Advances in 3D Geoinformation Systems pp 213-227 | Cite as

# 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 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* _{ xx } ^{2} *x,y*) + 2*s* _{ xy } ^{2} (*x,y*)+*S* _{ yy } ^{2} (*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.

## Keywords

Contour Line Reconstructed Surface Elevation Data Digital Elevation Model Spline Surface## Preview

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## References

- [1]Gousie MB (1998) Contours to digital elevation models: Grid-based surface reconstruction methods. PhD Thesis, Rensselaer Polytechnic Institute, Troy, New YorkGoogle Scholar
- [2]Franke R (1982) Scattered data interpolation: tests of some methods. Math Comp 38(157):181–200CrossRefGoogle Scholar
- [3]Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76:1905–1915CrossRefGoogle Scholar
- [4]Pouderoux J, Gonzato JC, Tobor I, Guitton P (2004) Adaptive hierarchical RBF interpolation for creating smooth digital elevation models. Proc 12th Ann ACM Intl Workshop GIS 2004, Washington, DC, USA, 232–240Google Scholar
- [5]Franklin WR (2000) Applications of analytical cartography. Carto & GIS 27(3):225–237Google Scholar
- [6]Goncalves G, Julien P, Riazanoff S, Cervelle B (2002) Preserving cartographic quality in DTM interpolation from contour lines. ISPRS J Photogram & Remote Sens 56:210–220CrossRefGoogle Scholar
- [7]Dakowicz M, Gold CM (2003) Extracting meaningful slopes from terrain contours. Intl J Comput Geom & Applns 13(4):339–357.CrossRefGoogle Scholar
- [8]Floater MS (2000) Meshless parameterization and B-spline surface approximation. In: Cipolla R, Martin R (eds) The mathematics of surfaces IX. Springer, Berlin Heidelberg New YorkGoogle Scholar
- [9]Brovelli MA, Cannata M, Longoni UM (2004) LIDAR data filtering and DTM interpolation within GRASS. Trans in GIS 8(2):155–174CrossRefGoogle Scholar