Terrain inclination and curvature from wavelet coefficients

Abstract.

 The wavelet transformation (WT) of a function y=f(x) with a wavelet of the nth order approximates the nth derivative of the function except for a constant scaling factor and a frequency-dependent phase shift. In the two-dimensional WT, the equivalent applies to the partial derivatives of a function z=f(x,y). If a digital terrain model (DTM) has been stored in form of wavelet coefficients (WCs), then the gradient and, if required, also curvature values can be directly deduced from the WCs. By means of special functions (test functions) whose derivatives are known, the scaling (`amplitude correction') and the displacement (`phase correction in the space domain') can be determined. The moments of the wavelets and the scaling functions (high and low-pass filters) make it possible to derive the approximation formulae in a clear and wavelet-independent manner.