Picture Reconstruction / Multigrid

Group 8

Stefan Spielvogel

Alexander Piazza

Alexander Kosukhin

218/04/23

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

318/04/23

TaskReconstruction from partly destroyed image

418/04/23

Task

518/04/23

Task

618/04/23

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.

718/04/23

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( 4h2 ) instead of 1 − O( h2 )

818/04/23

Multigrid – Coarse Grids

smooth error is (relatively) more oscillatory there!

918/04/23

Multigrid – Coarse Grids

1018/04/23

Multigrid – V-cycle

1118/04/23

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

1218/04/23

Our Approach

1318/04/23

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)

1418/04/23

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.

1518/04/23

Our Approach

Difficulties:Finding suitable data structures for

implementing MGBugs hard to find

Observations:MG not much faster than using R-B-Gauss-

Seidel

1618/04/23

Outcome

View Reconstruction Steps Reconstruction error:

Reconstruction Error

0

0,05

0,1

0,15

0,2

0,25

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

1718/04/23

Q & A

Feel free to ask your questions! Thank you for your attention!