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

2. DIGITAL IMAGE PROCESSING Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Gholizadeh M.Gholizadeh mhdgholizadeh@gmail.com ( J.Shanbehzadeh M.Gholizadeh )

3. Road map of chapter 5 5.1 5.3 5.45.5 5.1 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 A Model of the Image Degradation/Restoration Process 5.2 Noise Models Restoration in the Presence of Noise Only-Spatial Filtering 5.3 5.4 Periodic Noise Reduction by Frequency Domain Filtering 5.5 Linear, Position-Invariant Degradations 5.6 Estimating the degradation Function 5.7 5.8 Inverse Filtering Minimum Mean Square Error (Wiener) Filtering ( J.Shanbehzadeh M.Gholizadeh )

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

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

6. Restoration in the Presence of Noise Only - Spatial Filtering 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 Mean FiltersOrdered-Statistic FiltersAdaptive Filters ( J.Shanbehzadeh M.Gholizadeh )

7. Restoration in the Presence of Noise Only - Spatial Filtering 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 Mean FiltersOrdered-Statistic FiltersAdaptive FiltersMean Filters ( J.Shanbehzadeh M.Gholizadeh )

8. Mean Filters Performance superior to the filters discussed in Section 3.6 Degradation 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 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 )

9. 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 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 ( J.Shanbehzadeh M.Gholizadeh )

10. Contra harmonic Mean Filter: Q = the filter order Positive Q is suitable for eliminating pepper noise. Negative Q is suitable for eliminating salt noise. For Q = 0, the filter reduces to an arithmetic mean filter. For Q = -1, the filter reduces to a harmonic mean filter. 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 ( 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 Original Image Corrupted by AWGN Image obtained using a 3x3 geometric mean filter Image obtained using a 3x3 arithmetic mean filter AWGN: Additive White Gaussian Noise 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 )

15. 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 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 ( J.Shanbehzadeh M.Gholizadeh )

16. Contraharmonic Filters: Incorrect Use 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 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 )

17. Contra harmonic Filters - Example The Contraharmonic mean filter works well for images containing salt or pepper type noise 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 )

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. Restoration in the Presence of Noise Only - Spatial Filtering 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 Mean FiltersOrdered-Statistic FiltersAdaptive FiltersOrdered-Statistic Filters ( J.Shanbehzadeh M.Gholizadeh )

20. Order-Statistic Filters: Revisit Subimage Original image Moving Window Statistic parameters Mean, Median, Mode, Min, Max, Etc. Output image 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 )

21. Order-Statistics Filters 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) 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 )

22. Midpoint filter Combines order statistics and averaging 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 ( J.Shanbehzadeh M.Gholizadeh )

23. Median Filter: How it works Degraded image Salt noise Pepper noise Moving Window Sorted Array Salt noise Pepper noise Median 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. A median filter is good for removing impulse, isolated noise 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 )

24. Median Filter: How it works Image corrupted by salt and pepper noise with p a =p b = 0.1 Images obtained using a 3x3 median filter 1 4 2 3 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 )

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 Image corrupted by pepper noise with prob. = 0.1 Image corrupted by salt noise with prob. = 0.1 Image obtained using a 3x3 max filter Image obtained using a 3x3 min 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 ( J.Shanbehzadeh M.Gholizadeh )

28. Max and Min Filters - Example Various Windows size for Max and Min 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 )

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 Formula: where g r (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. 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 )

33. Alpha-trimmed Mean Filter - Example Image corrupted by additive uniform noise Image 2 obtained using a 5x5 arithmetic mean filter Image additionally corrupted by additive salt-and- pepper noise Image 2 obtained using a 5x5 geometric mean filter 12 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 )

34. Image corrupted by additive uniform noise Image 2 obtained using a 5x5 median filter Image additionally corrupted by additive salt-and- pepper noise 1 2 Image 2 obtained using a 5x5 alpha- trimmed mean filter with d = 5 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 )

35. Image obtained using a 5x5 arithmetic mean filter Image obtained using a 5x5 geometric mean filter Image obtained using a 5x5 median filter Image obtained using a 5x5 alpha- trimmed mean filter with d = 5 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 )

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. Restoration in the Presence of Noise Only - Spatial Filtering 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 Mean FiltersOrdered-Statistic FiltersAdaptive Filters ( J.Shanbehzadeh M.Gholizadeh )

39. Adaptive Filters 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 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 )

40. Adaptive, Local Noise Reduction Filters 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 S xy The local variance of the pixels in S xy The filter can be proceeded as follows: The variance of the noise s h 2 is zero (zero noise): g(x,y)=f(x,y) High local variance (edges) relative to s h 2 : 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 S xy 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 )

41. Purpose: 1. If s h 2 is zero,  No noise the filter should return g(x,y) because g(x,y) = f(x,y) 2. If s L 2 is high relative to s h 2,  Edges (should be preserved), the filter should return the value close to g(x,y) 3. If s L 2 = s h 2,  Areas inside objects the filter should return the arithmetic mean value m L Formula: Concept: Want to preserve edges 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 ( J.Shanbehzadeh M.Gholizadeh )

42. 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 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 ( J.Shanbehzadeh M.Gholizadeh )

43. Adaptive Noise Reduction Filter - Example Image corrupted by additive Gaussian noise with zero mean and  2 =1000 Image obtained using a 7x7 arithmetic mean filter Image obtained using a 7x7 geometric mean filter Image obtained using a 7x7 adaptive noise reduction 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 ( 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 Median filter perform well as long as spatial density of the impulse noise is not vary large (P a and P b <0.2 ) The adaptive filter Can handle impulse noise with probabilities even larger than P a and P b >0.2 Preserve detail while smoothing non-impulse noise Change (increase) S xy during filter operation 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 )

46. Algorithm: Level A: A1= z median – z min A2= z median – z max If A1 > 0 and A2 < 0, go to level B Else increase window size If window size <= S max repeat level A Else return z xy Level B: B1= z xy – z min B2= z xy – z max If B1 > 0 and B2 < 0, return z xy Else return z median z min = minimum gray level value in S xy z max = maximum gray level value in S xy z median = median of gray levels in S xy z xy = gray level value at pixel (x,y) S max = maximum allowed size of S xy where Purpose: Want to remove impulse noise while preserving edges 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 ( J.Shanbehzadeh M.Gholizadeh )

47. Adaptive Median Filter: How it works Level A: A1= z median – z min A2= z median – z max Determine whether z median is an impulse or not If A1 > 0 and A2 < 0, go to level B Else  Window is not big enough increase window size If window size <= S max repeat level A Else return z xy 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 )

48. Determine whether z xy is an impulse or not Level B:  z median is not an impulse B1= z xy – z min B2= z xy – z max If B1 > 0 and B2 < 0,  z xy is not an impulse return z xy  to preserve original details Else return z median  to remove impulse 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 ( J.Shanbehzadeh M.Gholizadeh )

49. Adaptive Median Filter - Example Image corrupted by salt-and-pepper noise with p a =p b = 0.25 Image obtained using a 7x7 median filter Image obtained using an adaptive median filter with S max = 7 More small details are preserved 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 )