Stochastic Motion and the Level Set Method in Computer Vision: Stochastic Active Contours

Abstract

Based on recent work on Stochastic Partial Differential Equations (SPDEs), this paper presents a simple and well-founded method to implement the stochastic evolution of a curve. First, we explain why great care should be taken when considering such an evolution in a Level Set framework. To guarantee the well-posedness of the evolution and to make it independent of the implicit representation of the initial curve, a Stratonovich differential has to be introduced. To implement this differential, a standard Ito plus drift approximation is proposed to turn an implicit scheme into an explicit one. Subsequently, we consider shape optimization techniques, which are a common framework to address various applications in Computer Vision, like segmentation, tracking, stereo vision etc. The objective of our approach is to improve these methods through the introduction of stochastic motion principles. The extension we propose can deal with local minima and with complex cases where the gradient of the objective function with respect to the shape is impossible to derive exactly. Finally, as an application, we focus on image segmentation methods, leading to what we call Stochastic Active Contours.