1. Multigrid Computation for Variational Image Segmentation Problems: Multigrid approach  Rosa Maria Spitaleri Istituto per le Applicazioni del Calcolo-CNR Viale del Policlinico 137, 00161 Rome, Italy e-mail: spitaleri@ iac.rm.cnr.it Advances in Numerical Algorithms, GRAZ, September 10- 13, 2003

2. A few research topics  numerical simulation (PDE approximation)  linear and non-linear multigrid computation  applications: –grid generation –shallow water equations –image segmentation  numerical results  visualization  interactivity

3. Multigrid Computation  In the multigrid computation a system of finite-difference or finite-element equations is solved on a given grid by continuous interactions with a hierarchy of coarser grids  Main components of the multigrid computation: relaxation for error smoothing restriction (fine-to-coarser transfer operator) prolongation (coarse-to fine transfer operator) MG cycling ( cycling by grid change)

4.  Let G 0, G 1,..., G M be a sequence of grids, all approximating the domain , with mesh sizes h 0, h 1,..., h M such that h 0 > h 1 >... > h M.  Let us assume the system Lw = F in ,  w =  on  and the difference equations along with the associated boundary conditions L M w M = F M on G M,  M w M =  M on  M

5.  Full Approximation Storage ( FAS ) algorithm for non linear problems  The general form of the correction equation is L l w l = F l,  l w l =  l where F l = L l (I l l+1 w a l+1 )+I l l+1 (F l+1 -L l+1 w a l+1 )  l =  l (I l l+1 w a l+1 )+I l l+1 (  l+1 -  l+1 w a l+1 ), w a l+1 is the computed approximation on the finer grid and I l l+1 an appropriate fine-to-coarse transfer operator (restriction)

6.  In the fine grid correction step ( w a l+1 ) new = ( w a l+1 ) old + I l l+1 (w a l - I l l+1 w a l+1 ) where I l l+1 is an appropriate coarse-to-fine transfer operator (prolongation)  In the FAS algorithm the idea is to store the full current approximation w a l instead of storing just the correction

7. Pre-smoothing  compute by applying 1 (≥0) sweeps of a given (non-linear) relaxation method with starting guess One iteration step of the (l+1,l) two-grid method, which computes from, is composed of the following steps: FAS two-grid method

8. Post-smoothing  compute by applying 2 (≥0) sweeps of a given (non-linear) relaxation method with starting guess The non-linear relaxation procedure, suitable for error smoothing, is applied 1 times before and 2 times after the coarse grid correction step

9. Coarse-grid correction u computation of the residual u restriction of the residual u restriction of the current approximation u computation of the exact solution of u computation of the correction u interpolation of the correction u computation of the correct approximation

10. Multigrid Components  coarse grid construction (hierarchy of grids): meshsize doubling  relaxation for error smoothing (, ): Gauss-Seidel relaxation-non collective  relaxation ordering : rotated lexicographical  multigrid cycling : V, W cycles

11.  restriction ( fine-to-coarse-grid transfer operator )(, ) : straight injection, weighted --> nine-point  prolongation (coarse-to-fine-grid transfer operator ) ( ) : piecewise linear interpolation, weighted-->nine-point

12. References á R. M. Spitaleri, A Scientific Computing Environment for Differential Modeling, Mathematics and Computer in Simulation, 62, 4: (2003) á R.M. Spitaleri, R. March, D. Arena, A Multigrid Finite Difference Method for the Solution of Euler Equations of the Variational Image Segmentation Applied Numerical Mathematics, 39: 181-189 (2001) á R. M. Spitaleri, Numerical Insights in the Solution of Euler Equations of the Variational Image Segmentation, Non Linear Analysis, 47: 3333-3344 (2001) á Spitaleri R.M., Full-FAS Multigrid Grid Generation Algorithms, Applied Numerical Mathematics 32, 4: 483-494 (2000) á R.M Spitaleri, R. March and D. Arena, Finite Difference Solution of Euler Equations Arising in Variational Image Segmentation, Numerical Algorithms, 21: 353-365 (1999) á R.M. Spitaleri (Ed.) Special issue on Applied Scientifc Computing-Grid Generation, Approximated Solutions and Visualization, Applied Numerical Mathematics, (2003- to appear) á R.M. Spitaleri (Ed.) Special Issue on Applied Numerical Computing: Grid Generation and Solution Methods for Advanced Simulations, Applied Numerical Mathematics 46, 3-4, ( 2003)

13. Organization 2003 u MASCOT 03 IMACS/ISGG Worshop 3 rd Meeting on Applied Scientific Computing and Tools, Grid generation, Approximation, Simulation and Visualization FORTE VILLAGE RESORT, Sardinia, October 3-4, 2003 in TCN-CAE International Conference on CAE and Computational Technologies for Industry, October 2-5, 2003 u EWM2003-Annual Conference of EWM-European Women in Mathematics, Luminy, France, November 3-7, 2003 Session ‘Numerical Methods’