A fast Gibbs sampler for synthesizing constrained fractals

It is well known that the spatial frequency spectrum of membrane and thin plate splines exhibit self-affine characteristics and, hence, behave as fractals. This behavior was exploited in generating the constrained fractal surfaces, which were generated by using a Gibbs sampler algorithm in the work of Szeliski and Terzopoulos (1989). The algorithm involves locally perturbing a constrained spline surface with white noise until the spline surface reaches an equilibrium state. We introduce a fast generalized Gibbs sampler that combines two novel techniques, namely, a preconditioning technique in a wavelet basis for constraining the splines and a perturbation scheme in which, unlike the traditional Gibbs sampler, all sites (surface nodes) that do not share a common neighbor are updated simultaneously. In addition, we demonstrate the capability to generate arbitrary order fractal surfaces without resorting to blending techniques. Using this fast Gibbs sampler algorithm, we demonstrate the synthesis of realistic terrain models from sparse elevation data.