An efficient algorithm for terrain simplification
Pankaj K. Agarwal,Pavan K. Desikan +1 more
- pp 139-147
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A randomized algorithm is presented that computes an {epsilon}-approximation of size O(c{sup 2} log{Sup 2} c) in O(n {sup 2+{delta}} + c{sup 3} log {Sup 3}c log n/c) expected time, where c is the size of the {Epsilon]- approximation with the minimum number of vertices and {delta} is any arbitrarily small positiveAbstract:
Given a set S of n points in {Re}{sup 3}, sampled from an unknown bivariate function f (x, y) (i.e., for each point p {element_of} S, z{sub p} = f (x{sub p}, y{sub p})), a piecewise-linear function g(x, y) is called an {epsilon}-approximation of f (x, y) if for every p {element_of} S, {vert_bar}f (x, y) - g (x, y){vert_bar} {le} {epsilon}. The problem of computing an {epsilon}-approximation with the minimum number of vertices is NP-Hard. We present a randomized algorithm that computes an {epsilon}-approximation of size O(c{sup 2} log{sup 2} c) in O(n{sup 2+{delta}} + c{sup 3} log{sup 2}c log n/c) expected time, where c is the size of the {epsilon}-approximation with the minimum number of vertices and {delta} is any arbitrarily small positive number. Under some reasonable assumptions, the size of the output is close to O(c log c) and the expected running time is O(n{sup 2+{delta}}). We have implemented a variant of this algorithm and include some empirical results.read more
Citations
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Book
Geometric Approximation Algorithms
TL;DR: This book is the first to cover geometric approximation algorithms in detail, and topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings.
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Efficient algorithms for geometric optimization
Pankaj K. Agarwal,Micha Sharir +1 more
TL;DR: A wide range of applications of parametric searching and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.
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Multiresolution Modeling: Survey and Future Opportunities
TL;DR: This report begins with a survey of the most notable available algorithms for automatic simplification of polygonal models, and considers the most significant directions in which existing simplification methods can be improved.
Quadric-based polygonal surface simplification
Michael Garland,Paul S. Heckbert +1 more
TL;DR: This dissertation presents a simplification algorithm, based on iterative vertex pair contraction, that can simplify both the geometry and topology of manifold as well as non-manifold surfaces, and proves a direct mathematical connection between the quadric metric and surface curvature.
Journal ArticleDOI
Optimal triangulation and quadric-based surface simplification
Paul S. Heckbert,Michael Garland +1 more
TL;DR: It is shown that in the limit as triangle area goes to zero on a differentiable surface, the quadric error is directly related to surface curvature, and in this limit, a triangulation that minimizes the Quadric error metric achieves the optimal triangle aspect ratio in that it minimized theL2 geometric error.
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