Surface Reconstruction from Contour Lines or LIDAR elevations by Least Squared-error Approximation using Tensor-Product Cubic B-splines

Abstract

We consider, in this paper, the problem of reconstructing the surface from contour lines of a topographic map. We reconstruct the surface by approximating the elevations, as specified by the contour lines, by tensor-product cubic B-splines using the least squared-error criterion. The resulting surface is both accurate and smooth and is free from the terracing artifacts that occur when thin-plate splines are used to reconstruct the surface. The approximating surface, S(x,y), is a linear combination of tensor-product cubic B-splines. We denote the second-order partial derivatives of S by Sxx, Sxy and Syy. Let hk be the elevations at the points (xk,yk) on the contours. S is found by minimising the sum of the squared-errors {S(xk,yk) -hk} 2 and the quantity ∫ ∫ S2xx(x,y) + 2S2y(x,y) + S2yy(x,y)dydx, the latter weighted by a constant λ. Thus, the coefficients of a small number of tensor-product cubic B-splines define the reconstructed surface. Also, since tensor-product cubic B-splines are non-zero only for four knot-intervals in the x-direction and y-direction, the elevation at any point can be found in constant time and a grid DEM can be generated from the coefficients of the B-splines in time linear in the size of the grid.