1. Picture Reconstruction / Multigrid Group 8 Stefan Spielvogel Alexander Piazza Alexander Kosukhin

2. 2 28/06/2015 Agenda What was the task to be solved? Why using multigrid algorithm for this problem? Our approach to implement mutigrid! Some nice outcome of our programm! Q and hopefully A! Literature: Briggs tutorial – helped us a lot!  http://www.math.ust.hk/~mawang/teaching/math532/ mgtut.pdf

3. 3 28/06/2015 Task Reconstruction from partly destroyed image

4. 4 28/06/2015 Task

5. 5 28/06/2015 Task

6. 6 28/06/2015 Multigrid Many relaxation schemes have the smoothing property, where oscillatory modes of the error are eliminated effectively, but smooth modes are damped very slowly. This might seem like a limitation, but by using coarse grids we can use the smoothing property to good advantage.

7. 7 28/06/2015 Multigrid – Coarse Grids Coarse grids can be used to compute an improved initial guess for the fine-grid relaxation. This is advantageous because: Relaxation on the coarse-grid is much cheaper (1/2 as many points in 1D, 1/4 in 2D, 1/8 in 3D) Relaxation on the coarse grid has a marginally better convergence rate, for example  1 − O( 4h 2 ) instead of 1 − O( h 2 )

8. 8 28/06/2015 Multigrid – Coarse Grids smooth error is (relatively) more oscillatory there!

9. 9 28/06/2015 Multigrid – Coarse Grids

10. 10 28/06/2015 Multigrid – V-cycle

11. 11 28/06/2015 Our Approach Software Design: image.cpp image.h: holds the Image class im_matrix.cpp im_matrix.h: holds the matrix class impaint.cpp: main programm with V-cycle and GS-solver

12. 12 28/06/2015 Our Approach

13. 13 28/06/2015 Our Approach We used different levels of maps, representing the coarse grids These maps were used as lookup-tables to store, which of the pixels are known or unknown on the certain level Only unknown pixels have to be treated in iterations (RED BLACK GS)

14. 14 28/06/2015 Our Approach We computed the reconstruction error by calculating the L2-Norm of the difference between original image and temporary solution after each V-cycle, normalized by the number of pixels.

15. 15 28/06/2015 Our Approach Difficulties:  Finding suitable data structures for implementing MG  Bugs hard to find Observations:  MG not much faster than using R-B-Gauss- Seidel

16. 16 28/06/2015 Outcome View Reconstruction Steps Reconstruction error:

17. 17 28/06/2015 Q & A Feel free to ask your questions! Thank you for your attention!