Adaptive Finite Volume Method For Solving Diffusion Equations On A Consistent Quadtree Grid
Presented by Mrs. Zuzana KRIVá
Type: Oral presentation
Track: Mathematical Methods in Image Processing
We present new finite volume method for solving diffusion PDEs used in image processing on adaptive grids, i.e. grids adapted to the data with the aim to decrease the number of unknowns in computations , . The diffusion equations are solved on the consistent adaptive grid built by modifying the quadtree structure in such way, that the connection of representative points of two adjacent finite volumes is perpendicular to their common boundary. First we present the scheme for the linear heat equation and evaluate its EOC by subsequent refinement of a selected fixed initial grid. To solve more general diffusion equations we adjust the recently developed methods  which utilizes all necessary information locally using the values on edges which are updated using the conservation law principles. Such methods evaluate the gradients of solution by using representative points which are midpoints of the edges. This requirement if not fulfilled in our consistent adaptive grid, so we present the method how to evaluate solution gradient on this type of grids. We apply the newly developed approaches to numerical solution of the regularized Perona-Malik equation . [1.] Bänsch E., Mikula K., A coarsening finite element strategy in image selective smoothing, Computing and Visualization in Science, Vol. 1 (1997), pp. 53-61. [2.] Catté al.(1992)] Catté, F., Lions, P.L., Morel, J.M., Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., Vol. 29 (1992), pp. 182-193. [3.] Eymard R., Handlovičová A., Mikula K., Regularized mean curvature flow level set equation, IMA Journal of Numerical Analysis,Vol. 31, No. 3, pp. 813-846 [4.] Krivá Z., Mikula K., An Adaptive Finite Volume Scheme for Solving Nonlinear Diffusion Equations in Image Processing, J. of Visual Communication and Image Representation, Vol. 13, (2002), pp. 22-35.