Using Langevin-Type Stochastic-Dynamical Particles for Sampling and Rendering Implicit Surfaces

We propose a new sampling method for 2D and 3D implicit surfaces. The method is based on a stochastic process defined by the Langevin equation with a Gaussian random-force term. Our Langevin equation describes a stochastic-dynamical particle, which develops in time confined around the sampled implicit surface with small width. Its numerically generated solutions can be easily moved onto the surface strictly with very few iteration of the Newton correction. The method is robust in a sense that an arbitrary number of sample points can be obtained starting from one simple initial condition. It is because (1) the time development of the stochastic-dynamical particle does not terminate even when it reaches the sampled implicit surface, and (2) there is non-zero transition probability from one disconnected component to another. The method works very well for implicit surfaces which are complicated topologically, mathematically, and/or in shape. It also has some advantageous features in rendering 3D implicit surfaces. Many examples of applying our sampling method to real 2D and 3D implicit surfaces are presented. key words: implicit surface, sampling, stochastic process, Langevin equation, particle