### Abstract

Given a polygonal curve P = [p_{1}, p_{2}, · · ·, p_{n}], the polygonal approximation problem considered in this paper calls for determining a new curve P' = [p_{1}, p_{2}> · · ·, p_{m}] 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 [p_{A}, p_{A+1}] of P' that substitutes a chain [p_{B},..., p_{C}] in P is such that for all i where B ≤ i ≤ C, the approximation error of p_{i} with respect to [p_{A}, P_{A+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 R^{d}, 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 L_{1}, L_{2} or L_{∞} distance metrics, we give algorithms to solve the min-# problem in 0(n^{2}) time and the min-∈ problem in 0(n^{2} log n) time, improving the best known algorithms to date by a factor of log n. When P is a polygonal curve in R^{3} that is strictly monotone with respect to one of the three axes, we show that if the L_{1} and L_{∞} metrics are used then the min-# problem can be solved in O(n^{2}) time and the min-∈ problem can be solved in O(n^{3}) time. If distances are computed using the L_{2} metric then the min-# and min-∈ problems can be solved in O(n^{3}) and O(n^{3} log n) time respectively. All our algorithms exhibit O(n^{2}) space complexity.

Original language | English (US) |
---|---|

Pages (from-to) | 97-110 |

Number of pages | 14 |

Journal | Proceedings of SPIE - The International Society for Optical Engineering |

Volume | 1832 |

DOIs | |

State | Published - Apr 9 1993 |

Event | Vision Geometry 1992 - Boston, United States Duration: Nov 16 1992 → … |

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### 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.

Research output: Contribution to journal › Conference article

}

TY - JOUR

T1 - On approximating polygonal curves in two and three dimensions

AU - Eu, David

AU - Toussaint, Godfried T.

PY - 1993/4/9

Y1 - 1993/4/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84919238436&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84919238436&partnerID=8YFLogxK

U2 - 10.1117/12.142160

DO - 10.1117/12.142160

M3 - Conference article

AN - SCOPUS:84919238436

VL - 1832

SP - 97

EP - 110

JO - Proceedings of SPIE - The International Society for Optical Engineering

JF - Proceedings of SPIE - The International Society for Optical Engineering

SN - 0277-786X

ER -