Image Reconstruction

1
Image Reconstruction

• Group 6
• Zoran Golic

2
Overview

• Problem
• Multigrid-Algorithm
• Results
• Aspects worth mentioning

3
Problem
Original
Disturbed

• Imagine an image being sent over a noisy channel
  -gt

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 -gt
  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 ? x - x?

12
Idea of Multigrid

• Discretization of PDE (SiWiR 1) leads to matrix
  equation Ax b
• The Algebraic error is ? x - x?
• To obtain ? we solve the equivalent equation A?
  r (r is the residual of Axb
• -gt r b Ax)

13
Algorithm

• Step 1 Presmoothing
• Step 2 Coarse grid correction
• Fine grid residual calculation
• Restriction to coarser grid
• Solve e A-1r 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 -gt can also be
  solved by multigrid.
• Recursive Call needed

15
V- and W-Cycles
16
Results
17
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