Shape Recovery Using Stochastic Heat Flow

Abstract

We consider the problem of depth estimation from multiple images based on the defocus cue. For a Gaussian defocus blur, the observations can be shown to be the solution of a deterministic but inhomogeneous diffusion process. However, the diffusion process does not sufciently address the case in which the Gaussian kernel is deformed. This deformation happens due to several factors like self-occlusion, possible aberrations and imperfections in the aperture. These issues can be solved by incorporating a stochastic perturbation into the heat diffusion process. The resultant o w is that of an inhomogeneous heat diffusion perturbed by a stochastic curvature driven motion. The depth in the scene is estimated from the coefcient of the stochastic heat equation without actually knowing the departure from the Gaussian assumption. Further, the proposed method also takes into account the non-convex nature of the diffusion process. The method provides a strong theoretical framework for handling the depth from defocus problem. The limited depth of eld introduces a defocus blur in images captured with conventional lenses based on the range of depth variation in a scene. This artifact has been used in computer vision for estimating depth in the scene. As discussed in [7], this method of shape recovery is particularly relevant for complex scenes which have a large amount of geometric detail and complex self occlusion relationships which make it difcult to estimate the shape using stereo based methods. In this paper we introduce a new technique for recovering the structure based on the defocus blur. The principal idea that enables our work is that the defocus effect can be modeled in terms of inhomogeneous diffusion (e.g., spatially varying coefcients) of heat using the heat equation. This is because the defocus blur can be modeled as a Gaussian blur, which forms a temporally evolving kernel for the isotropic heat equation. This approach was explored by Favaro et al.[4]. Their method had two main shortcomings. First, it could not handle departure from Gaussian assumption in case of self-occlusions. Second, it made an assumption that the diffusion coefcient is a convex function and the solution was based on conjugate gradient based method. In this paper we address both these shortcomings. Here, we propose a model wherein the heat equation is perturbed stochastically. In this approach the departure from the Gaussian blur model is implicitly accounted for in the stochastic perturbation of diffusion. The mathematical existence for the stochastically perturbed heat equation, which is used