2.2 Spectral synthesis

Spectral synthesis (also called Fourier transform filtering) is another representative of how fBm surface can be obtained [1].

First step we compute the complex coefficients tex2html_wrap_inline613 of inverse Fourier transformation. In order to generate fBm spectral density function S(f) (c.f. [9]) of these coefficients must be proportional to tex2html_wrap_inline617


where tex2html_wrap_inline621 controls the fractal dimension of the final object according to


and H denotes Hurst exponent. Then we calculate the inverse Fourier transformation in two dimensions according to formula


The function f(x,y) is then fBm with its fractal dimension


The coefficients of Fourier transformation with lowest indices have the biggest influence to resulting shape of the surface (see in Figure 4) and the higher indices manifest themselves just locally.

Figure: Surfaces obtained with the spectral synthesis using a) 2, b) 4, c) 8, d) 16 and e) 32 respectively members of Fourier series.

Ivo Marak - marak@sgi.felk.cvut.cz