On approximating polygonal curves in two and three dimensions

Research output: Contribution to journalConference article

Abstract

Given a polygonal curve P = [p1, p2, · · ·, pn], the polygonal approximation problem considered in this paper calls for determining a new curve P' = [p1, p2> · · ·, pm] such that (i) m is significantly smaller than n, (ii) the vertices of P are a subset of the vertices of P and (iii) any line segment [pA, pA+1] of P' that substitutes a chain [pB,..., pC] in P is such that for all i where B ≤ i ≤ C, the approximation error of pi with respect to [pA, PA+1], according to some specified criterion and metric, is less than a predetermined error tolerance. Using the parallel-strip error criterion, we study the following problems for a curve P in Rd, where d ≥ 2: (i) minimize m for a given error tolerance and (ii) given m, find the curve P' that has the minimum approximation error over all curves that have at most m vertices. These problems are called the min-# and min-∈ problems, respectively. For R2 and with any one of the L1, L2 or L distance metrics, we give algorithms to solve the min-# problem in 0(n2) time and the min-∈ problem in 0(n2 log n) time, improving the best known algorithms to date by a factor of log n. When P is a polygonal curve in R3 that is strictly monotone with respect to one of the three axes, we show that if the L1 and L metrics are used then the min-# problem can be solved in O(n2) time and the min-∈ problem can be solved in O(n3) time. If distances are computed using the L2 metric then the min-# and min-∈ problems can be solved in O(n3) and O(n3 log n) time respectively. All our algorithms exhibit O(n2) space complexity.

Original languageEnglish (US)
Pages (from-to)97-110
Number of pages14
JournalProceedings of SPIE - The International Society for Optical Engineering
Volume1832
DOIs
StatePublished - Apr 9 1993
EventVision Geometry 1992 - Boston, United States
Duration: Nov 16 1992 → …

Fingerprint

Three-dimension
Two Dimensions
Curve
curves
apexes
Approximation Error
approximation
Metric
Tolerance
Polygonal Approximation
set theory
Distance Metric
strip
Space Complexity
Approximation Problem
p.m.
Line segment
Substitute
Pi
substitutes

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics
  • Computer Science Applications
  • Applied Mathematics
  • Electrical and Electronic Engineering

Cite this

On approximating polygonal curves in two and three dimensions. / Eu, David; Toussaint, Godfried T.

In: Proceedings of SPIE - The International Society for Optical Engineering, Vol. 1832, 09.04.1993, p. 97-110.

Research output: Contribution to journalConference article

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