We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε = ε(∆t, ∆x) > 0 which goes to 0 as (∆t, ∆x) is going to 0 and does not depend on the time t n . The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of ε = ε n (∆t, ∆x) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold ε n . A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.

In this paper we propose an accurate method for image segmentation following the level-set approach. The method is based on an adaptive "filtered" scheme recently introduced by the authors. The main feature of the scheme is the possibility to stabilize an a priori unstable high-order scheme via a filter function which allows to combine a high-order scheme in the regularity regions and a monotone scheme elsewhere, in presence of singularities. The filtered scheme considered in this paper uses the local Lax-Friedrichs scheme as monotone scheme and the Lax-Wendroff scheme as high-order scheme but other couplings are possible. Moreover, we introduce also a modified velocity function for the level-set model used in segmentation, this velocity allows to obtain more accurate results with respect to other velocities proposed in the literature. Some numerical tests on synthetic and real images confirm the accuracy of the proposed method and the advantages given by the new velocity.

The lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the approximate solutions, in order to detect the most troublesome regions of the domain. This information helps to adapt the approximation scheme in order to avoid spurious oscillations when using high-order schemes. In this paper we propose a genuinely multidimensional extension of the WENO procedure in order to overcome the limitations of indicators based on dimensional splitting. Our aim is to obtain new regularity indicators for problems in 2D and apply them to a class of "adaptive filtered" schemes for first order evolutive Hamilton-Jacobi equations. According to the usual procedure, filtered schemes are obtained by a simple coupling of a high-order scheme and a monotone scheme. The mixture is governed by a filter function F and by a switching parameter ε n = ε n (∆t, ∆x) > 0 which goes to 0 as (∆t, ∆x) is going to 0. The adaptivity is related to the smoothness indicators and allows to tune automatically the switching parameter ε n j in time and space. Several numerical tests on critical situations in 1D and 2D are presented and confirm the effectiveness of the proposed indicators and the efficiency of our scheme.

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