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

  • Shyamalee Mukherji
Part of the Lecture Notes in Geoinformation and Cartography book series (LNGC)


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) + 2s 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.


Contour Line Reconstructed Surface Elevation Data Digital Elevation Model Spline Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Gousie MB (1998) Contours to digital elevation models: Grid-based surface reconstruction methods. PhD Thesis, Rensselaer Polytechnic Institute, Troy, New YorkGoogle Scholar
  2. [2]
    Franke R (1982) Scattered data interpolation: tests of some methods. Math Comp 38(157):181–200CrossRefGoogle Scholar
  3. [3]
    Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76:1905–1915CrossRefGoogle Scholar
  4. [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. [5]
    Franklin WR (2000) Applications of analytical cartography. Carto & GIS 27(3):225–237Google Scholar
  6. [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. [7]
    Dakowicz M, Gold CM (2003) Extracting meaningful slopes from terrain contours. Intl J Comput Geom & Applns 13(4):339–357.CrossRefGoogle Scholar
  8. [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. [9]
    Brovelli MA, Cannata M, Longoni UM (2004) LIDAR data filtering and DTM interpolation within GRASS. Trans in GIS 8(2):155–174CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Shyamalee Mukherji
    • 1
  1. 1.Centre of Studies in Resources EngineeringIndian Institute of Technology, BombayMumbaiIndia

Personalised recommendations