Polygonal approximation by the minimax method

doi:10.1016/0146-664X(82)90011-9

Computer Graphics and Image Processing

Volume 19, Issue 3, July 1982, Pages 248-264

Computer Graphics an...

https://doi.org/10.1016/0146-664X(82)90011-9 Get rights and content

Abstract

This paper is concerned with the problem of approximating digitized pictures by polygons. The digitized picture is represented by a two-dimensional array of points, and its is desired to convert the given array into a set of polygons, such that each polygon has the least number of sides and the error between the initial points and the approximated lines is less than a given constant (E). There are many other solutions to this problem, but to evaluate the error, they use either the least-squares method or the cone intersection method. In this paper, it is shown that the minimax approximation that minimizes the maximum distance between the given points and the approximated line is the best approximation for the problem. A method is presented for obtaining the minimax approximated lines from the given N points in time proportional to N ∗ log N. From the obtained lines a polygon is then found using another algorithm. The polygon satisfies the condition that the number of sides is minimum and the maximum distance between the given points and the sides is less than the given E.

References (24)

C.M Williams
An efficient algorithm for the piecewise linear approximation of planar curves
Computer Graphics and Image Processing
(1978)
A Cantoni
Optimal curve-fitting with piecewise linear functions
IEEE Trans. Comput.
(1971)
H Freeman et al.
On quantization of line-drawing data
IEEE Trans. Comput.
(1969)
B Gluss
Further remarks on line segment curve-fitting using dynamic programming
Comm. ACM
(1962)
T Pavlidis
Algorithms for shape analysis of contours and waveforms
IEEE Trans. Pattern Anal. Mach. Intell.
(1980)
H.H Johnson et al.
A geometric method for approximating convex arc
SIAM J. Appl. Math.
(1980)
J Sklansky et al.
Fast polygonal approximation of digitized curves
T Pavlidis
Polygonal approximations by Newton's method
IEEE Trans. Comput.
(1977)
T Pavlidis
Structural Pattern Recognition
T Pavlidis
Structural Pattern Recognition
J Sklansky et al.
A theory of nonuniformity digitized binary pictures
IEEE Trans. Systems, Man, Cybernet.
(1976)

K Ichida et al.

Segmentation of planar curve

Electronics and Communication in Japan

(1975)

D.E McClure

Nonlinear segmented function approximation and analysis of line patterns

Quart. Appl. Math.

(1975)

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Polygonal approximation by the minimax method

Abstract

Computer Graphics and Image Processing

IEEE Trans. Comput.

IEEE Trans. Comput.

Comm. ACM

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IEEE Trans. Pattern Anal. Mach. Intell.

A geometric method for approximating convex arc

SIAM J. Appl. Math.

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IEEE Trans. Comput.

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Structural Pattern Recognition

A theory of nonuniformity digitized binary pictures

IEEE Trans. Systems, Man, Cybernet.

Segmentation of planar curve

Electronics and Communication in Japan

Nonlinear segmented function approximation and analysis of line patterns

Quart. Appl. Math.