Sampling implicit surfaces based on stochastic differential equations with converging constraint

Abstract

We propose a new stochastic sampling method for implicit surfaces based on an Ito-type stochastic differential equation with the converging constraint. The equation can realize a stationary state in which the probability distribution becomes uniform over the whole area of a constraint surface. A hypothetical stochastic particle described by the equation performs Brownian motion, i.e. random walk, being confined on a constraint surface, i.e. an implicit surface. Therefore its trajectory gives us sample points on it. The stochastic sampling method is fast and widely applicable to implicit surfaces defined with twice differentiable constraint functions. It is applicable to non-polynomial surfaces and surfaces with branches, too. It does not require an initial sample point on a sampled surface to start sampling. It works well with a single-particle system without necessity of introducing interaction between particles. It also has advantageous features at the rendering stage.