Discontinuity-Preserving Surface Reconstruction Using Stochastic Differential Equations

We address the problem of reconstructing a surface from irregularly spaced sparse and noisy range data while concurrently identifying and preserving the significant discontinuities in depth. It is well known that, starting from either the probabilistic Markov random field model or the mechanical membrane or thin plate model for the surface, the solution of the reconstruction problem can be eventually reduced to the global minimization of a certain “energy” function. Requiring the preservation of depth discontinuities makes the energy function nonconvex and replete with multiple local minima. We present a new method for obtaining discontinuity-preserving reconstruction based on the numerical solution of an appropriate Ito vector stochastic differential equation (SDE). The reconstructed surface is found by following the sample path of the (stochastic) diffusion process that solves the SDE in question. Our central contribution is the demonstration of the efficacy of the stochastic differential equation technique for solving a vision problem. Through comparisions of the results of our method to those of the two well-known existingglobalminimization based reconstruction techniques, we show a significant improvement in the final reconstructions obtained.