A fixed grid curve representation for efficient processing

Abstract

A compact form of representation for curves and regional boundaries is described.

A curve is approximated by a polygonal line where each side joins a grid point to one of its eight nearest neighbors (as is done in the commonly used chain-encoded representation). Vertex number (2k + 1)2^m is defined relative to vertex number k2^m+l. The resulting representation requires about 5 bits of storage per vertex, irrespective of the total length of the curve.

Sections of a curve may be collectively examined. Typically, that part of the curve between vertices number k2^m and (k + 1)2^m, for some k and m, is considered. Given the locations of the endpoints of a section and the number of sides in the section, it is easy to compute a bounding rectangle which must completely contain the section. Often, this is all the information we need to know about a section. For example, two sections cannot intersect if their bounding rectangles do not overlap. When necessary, a section can be easily divided into two smaller sections, each of which can be examined in turn. This process may be continued down to single sided sections if necessary.

Under favorable circumstances, for an n sided approximation to a closed curve, the time required to determine if a point is inside the curve is independent of n. The intersection of n sided approximations to two curves can usually be found in 0(1n n) time.

References

1. 1 Boyell, Roger L. The method of successive grids for reduction of function storage requirements. Comp. J. 5 (Jan. 1963), 320-321.
2. 2 Burton, Warren. Representation of many-sided polygons and polygonal lines for rapid processing. Comm. ACM 20, 3 (March 1977), 166-171.
3. 3 Burton, Warren. A fixed grid curve or region representation for rapid processing. Report CS78-1. Dept. Math. and Comp. Sci., Michigan Technological Univ., January 1978.
4. 4 Freeman, Herbert. Computer processing of line-drawing images. Comp. Surveys 6, 1 (March 1974), 57-97.,
5. 5 Mandelbrot, Benoit. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156 (May 5,1967), 636-638.
6. 6 Mandelbrot, Benoit. Fractals: Form, Chance and Dimension. W. H. Freeman, San Francisco, 1977.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Parametric curve and surface models

      • Point-based models

      • Volumetric models

• Mathematics of computing

  • Discrete mathematics

    • Graph theory

      • Approximation algorithms