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Foundations of Image Processing: Convolution

Foundations of Image Processing: Convolution. September 29 , 2006 Fourier Transformation (Earl Glynn) October 27, 2006 De-Convolution (Christopher Wood). Degradation function H: Describes degradation through imaging process (e.g. point-spread function of microscope)

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Foundations of Image Processing: Convolution

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  1. Foundations of Image Processing:Convolution September 29 , 2006Fourier Transformation (Earl Glynn) October 27, 2006De-Convolution (Christopher Wood)

  2. Degradation function H: Describes degradation through imaging process (e.g. point-spread function of microscope) Linear and position-invariant Typically in frequency domain h (x,y): H (u,v) as spatial representation Image Degradation Degradation function H + f(x,y) g(x,y) Noise η(x,y)

  3. Example for Degradation: Point-Spread Function

  4. Different Types of Noise Gaussian Rayleigh Uniform Salt & Pepper Image Frequency Intensity © 2002 R. C. Gonzalez & R. E. Woods

  5. Image Enhancement Choose Restoration filter to visualize (highlight) important features Subjective judgment of result by humans Image Restoration Use knowledge about noise and degradation function to restore image Even complete knowledge about degradation function does not allow perfect restoration Quantification Visualization Image Processing Restoration filter(s) g(x,y) f restored(x,y)

  6. Spatial Filter

  7. Spatial Filter

  8. Spatial Filter

  9. Histogram

  10. Basic Gray-level Transformations © 2002 R. C. Gonzalez & R. E. Woods

  11. fly_max250=fly < 250 Linear Gray Level Transformation

  12. Power-Law (Gamma) Transformation © 2002 R. C. Gonzalez & R. E. Woods

  13. Gamma correction γ=0.01 Power-Law Transformation

  14. Spatial Kernel Filter

  15. Spatial Kernel Filter

  16. Principle Replace every pixel with the average of its neighborhood Retain low frequency information Reduces high frequency information Purpose Noise reduction (e.g. Gauss, not salt-and-pepper) Preparation for thresholding Advantages Straight forward implementation and application Disadvantages Blurs edges Reduces resolution Smoothing (Lowpass) Filters

  17. Overlay image at pixel position (x,y) with kernel Calculate mean of all pixels within area covered by kernel Move kernel to next pixel Mean (Box) Filter

  18. Box Filter Raw Data

  19. Box Filter Raw Data

  20. Box Filter 7x7 box filter

  21. Artifacts of Box Filter © 2002 R. C. Gonzalez & R. E. Woods

  22. Give center pixel highest weight Diagonal pixels are farthest away from center Often modeled after Gauss distribution Weighted Average

  23. Weighted Average (Lowpass Filter) Raw Data

  24. Principle Order pixels within neighborhood and replace center pixel with value based on ranking (e.g. median) Purpose Impulse noise reduction (salt-and-pepper) Contouring, posterization Advantages Minimal degradation and shift of edges Does not reduce brightness across steps Can be applied multiple times Disadvantages Non-linear filter Order-Statistics Filters

  25. Rank all pixel values covered by kernel Replace center pixel with median value Median Filter Median

  26. Median Filter Raw Data Median

  27. Median Filter Raw Data Median

  28. Principle Replace every pixel with first or second derivative Retains high frequency information Reduces low frequency information Purpose Edge detection Highlight fine details Advantages Does not blur image Second derivative is linear transformation Disadvantages Enhance noise Sharpening Spatial (Highpass) Filters

  29. Requirements for digital implementation Zero in flat segments Non-zero at onset of step or ramp Non-zero along ramp Detects edges Directional filter First Derivative (Slope)(e.g. Sobel Filter)

  30. Requirements for digital implementation Zero in flat segments Non-zero at onset and end of step or ramp Zero along ramp of constant slope Sharpens image Second Derivative (e.g. Laplacian Filter)

  31. Derivatives © 2002 R. C. Gonzalez & R. E. Woods

  32. Derivatives: Image Data

  33. Derivatives: First Derivative

  34. Derivatives: Second Derivative

  35. Based on first derivative (gradient) Apply filter for x-direction, than for y-direction Factor 2 adds some smoothing Sobel Operators X mask Y mask and

  36. Sobel Operator Raw data γ=0.7

  37. Based on second derivative Add original image to keep background features (subtract or add Laplacian depending on definition) Laplacian Filter Isotropic for: 90° 45° Image - or = or

  38. Laplacian Filter w/o background addition Maximum projection

  39. High-Boost Filter Maximum projection A=1: Laplacian

  40. September 8, 2006FCS User Club September 13-15, 2006: Advanced Microscopy Workshop September 29 , 2006Fourier Transformation (Earl Glynn) October 6, 2006 Illumination and Filters (Amanda Combs) October 27, 2006De-Convolution (Christopher Wood) Foundations of Microscopy, Image Processing, FCS User Club

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