Adaptive cell-centered finite volume method for diffusion equations on a consistent quadtree grid

Abstract

Models applied in image processing are often described by nonlinear PDEs in which a good approximation of gradient plays an important role especially in such cases where irregular finite volume grids are used. In image processing, such a situation can occur during a coarsening based on quadtree grids. We present a construction of a deformed quadtree grid in which the connection of representative points of two adjacent finite volumes is perpendicular to their common boundary enabling us to apply the classical finite volume methods. On the other hand, for such an adjusted grid, the intersection of representative points connection with a finite volume boundary is not a middle point of their common edge and standard methods cannot achieve a good accuracy. In this paper we present a new cell-centered finite volume method to evaluate solution gradients, which results into a solution of a simple linear algebraic system and we prove its unique solvability. Finally we present numerical experiments for the regularized Perona-Malik model in which we applied this new method.

Power diagrams and sparse paged grids for high resolution adaptive liquids

“…Building on this idea, de Goes et al [2015] developed a Lagrangian particle method that constructs a power diagram at each time step for enforcing incompressibility. Krivá et al [2016] used the power diagram of a quadtree in two dimensions to solve a Poisson problem in the context of image processing, although they did not identify their construction as a power diagram. Similarly, Sifounakis et al [2016] proposed a so-called virtual slanting approach for a Navier-Stokes solver on quad/octrees; however, their strategy of simply slanting faces breaks down in three dimensions, where proper orthogonality cannot be maintained without (explicitly or implicitly) changing the local cell connectivity.…”