Abstract
The use of curves to represent two-dimensional structures is an important part of many scientific investigations. For example, geographers use curves extensively to represent map features such as contour lines, roads, and rivers. Circuit layout designers use curves to specify the wiring between circuits. Because of the very large amount of data involved and the need to perform operations on this data efficiently, the representation of such curves is a crucial issue. A hierarchical representation consisting of binary trees with a special datum at each node is described. This datum is called a strip and the tree that contains such data is called a strip tree. Lower levels in the tree correspond to finer resolution representations of the curve. The strip tree structure is a direct consequence of using a special method for digitizing lines and retaining all intermediate steps. This gives several desirable properties. For curves that are well-behaved, intersection and point-membership (for closed curves) calculations can be solved in 0(log n) where n is the number of points describing the curve. The curves can be efficiently encoded and displayed at various resolutions. The representation is closed under intersection and union and these operations can be carried out at different resolutions. All these properties depend on the hierarchical tree structure which allows primitive operations to be performed at the lowest possible resolution with great computational time savings.
Strip trees is a linear interpolation scheme which realizes an important space savings by not representing all the points explicitly. This means that even when the overhead of the tree indexing is added, the storage requirement is comparable to raster representations which do represent most of the points explicitly.
- 1 Barrow, H.G. Interactive aids for cartography and photo interpretation. Semiannual, Tech. Rep, DARPA Contract DAAG 29-76-C-0057, SRI Int., Nov 1977.Google Scholar
- 2 Bentley, J.L. Multidimensional search trees used for associative searching. Comm. A CM 18, 9 (Sept. 1975), 509-517. Google ScholarDigital Library
- 3 Burton, W. Representation of many-sided polygons and polygonal lines for rapid processing. Comm. ,4 CM 20, 3 (March 1977), 166-171. Google ScholarDigital Library
- 4 Douglas, D.H. and Peucker, T. Algorithms for the reduction of the number of points required to represent a line or its caricature. Can. Cartogr. 10, 2 (Dec. 1973).Google ScholarCross Ref
- 5 Duda, R.O. and Hart, P.E. Pattern Classification and Scene Analysis. Wiley-Interscience, New York, 1973. Google ScholarDigital Library
- 6 Dyer, C.R., Rosenfeld, A., and Samet, H. Region representation: Boundary codes from quadtrees. Comm. ACM 23, 3 (March 1980), 171-178. Google ScholarDigital Library
- 7 Mandelbrot, B.B. Fractals: Form, Chance and Dimension. W.H. Freeman, San Francisco, 1977.Google Scholar
- 8 Merrill, R.D. Representation of contours and regions for efficient computer search. Comm. ACM 16, 2 (Feb. 1973), 69-82. Google ScholarDigital Library
- 9 Minsky, M.L. and Papert, S. Perceptrons: An introduction to computational geometry. MIT Press, Cambridge, MA, 1969. Google ScholarDigital Library
- 10 Peucker, T. A theory of the cartographic line. International Yearbook of Cartography 16, 1976.Google Scholar
- 11 Peuquet, D.J. Raster processing: An alternative approach to automated cartographic data handling. Am. Cartogr. 6, 2 (April, 1979), 129-139.Google ScholarCross Ref
- 12 Samet, H. Region representation: Quadtrees from boundary codes. Comm. ACM 23, 3 (March 1980), 163-170, Google ScholarDigital Library
- 13 Sloan, K.R., Jr. Maps and map data structures. Forthcoming Tech. Rept., Comptr. Sci Dept, Univ. of Rochester, Rochester, New York, 1981.Google Scholar
- 14 Tanaka, H. and Ballard, D.H. Extension of strip tree operations. Comptr Sci Dept, Univ. of Rochester, Rochester, New York, 1980.Google Scholar
- 15 Tanimoto, S. and Pavlidis, T. A hierarchical data structure for picture processing. Comptr Graphics and Image Processing 4, 2 (June 1975), 104-119.Google ScholarCross Ref
- 16 Turner, K.J. Computer perception of curved objects using a television camera. Ph.D. thesis, Univ. of Edinburgh, Scotland, 1974.Google Scholar
Index Terms
- Strip trees: a hierarchical representation for curves
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