1. Image Reconstruction Group 6 Zoran Golic

2. Overview Problem Multigrid-Algorithm Results Aspects worth mentioning

3. Problem Imagine an image being sent over a noisy channel -> OriginalDisturbed

4. Numeric Approach Assume smooth image Information flows from one pixel to all neighbors

5. Numeric Approach Assume smooth image Information flows from one pixel to all neighbors Diffusionequation: grad · g(u)grad(u) = 0

6. Numeric Approach Assume smooth image Information flows from one pixel to all neighbors Diffusionequation: grad · g(u)grad(u) = 0 g(u) is a real function that controls the flow (edge detection) For simplification set g(u) = 1

7. Intuitive Approach Solve equation by an iterative solver like Red- Black Gauß-Seidel, CG…

8. Multigrid Approach Solving an image of size 128x128 pixels results in ~16500 points per iteration

9. Multigrid Approach Solving an image of size 128x128 pixels results in ~16500 points per iteration Try to solve image on a coarser grid (e.g. every 2. point results in imagesize of 64x64 pixels -> 4096 points per Iteration

10. Idea of Multigrid Discretization of PDE (SiWiR 1) leads to matrix equation Ax = b

11. Idea of Multigrid Discretization of PDE (SiWiR 1) leads to matrix equation Ax = b The Algebraic error is ẽ = x - x

12. Idea of Multigrid Discretization of PDE (SiWiR 1) leads to matrix equation Ax = b The Algebraic error is ẽ = x - x To obtain ẽ we solve the equivalent equation Aẽ = r (r is the residual of Ax=b -> r = b – Ax)

13. Algorithm Step 1: Presmoothing Step 2: Coarse grid correction Fine grid residual calculation Restriction to coarser grid Solve e = A -1 r on coarse grid Prolongate (interpolate) to fine grid error Correct image Step 3: Postsmoothing

14. Multigrid-Multigrid Of course this is not enough !! Ae = r is again a matrix equation -> can also be solved by multigrid. Recursive Call needed

15. V- and W-Cycles

16. Results

17. Gauß-SeidelMultigrid 64x6410,00ms6,78ms 128x12842,07ms22,00ms 256x256176,20ms83,40ms 512x512911,93ms339,80ms Figures

18. Some side aspects Extension for images of non-quadratic size (effect on the depth of grid coarsening) Residual was not used as stopping criteria of Gauss-Seidel.

19. Thank You