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In this paper we present segmentation method for piecewise smooth pictures and our implemented software. We describe image segmentation problem and its difficulties in real applications. Since image segmentation is a complicated problem, we focus on the segmentation method only for piecewise smooth pictures based on the Mumford-Shah functional and its connection to spectral methods. We have developed software for the piecewise image segmentation; currently, we focus on decreasing the execution time of massively parallel computations and quality of results. The results conclude the paper. [ABSTRACT FROM AUTHOR]

The maximum a posteriori (MAP) estimates for linear inverse problems are studied using hierarchical Gaussian models. The stability of this point estimate is considered with respect to different discretizations. We analyze the phenomena which appear when the discretization becomes finer. An edge-preserving Bayesian reconstruction method for signal restoration problems is introduced and studied with arbitrarily fine discretization. Moreover, different noise asymptotics are considered for the inverse problem. We show that the maximum a posteriori and conditional mean estimates converge under different conditions. Finally, we discuss connection of this method to Mumford–Shah functional. This paper reviews results from [6]. [ABSTRACT FROM AUTHOR]

As results of 3d sensors using the ”time of flight” technology we get noise information of the shape of objects together with intensities. From the mathematical point of view these information are scalar fields d and I. For filtering and smoothing of images we minimize a Mumford-Shah functional by solving the boundary value problem of the relevant Euler-Lagrange equations. For the numerical solution of the boundary value problem we use a finite volume discretization and we solve the resulting nonlinear equation system on the finite volume grid by Newtons method and the steepest descent method. [ABSTRACT FROM AUTHOR]