Biharmonic Spline Wavelets versus Generalized Multi-quadrics for Continuous Surface Representations

In some respect, the continuous representation of surfaces such as the geoid, the topography, or other spatial phenomena, is superior to discrete forms like the TIN (Triangular Irregular Network) or a raster DEM (Digital Elevation Model) as long as these surfaces exhibit a certain degree of local smoothness. In this contribution, we shall concentrate on the special study of biharmonic spline wavelets and of generalized multi-quadrics, with the emphasis on increased efficiency while maintaining the local approximation quality up to the desired resolution.To support these interpolation techniques a newly developed global search algorithm will be adapted to this problem. It is based on heuristic methods that would allow the user to handle gridded as well as arbitrarily scattered data. The number of coefficients for any surface representation using the global optimization will be extremely small while maintaining a magnitude of deviations that is still acceptable. On the other hand, long computation times have to be taken into account. Concerning the functional model one may face fewer restrictions for its structure when using global methods because no derivatives have to be computed and no approximate values need to be provided.Some geodetic examples will show the potential of the new techniques, particularly in view of the more classical Fourier analysis.