Evaluation of Gradient Norms on a Consistent Quadtree Grid in 2D

This paper solves the problem of how to evaluate gradients and their norms on a quadtree grid, which is deformed in such a way that the connections of centers of its adjacent elements are perpendicular to their common boundaries. On the grid, we solve the parabolic PDEs representing the curvature-driven filters based on the mean curvature flow and geodetic mean curvature flow equations in a level set formulation. The numerical solution of these equations is based on the finite volume method, where the finite volumes correspond to elements of the deformed quadtree. The described method utilizes representative points not only for the finite volumes but also for the edges forming the boundaries of grid elements. Using these points we evaluate the gradients locally. Solution values in the edge representative points are updated by balancing the fluxes in such a way that we always need only neighbors of a finite volume sharing a common edge with it, not only a vertex. This fact is important for the efficiency of the algorithm. The edge representative points have been chosen in such a way that they lie on a connection of volume representative points enabling to derive a special formula for the gradient norm. We discuss the ways of approximating the norms of the gradients, and on selected examples we show their properties.