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Diffusion Filters

ECE 463 Term Project. Diffusion Filters. S. Derin Babacan Department of Electrical and Computer Engineering Northwestern University March 7 , 200 6. Introduction. Filters for enhancement, restoration, smoothing and feature extraction Varying for each data entry based on local features

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Diffusion Filters

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  1. ECE 463 Term Project Diffusion Filters S. Derin Babacan Department of Electrical and Computer Engineering Northwestern University March7, 2006

  2. Introduction • Filters for enhancement, restoration, smoothing and feature extraction • Varying for each data entry based on local features • u = [u(0),u(1),..,u(n-1),u(n)] • w = [w0 , w1 , .. , wn-1 , wn ] • wi = [wi(0), wi(1), .. , wi(m-1), wi(m) ] • Mean-value preserving

  3. Physics of Diffusion Flux Diffusion Tensor • Fick’s Law • Continuity • Diffusion Equation • Gradient : (ux , uy) • Divergence : div Ψ = Ψx + Ψy • Concentration replaced by data values Concentration Gradient

  4. Diffusion Filters • D : a scalar (flux and gradient are parallel) • Isotropic diffusion • D spatially invariant (fixed): Homogeneous linear • D = g(·): Inhomogeneous (linear/nonlinear) • D : a matrix (rotation and/or scaling), flux and gradient are generally not parallel • Anisotropic diffusion • Eigenvectors of D determine the diffusion direction

  5. Linear Diffusion Filters • Homogeneous case D = 1 • Initial condition u(x,0) = f(x) • Unique solution:

  6. Linear Diffusion Filters HL IHL →

  7. Linear Diffusion Filters • Inhomogeneous case • Decrease blurring of the important features • Spatial adaptation → Reduce diffusion at the edges • D = g(·), monotonically decrease with increasing parameter • Gradient of f as the fuzzy edge detector Result

  8. Nonlinear Isotropic Filters • Incorporate adaptation based on the current filtered image instead of the initial image • Spatial and temporal adaptation • D = g(·) • g : a function of u • High contrast region : low diffusion • Low contrast region : high diffusion

  9. Nonlinear Isotropic Filters • Charbonnier • Perona-Malik • λ : contrast parameter to discriminate noise from edges

  10. Nonlinear Isotropic Filters • High noise in gradient • Instability, suppresion of important image features • Unnecessary suppression of diffusion (noise remains) • Regularization : Smooth the gradient before adaptation

  11. Nonlinear Isotropic Filters → R-PM t = 0 t = 40 t = 400 t = 1500 EE R-PM Original PM

  12. Nonlinear Isotropic Filters Original PM Canny PM

  13. Nonlinear Anisotropic Filters • So far, adaptation is only done spatially and/or temporally, but diffusion direction is fixed • Adaptation of the diffusion orientation • D is chosen as a matrix (scaling and/or rotation) • Varying the diffusion direction over the image can preserve/enhance local/semilocal features • Intraregion smoothing instead of interregion smoothing

  14. Edge Enhancing Filter • Change the orientation and the strength of the diffusion at an edge • Smooth along the edge boundary • Diffusion along the edges • Reduce/Stop diffusion across the edge (e.g. Orthogonal to the edge) • Intraregion smoothing

  15. Edge Enhancing Filter • Create a positive semidefinite diffusion tensor D • Orthonormal eigenvectors of D : υ1 ,υ2 • Eigenvalues control the diffusion strength • 1) • 2)

  16. t = 0 t = 40 t = 400 t = 1500 Edge Enhancing Filter R-PM EE t = 0 t = 250 t = 875 t = 3000 ←

  17. Coherence Enhancing Filter • Interrupted lines, flows in addition to noise • Need to process line or coherent flow-like structures, complete gaps

  18. Coherence Enhancing Filter • Orientations instead of directions (invariance to sign changes in gradient) • Orientation feature: Structure tensor • Create diffusion tensor as a function of structure tensor instead of the gradient

  19. Coherence Enhancing Filter • Jρpositive semidefinite → orthonormal eigenvectors υ1 ,υ2 ; eigenvalues μ1≥ μ2 ≥ 0 • υ1 : orientation of the highest grey value fluctuation • υ2 : preferred local orientation (lowest fluctuation) • (μ1 - μ2)2 : a measure of local coherence • Suppress diffusion along υ1 • Smooth along the direction of υ2

  20. Coherence Enhancing Filter • Choose the same set of eigenvectors for D • Set eigenvalues λ1, λ2 as: • (μ1 - μ2 )2 large : λ2 ≈ 1 • (μ1 - μ2 )2 small : λ2 ≈ ά

  21. Coherence Enhancing Filter

  22. Coherence Enhancing Filter Original PM CE

  23. Conclusions • Fast and stable filters • Highly flexible • Varying the parameters of the diffusion tensor (orientation, rotation, strength) • Incorporation of new features (texture, color etc.) • Separate filters for each data entry • High performance

  24. Thank you Questions ?

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