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 tensorproduct cubic B-splines. We denote the second-order partial derivatives of S by S xx , S xy and S yy . Let h k be the elevations at the points (x k ,y k ) on the contours. S is found by minimising the sum of the squared-errors {S(x k ,y k )?h k }2 and the quantity ∫∫ S 2 xx x,y) + 2s 2 xy (x,y)+S 2 yy (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.
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Mukherji, S. (2008). Surface Reconstruction from Contour Lines or LIDAR elevations by Least Squared-error Approximation using Tensor-Product Cubic B-splines. In: van Oosterom, P., Zlatanova, S., Penninga, F., Fendel, E.M. (eds) Advances in 3D Geoinformation Systems. Lecture Notes in Geoinformation and Cartography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72135-2_13
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DOI: https://doi.org/10.1007/978-3-540-72135-2_13
Publisher Name: Springer, Berlin, Heidelberg
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