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Image Reconstruction Group 6 Zoran Golic
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Overview Problem Multigrid-Algorithm Results Aspects worth mentioning
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Problem Imagine an image being sent over a noisy channel -> OriginalDisturbed
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Numeric Approach Assume smooth image Information flows from one pixel to all neighbors
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Numeric Approach Assume smooth image Information flows from one pixel to all neighbors Diffusionequation: grad · g(u)grad(u) = 0
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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
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Intuitive Approach Solve equation by an iterative solver like Red- Black Gauß-Seidel, CG…
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Multigrid Approach Solving an image of size 128x128 pixels results in ~16500 points per iteration
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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
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Idea of Multigrid Discretization of PDE (SiWiR 1) leads to matrix equation Ax = b
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Idea of Multigrid Discretization of PDE (SiWiR 1) leads to matrix equation Ax = b The Algebraic error is ẽ = x - x
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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)
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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
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Multigrid-Multigrid Of course this is not enough !! Ae = r is again a matrix equation -> can also be solved by multigrid. Recursive Call needed
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V- and W-Cycles
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Results
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Gauß-SeidelMultigrid 64x6410,00ms6,78ms 128x12842,07ms22,00ms 256x256176,20ms83,40ms 512x512911,93ms339,80ms Figures
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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.
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Thank You
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