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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 September 29 , 2006Fourier Transformation (Earl Glynn) October 27, 2006De-Convolution (Christopher Wood)
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)
Different Types of Noise Gaussian Rayleigh Uniform Salt & Pepper Image Frequency Intensity © 2002 R. C. Gonzalez & R. E. Woods
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)
Basic Gray-level Transformations © 2002 R. C. Gonzalez & R. E. Woods
fly_max250=fly < 250 Linear Gray Level Transformation
Power-Law (Gamma) Transformation © 2002 R. C. Gonzalez & R. E. Woods
Gamma correction γ=0.01 Power-Law Transformation
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
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
Box Filter Raw Data
Box Filter Raw Data
Box Filter 7x7 box filter
Artifacts of Box Filter © 2002 R. C. Gonzalez & R. E. Woods
Give center pixel highest weight Diagonal pixels are farthest away from center Often modeled after Gauss distribution Weighted Average
Weighted Average (Lowpass Filter) Raw Data
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
Rank all pixel values covered by kernel Replace center pixel with median value Median Filter Median
Median Filter Raw Data Median
Median Filter Raw Data Median
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
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)
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)
Derivatives © 2002 R. C. Gonzalez & R. E. Woods
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
Sobel Operator Raw data γ=0.7
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
Laplacian Filter w/o background addition Maximum projection
High-Boost Filter Maximum projection A=1: Laplacian
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