RECURSIVE APPROXIMATION OF TOPOGRAPHIC DATA USING QUADTREES AND ORTHOGONAL POLYNOMIALS

Orthogonal polynomials (OP) are used to estimate polynomial coefficients and root-mean-square deviations (RMSD) for gridded elevation data wjthin quadtree subquadrants. These subqua.drants are recursively subdivided into four if the RMSD exceeds some threshold. Polynomials of orders one through six are fitted to three 256 by 256 OEMs, using RMSD thresholds of 1, 3.5, and 7 meters. The OP-quadtrees required from 9 to 20 percent of original grid space when the RMSD was set at 7 meters, but between 48 and 99 percent of that space for an RMSD of 1 meter. For a fixed RMSD, the total space required appears to be independent of polynomial order. If this effect is true in general, the obvious implication is that order does not matter. In that case, low-order polynomials could be used, saving computation time. When order is held constant, the space required by the OP-quadtree appears to be an inverse power function of the RMSD criterion.