Given a polygonal curve P equals [p1, p2, ..., pn], the polygonal approximation problem considered in this paper calls for determining a new curve P' equals [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 [pB, ..., pC] in P is such that for all i where B <= i <= C, the approximation error of Pi 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 Rd, where d >= 2: (1) 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-(epsilon) problems, respectively. For R2 and with any one of the L1, L2 or L(infinity) distance metrics, we give algorithms to solve the min-# problem in O(n2) time and the min-(epsilon) problem in O(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 LINF metrics are used then the min-# problem can be solved in O(n2) time and the min-(epsilon) problem can be solved in O(n3) time. All our algorithms exhibit O(n2) space complexity.
[1]
Luigi P. Cordella,et al.
An O(N) algorithm for polygonal approximation
,
1985,
Pattern Recognit. Lett..
[2]
Theodosios Pavlidis,et al.
Segmentation of Plane Curves
,
1974,
IEEE Transactions on Computers.
[3]
D. Avis,et al.
Simple On-Line Algorithms for Convex Polygons
,
1985
.
[4]
S. Louis Hakimi,et al.
Fitting polygonal functions to a set of points in the plane
,
1991,
CVGIP Graph. Model. Image Process..
[5]
Hiroshi Imai,et al.
Computational-geometric methods for polygonal approximations of a curve
,
1986,
Comput. Vis. Graph. Image Process..
[6]
Jack Sklansky,et al.
Fast polygonal approximation of digitized curves
,
1980,
Pattern Recognit..
[7]
Michael Ian Shamos,et al.
Computational geometry: an introduction
,
1985
.
[8]
Jack Sklansky,et al.
Minimum-Perimeter Polygons of Digitized Silhouettes
,
1972,
IEEE Transactions on Computers.
[9]
M. Iri,et al.
Polygonal Approximations of a Curve — Formulations and Algorithms
,
1988
.
[10]
A. Melkman,et al.
On Polygonal Chain Approximation
,
1988
.
[11]
Kenneth Steiglitz,et al.
Combinatorial Optimization: Algorithms and Complexity
,
1981
.
[12]
Charles M. Williams,et al.
An Efficient Algorithm for the Piecewise Linear Approximation of Planar Curves
,
1978
.