithms that search for the best mesh by taking the current best step toward the global minimum. Like many mesh simplification algorithms, our algorithm is based on the iterative application of local mesh operations to ....

P. Agarwal and S. Suri. "Surface approximation and geometric partitions." Proc. Fifth Symp. Discrete Algorithms, pp. 24-33, 1994,

....connectivity and then multiresolution wavelet analysis is used over each patch. This wavelet approach preserves global topology. In computational geometry, it has been shown that computing the minimal facet ffl approximation is NP hard for both convex polytopes [DJ90] and polyhedral terrains [AS94] Thus, algorithms to solve these problems have evolved around finding polynomial time approximations that are close to the optimal. Let k o be the size of a min # approximation. An algorithm has been given in [MS92] for computing an ffl approximation of size O(k o log n) for convex polytopes. ....

.... constant of proportionality depends on ffi , and tends to 1 as ffi tends to 0) In [BG94] 3 Bronnimann and Goodrich observed that a variant of Clarkson s algorithm yields a polynomial time deterministic algorithm that computes an approximation of size O(k 0 ) Working with polyhedral terrains, AS94] present a polynomial time algorithm that computes an ffl approximation of size O(k o log k o ) to a polyhedral terrain. Our work is different from the above in that it allows adaptive, genus preserving, ffl approximation of polygonal objects, such that the complexity of the approximate solution ....

P. Agarwal and S. Suri. Surface approximation and geometric partitions. In Proceedings Fifth Symposium on Discrete Algorithms, pages 24--33, 1994.

....connectivity and then multiresolution wavelet analysis is used over each patch. This wavelet approach preserves global topology. In computational geometry, it has been shown that computing the minimal facet # approximation is NP hard for both convex polytopes [5] and polyhedral terrains [1]. Thus, algorithms to solve these problems have evolved around finding polynomial time approximations that are close to the optimal. Let k o be the size of a min # approximation. An algorithm has been given in [16] for computing an # approximation of size O#k o log n# for convex polytopes. This ....

....any ##0 (the constant of proportionality depends on #, and tends to 1 as # tends to 0) In [2] Bronnimann and Goodrich observed that a variant of Clarkson s algorithm yields a polynomial time deterministic algorithm that computes an approximation of size O#k 0 #. Working with polyhedral terrains, [1] present a polynomial time algorithm that computes an # approximation of size O#k o log k o # to a polyhedral terrain. Our work is different from the above in that it allows adaptive, genus preserving, # approximation of arbitrary polygonal objects. Additionally, we can simplify bordered meshes ....

P. Agarwal and S. Suri. Surface approximation and geometric partitions. In Proceedings Fifth Symposium on Discrete Algorithms, pages 24--33, 1994.

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P. Agarwal and S. Suri. Surface approximation and geometric partitions. Proc. 5th Annu. ACM Sympos. Discrete Algorithms, 1994. pp 24-33.

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Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proceedings 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 24-- 33, Arlington, Virginia, USA, January 1994.

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Agarwal, P.K., Suri, S., 1994, Surface approximation and geometric partitions, Proceedings 5th ACM-SIAM Symposium on Discrete Algorithms, pp. 24--33.

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P. Agarwal and S. Suri, Surface approximation and geometric partitions, in Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 1994, pp. 24--33.

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Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 24--33, 1994.

No context found.

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 24--33, 1994.

No context found.

Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 24-- 33, 1994. (Also available as DukeU. CS tech report, ftp: //ftp.cs.duke.edu/dist/techreport/1994/1994-21.ps.Z).

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P.K. Agarwal and S. Suri. Surface approximation and geometric partitions. In Proceedings 5th ACM-SIAM Symposium On Discrete Algorithms, pages 24--33, 1994.

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