Interior and exterior shape representations using the screened poisson equation

Abstract

Shape classification is a required task in many systems for image and video understanding. Implicit shape representations, such as the solutions to the Eikonal or Poisson equations defined on the shape, have been shown to be particularly effective for generating features that are useful for classification. The Poisson-based shape representation can be derived at each point inside the shape as the expected time for a particle undergoing Brownian motion to hit the shape boundary. This representation has no natural generalization when considering points outside of a shape, however, because the corresponding Brownian motion would have infinite expected hitting time. In this article, we modify the Brownian motion model by introducing an exponential lifetime for the particle, yielding a random variable whose expected value satisfies a screened Poisson equation that can be solved at points both interior and exterior to the shape. We then show how moments of this new random variable can be used to improve classification results on experiments with natural silhouettes and handwritten numerals.