ACM SIGACT News
Optimal Net Surface Problems with Applications
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Piecewise Linear Image Coding Using Surface Triangulation and Geometric Compression
DCC '00 Proceedings of the Conference on Data Compression
Efficient algorithms for bichromatic separability
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Computing large planar regions in terrains, with an application to fracture surfaces
Discrete Applied Mathematics - The 2001 international workshop on combinatorial image analysis (IWCIA 2001)
Variational shape approximation
ACM SIGGRAPH 2004 Papers
Surface compression with geometric bandelets
ACM SIGGRAPH 2005 Papers
Online geometric reconstruction
Proceedings of the twenty-second annual symposium on Computational geometry
Approximate range searching using binary space partitions
Computational Geometry: Theory and Applications
Efficient algorithms for bichromatic separability
ACM Transactions on Algorithms (TALG)
Algorithmic aspects of proportional symbol maps
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Polyhedral approximation and practical convex hull algorithm for certain classes of voxel sets
Discrete Applied Mathematics
Approximate range searching using binary space partitions
Computational Geometry: Theory and Applications
Polyhedral surface approximation of non-convex voxel sets through the modification of convex hulls
IWCIA'08 Proceedings of the 12th international conference on Combinatorial image analysis
Geodesic Methods in Computer Vision and Graphics
Foundations and Trends® in Computer Graphics and Vision
Online geometric reconstruction
Journal of the ACM (JACM)
Cup products on polyhedral approximations of 3D digital images
IWCIA'11 Proceedings of the 14th international conference on Combinatorial image analysis
Approximate range searching using binary space partitions
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
Operations Research Letters
The complexity of separating points in the plane
Proceedings of the twenty-ninth annual symposium on Computational geometry
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Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: given a set S of n points sampled from a bivariate function f(x,y) and an input parameter $\eps 0$, compute a piecewise-linear function $\Sigma(x,y)$ of minimum complexity (that is, an xy-monotone polyhedral surface, with a minimum number of vertices, edges, or faces) such that $| \Sigma(x_p, y_p) \; - \; z_p | \:\:\leq\:\: \eps$ for all $(x_p, y_p, z_p) \in S$. We give hardness evidence for this problem, by showing that a closely related problem is NP-hard. The main result of our paper is a polynomial-time approximation algorithm that computes a piecewise-linear surface of size O(Ko log Ko), where Ko is the complexity of an optimal surface satisfying the constraints of the problem.The technique developed in our paper is more general and applies to several other problems that deal with partitioning of points (or other objects) subject to certain geometric constraints. For instance, we get the same approximation bound for the following problem arising in machine learning: given n "red" and m "blue" points in the plane, find a minimum number of pairwise disjoint triangles such that each blue point is covered by some triangle and no red point lies in any of the triangles.