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Image Reconstruction

Image Reconstruction. Group 6 Zoran Golic. Overview. Problem Multigrid-Algorithm Results Aspects worth mentioning. Problem. Imagine an image being sent over a noisy channel ->. Original. Disturbed. Numeric Approach. Assume smooth image

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Image Reconstruction

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  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 -> Original Disturbed

  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-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 -> 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

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