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DIGITAL IMAGE PROCESSING

DIGITAL IMAGE PROCESSING. Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Gholizadeh mhdgholizadeh@gmail.com. DIGITAL IMAGE PROCESSING. Chapter 5 - Image Restoration and Reconstruction. Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Gholizadeh

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DIGITAL IMAGE PROCESSING

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  1. DIGITAL IMAGE PROCESSING Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Gholizadeh mhdgholizadeh@gmail.com

  2. DIGITAL IMAGE PROCESSING Chapter 5 - Image Restoration and Reconstruction Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Gholizadeh mhdgholizadeh@gmail.com ( J.ShanbehzadehM.Gholizadeh )

  3. Road map of chapter 5 5.5 5.3 5.3 5.8 5.4 5.6 5.1 5.2 5.4 5.5 5.1 5.2 5.8 5.7 5.7 5.6 • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • Estimating the degradation Function • Noise Models • A Model of the Image Degradation/Restoration Process • Minimum Mean Square Error (Wiener) Filtering • Periodic Noise Reduction by Frequency Domain Filtering • Restoration in the Presence of Noise Only-Spatial Filtering • Linear, Position-Invariant Degradations • Inverse Filtering ( J.Shanbehzadeh M.Gholizadeh )

  4. Road map of chapter 5 5.11 5.11 5.9 5.10 5.10 5.9 • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Geometric Mean Filter • Constrained Least Square Filtering • Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  5. 5.3 Restoration in the Presence of Noise Only - Spatial Filtering ( J.Shanbehzadeh M.Gholizadeh )

  6. Mean Filters Restoration in the Presence of Noise Only - Spatial Filtering • Ordered-Statistic Filters Adaptive Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  7. Mean Filters • Mean Filters Restoration in the Presence of Noise Only - Spatial Filtering • Ordered-Statistic Filters Adaptive Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  8. Mean Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Performance superior to the filters discussed in Section 3.6Degradation Model: • Arithmetic Mean Filter(Moving Average Filter): • Computes the average value of the corrupted image g(x,y) • The value of the restored image f To remove this part mn = size of moving window ( J.Shanbehzadeh M.Gholizadeh )

  9. Mean Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Geometric Mean Filter: • Smooth comparable to the arithmetic mean filter, but it tends to loss less detail. • Harmonic Mean Filter: • Work well for salt noise and Gaussian noise, but fails for pepper noise Works well for salt noise but fails for pepper noise ( J.Shanbehzadeh M.Gholizadeh )

  10. Mean Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Contra harmonic Mean Filter: Positive Q is suitable for eliminating pepper noise. Negative Q is suitable for eliminating salt noise. Q = the filter order For Q = 0, the filter reduces to an arithmetic mean filter. For Q = -1, the filter reduces to a harmonic mean filter. ( J.Shanbehzadeh M.Gholizadeh )

  11. Arithmetic Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  12. Arithmetic Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  13. Arithmetic Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  14. Geometric Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image Corrupted by AWGN Original Image Image obtained using a 3x3 arithmetic mean filter Image obtained using a 3x3 geometric mean filter AWGN: Additive White Gaussian Noise ( J.Shanbehzadeh M.Gholizadeh )

  15. Geometric Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image corrupted by salt noise with prob. = 0.1 Image corrupted by pepper noise with prob. = 0.1 Image obtained using a 3x3 contra- harmonic mean filter With Q = 1.5 Image obtained using a 3x3 contra-harmonic mean filter With Q=-1.5 ( J.Shanbehzadeh M.Gholizadeh )

  16. ContraharmonicFilters: Incorrect Use • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image corrupted by pepper noise with prob. = 0.1 Image corrupted by salt noise with prob. = 0.1 Image obtained using a 3x3 contra- harmonic mean filter With Q=-1.5 Image obtained using a 3x3 contra- harmonic mean filter With Q=1.5 ( J.Shanbehzadeh M.Gholizadeh )

  17. Contra harmonic Filters - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • The Contraharmonic mean filter works well for images containing salt or pepper type noise ( J.Shanbehzadeh M.Gholizadeh )

  18. Contra harmonic Filters - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  19. Mean Filters • Ordered-Statistic Filters Restoration in the Presence of Noise Only - Spatial Filtering • Ordered-Statistic Filters Adaptive Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  20. Order-Statistic Filters: Revisit • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Subimage Original image Statistic parameters Mean, Median, Mode, Min, Max, Etc. Moving Window ( J.Shanbehzadeh M.Gholizadeh ) Output image

  21. Order-Statistics Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Median filters: • Are particularly effective in the presence of both bipolar and unipolar impulse noise • Max filters (Fig. 5.8) • Max filter: reduce low values caused by pepper noise • Min filters (Fig. 5.8) • Min filter: reduce high values caused by salt noise Reduce “dark” noise (pepper noise) Reduce “bright” noise (salt noise) ( J.Shanbehzadeh M.Gholizadeh )

  22. Order-Statistics Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Midpoint filter • Combines order statistics and averaging ( J.Shanbehzadeh M.Gholizadeh )

  23. Median Filter: How it works • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections A median filter is good for removing impulse, isolated noise Salt noise Pepper noise Median Moving Window Degraded image Salt noise Sorted Array Pepper noise Filter output Normally, impulse noise has high magnitude and is solated. When we sort pixels in the moving window, noise pixels are usually at the ends of the array. Therefore, it’s rare that the noise pixel will be a median value. ( J.Shanbehzadeh M.Gholizadeh )

  24. Median Filter: How it works • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections 1 2 Image corrupted by salt and pepper noise with pa=pb= 0.1 3 4 Images obtained using a 3x3 median filter ( J.Shanbehzadeh M.Gholizadeh )

  25. Median Filter: How it works • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  26. Median Filter: How it works • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  27. Max and Min Filters - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image corrupted by salt noise with prob. = 0.1 Image corrupted by pepper noise with prob. = 0.1 Image obtained using a 3x3 min filter Image obtained using a 3x3 max filter ( J.Shanbehzadeh M.Gholizadeh )

  28. Max and Min Filters - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Various Windows size for Max and Min Filters: ( J.Shanbehzadeh M.Gholizadeh )

  29. Max and Min Filters - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  30. Max and Min Filters - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  31. Max and Min Filters - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  32. Alpha -trimmed Mean Filter • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Formula: • where gr(s,t) represent the remaining mn-d pixels after removing the d/2 highest and d/2 lowest values of g(s,t). • This filter is useful in situations involving multiple types of noise such as a combination of salt-and-pepper and Gaussian noise. ( J.Shanbehzadeh M.Gholizadeh )

  33. Alpha-trimmed Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image additionally corrupted by additive salt-and- pepper noise Image corrupted by additive uniform noise 1 2 Image 2 obtained using a 5x5 arithmetic mean filter Image 2 obtained using a 5x5 geometric mean filter ( J.Shanbehzadeh M.Gholizadeh )

  34. Alpha-trimmed Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image additionally corrupted by additive salt-and- pepper noise 1 2 Image corrupted by additive uniform noise Image 2 obtained using a 5x5 alpha- trimmed mean filter with d = 5 Image 2 obtained using a 5x5 median filter ( J.Shanbehzadeh M.Gholizadeh )

  35. Alpha-trimmed Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image obtained using a 5x5 arithmetic mean filter Image obtained using a 5x5 geometric mean filter Image obtained using a 5x5 alpha- trimmed mean filter with d = 5 Image obtained using a 5x5 median filter ( J.Shanbehzadeh M.Gholizadeh )

  36. Alpha-trimmed Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  37. Alpha-trimmed Mean Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  38. Mean Filters Adaptive Filters Restoration in the Presence of Noise Only - Spatial Filtering • Ordered-Statistic Filters Adaptive Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  39. Adaptive Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Global Concept: • Can apply to an image with regard to how image characteristics vary from one point to another • Filter behavior depends on statistical characteristics of local areas inside m×n moving window • More complex but superior performance compared with “fixed” filters • Statistical characteristics: • Local mean: • Local Variance: Noise variance ( J.Shanbehzadeh M.Gholizadeh )

  40. Adaptive, Local Noise Reduction Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • The response of the filter based on four quantities: • The value of the noise image • The variance of the noise corrupting f(x,y) to form g(x,y) • The local means of the pixel in Sxy • The local variance of the pixels in Sxy • The filter can be proceeded as follows: • The variance of the noise sh2 is zero (zero noise): g(x,y)=f(x,y) • High local variance (edges) relative to sh2 : edges that should be preserved; return a value close to g(x,y) • The two variances are equal: return the arithmetic mean value of the pixels in Sxy ( J.Shanbehzadeh M.Gholizadeh )

  41. Adaptive, Local Noise Reduction Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Purpose: Want to preserve edges Concept: • 1. If sh2 is zero,  No noise • the filter should return g(x,y) because g(x,y) = f(x,y) • 2. If sL2 is high relative to sh2,  Edges (should be preserved), • the filter should return the value close to g(x,y) • 3. If sL2 = sh2,  Areas inside objects • the filter should return the arithmetic mean value mL Formula: ( J.Shanbehzadeh M.Gholizadeh )

  42. Adaptive, Local Noise Reduction Filters • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • The variance of the overall noise needs to be estimated. • The only quantities that needs to be known or estimated: The variance of the overall noise • then the ratio (Eq. 5.3-12) is set to 1: Prevent negative gray levels • If negative values occur, rescale the gray values at the end (lost of the dynamic range) • Similar noise removal results compared to other mean filter ( J.Shanbehzadeh M.Gholizadeh )

  43. Adaptive Noise Reduction Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image corrupted by additive Gaussian noise with zero mean and s2=1000 Image obtained using a 7x7 arithmetic mean filter Image obtained using a 7x7 adaptive noise reduction filter Image obtained using a 7x7 geometric mean filter ( J.Shanbehzadeh M.Gholizadeh )

  44. Adaptive Noise Reduction Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  45. Adaptive Median Filter • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Median filter perform well as long as spatial density of the impulse noise is not vary large (Pa and Pb <0.2 ) • The adaptive filter • Can handle impulse noise with probabilities even larger than Pa and Pb >0.2 • Preserve detail while smoothing non-impulse noise • Change (increase) Sxy during filter operation ( J.Shanbehzadeh M.Gholizadeh )

  46. Adaptive Median Filter • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Purpose: Want to remove impulse noise while preserving edges Algorithm: Level A: A1= zmedian– zmin A2= zmedian– zmax If A1 > 0 and A2 < 0, go to level B Else increase window size If window size <= Smax repeat level A Else return zxy Level B: B1= zxy– zmin B2= zxy– zmax If B1 > 0 and B2 < 0, return zxy Else return zmedian where zmin = minimum gray level value in Sxy zmax = maximum gray level value in Sxy zmedian = median of gray levels in Sxy zxy = gray level value at pixel (x,y) Smax = maximum allowed size of Sxy ( J.Shanbehzadeh M.Gholizadeh )

  47. Adaptive Median Filter: How it works • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Determine whether zmedianis an impulse or not Level A: A1= zmedian– zmin A2= zmedian– zmax If A1 > 0 and A2 < 0, go to level B Else  Window is not big enough increase window size If window size <= Smax repeat level A Else return zxy ( J.Shanbehzadeh M.Gholizadeh )

  48. Adaptive Median Filter: How it works • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Level B: zmedian is not an impulse B1= zxy– zmin B2= zxy– zmax If B1 > 0 and B2 < 0,  zxy is not an impulse return zxy to preserve original details Else return zmedian to remove impulse Determine whether zxyis an impulse or not ( J.Shanbehzadeh M.Gholizadeh )

  49. Adaptive Median Filter - Example • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections Image corrupted by salt-and-pepper noise with pa=pb= 0.25 Image obtained using a 7x7 median filter Image obtained using an adaptive median filter with Smax = 7 More small details are preserved ( J.Shanbehzadeh M.Gholizadeh )

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