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.

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

“…After a solution u is obtained, a global solution to the original two-phase Mumford-Shah objective can be obtained by thresholding u with µ for almost every µ ∈ [0, 1], see [7]. Some other proposals for computing global solutions can be found in [14,16]. In this paper, we consider the discrete version given by…”

Shape Recovery Using Stochastic Heat Flow

“…In order to do this we first obtain an expression for the stochastic perturbation part of eqn(5). Here we consider the recent work done in stochastic level sets [8] and stochastic curvature driven motion [10]. As discussed by Yip[15] , the stochastic perturbation of the eqn(5) can be seen to be given by du(x,t) = n(x,t)dW (t).…”

“…This suffers from problems like it is not invariant to the parameterization of the curve, i.e., the evolution depends on the implicit representation of the initial curve and ill posedness, i.e., under certain conditions it approaches the inverse heat equation which is unstable [8], [10]. These difficulties are overcome by introducing the Stratonovich differential [11] given by…”

“…The numerical implementation of the scheme for evolution is done by considering a step δt in time and δ x in space and is given by [8] u…”

“…If the temperature decreases too fast the process may get stuck in a local minimum, else if it decreases slowly the convergence is delayed. Here we adopt T (n) = T 0 / √ n as suggested by Juan et al [8]. The stopping criterion is based on the Euclidean distance measure approaching zero.…”