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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. 5.4 Periodic Noise Reduction by Frequency Domain Filtering ( J.Shanbehzadeh M.Gholizadeh )

  6. Bandreject Filters Bandpass Filters Notch 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. Periodic Noise Reduction by Frequency Domain 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 Degraded Image d(r,c) D(U,V) Fourier Transform Frequency Domain Filter R(u,v) Degraded Function h(r,c) H(U,V) N(U,V) Noise Model n(r,c) Inverse Fourier Transform Restored Image ( J.Shanbehzadeh M.Gholizadeh )

  8. Bandreject Filters Bandreject Filters Bandpass Filters Notch 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 )

  9. BandrejectFilters • 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 • Use to eliminate frequency components in some bands. • Ideal Band-reject Filter: -D(u,v) =distance from the origin of the centered freq. rectangle -W=width of the band -D0=Radial center of the band. ( J.Shanbehzadeh M.Gholizadeh )

  10. BandrejectFilters • 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 Degraded image DFT Periodic noise can be reduced by setting frequency components corresponding to noise to zero. Bandreject filter Restored image ( J.Shanbehzadeh M.Gholizadeh )

  11. Bandreject Filters Bandpass Filters Restoration in the Presence of Noise Only - Spatial Filtering Bandpass Filters Notch 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 )

  12. BandpassFilters • 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 • Opposite operation of a band-reject filter: Periodic noise from the previous slide that is Filtered out. ( J.Shanbehzadeh M.Gholizadeh )

  13. Bandreject Filters Notch Filters Restoration in the Presence of Noise Only - Spatial Filtering Bandpass Filters Notch 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 )

  14. Notch 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 • A notch reject filter is used to eliminate some frequency components. • Rejects (or passes) frequencies in predefined neighborhoods about a center frequency. Ideal Must appear in symmetric pairs about the origin. Gaussian Butterworth ( J.Shanbehzadeh M.Gholizadeh )

  15. Notch reject 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 Notch filter (freq. Domain) Degraded image DFT ( J.Shanbehzadeh M.Gholizadeh ) Noise Restored image

  16. Notch reject 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 )

  17. Notch reject 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 )

  18. Bandreject Filters Restoration in the Presence of Noise Only - Spatial Filtering Bandpass Filters Notch 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 )

  19. Image Degraded by Periodic Noise Degraded image DFT (no shift) • 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 Several pairs of components are present  more than just one sinusoidal component DFT of noise Noise ( J.Shanbehzadeh M.Gholizadeh ) Restored image

  20. 5.6 Estimating the degradation Function ( J.Shanbehzadeh M.Gholizadeh )

  21. Estimation by Image Observation Estimation by Experimentation Estimation by Modeling • 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. Estimating the Degradation Function Degradation model: • 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 or Purpose: To estimateh(x,y) or H(u,v) Why? If we know exactly h(x,y), regardless of noise, we can do deconvolution to get f(x,y) back from g(x,y). Methods: 1. Estimation by Image Observation 2. Estimation by Experiment 3. Estimation by Modeling ( J.Shanbehzadeh M.Gholizadeh )

  23. Estimating the Degradation Function • 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 • Look for the information in the image itself: search for the small section of image containing simple structure (edge, point) • Select a small section from the degraded image • Reconstruct an unblurred image of the same size • The degradation function can be estimated by : ( J.Shanbehzadeh M.Gholizadeh )

  24. Estimation by Image Observation Estimation by Image Observation Restoration in the Presence of Noise Only - Spatial Filtering Estimation by Experimentation Estimation by Modeling • 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. Estimation by Image Observation • 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. Estimation by Image Observation Degraded image Original image (unknown) • 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 g(x,y) f(x,y)*h(x,y) Observation f(x,y) Subimage DFT Estimated Transfer function Restoration process by estimation DFT Reconstructed Subimage This case is used when we know only g(x,y) and cannot repeat the experiment! ( J.Shanbehzadeh M.Gholizadeh )

  27. Estimation by Image Observation Estimation by Experimentation Restoration in the Presence of Noise Only - Spatial Filtering Estimation by Experimentation Estimation by Modeling • 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. Estimation by Experiment • 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 • If we have the equipment used to acquire degraded image we can obtain accurate estimation of the degradation • Obtain an impulse response of the degradation using the same system setting • A linear space-invariant system is characterized completely by its impulse response ( J.Shanbehzadeh M.Gholizadeh )

  29. Estimation by Experiment • 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. Estimation by Experiment • Used when we have the same equipment set up and can repeat the experiment. • 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 Response image from the system Input impulse image System H( ) DFT DFT ( J.Shanbehzadeh M.Gholizadeh )

  31. Estimation by Image Observation Estimation by Modeling Restoration in the Presence of Noise Only - Spatial Filtering Estimation by Experimentation Estimation by Modeling • 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. Estimation by Modeling • 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 • Mathematical model of degradation can be for example atmosphere turbulence • Hufnagel & Stanley (1964) has established a degradation model due to atmospheric turbulence • K is a parameter to be determined by experiments because it changes with the nature of turbulence ( J.Shanbehzadeh M.Gholizadeh )

  33. Estimation by Modeling • 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. Estimation by Modeling • 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 • Used when we know physical mechanism underlying the image formation process that can be expressed mathematically. Example: Original image Severe turbulence Atmospheric Turbulence model k = 0.0025 Mild turbulence Low turbulence k = 0.00025 k = 0.001 ( J.Shanbehzadeh M.Gholizadeh )

  35. Estimation by Modeling: Motion Blurring • 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 Assume that camera velocity is The blurred image is obtained by where T = exposure time. ( J.Shanbehzadeh M.Gholizadeh )

  36. Motion Blurring - 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 For constant motion Original image Motion blurred image a = b = 0.1, T = 1 ( J.Shanbehzadeh M.Gholizadeh )

  37. Blur • 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 • Linear in one direction Horizontal Vertical Diagonal ( J.Shanbehzadeh M.Gholizadeh )

  38. PSF (Point Spread Function) • 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 • 2D Equivalent to Impulse Response • What happen to a single point of light when it passes through a system? • PSF describes a LSI system • In practise PSF should be estimated ( J.Shanbehzadeh M.Gholizadeh )

  39. Typical Blur Mask Coefficients • 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. 5.7 Inverse Filtering ( J.Shanbehzadeh M.Gholizadeh )

  41. Inverse 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 ( J.Shanbehzadeh M.Gholizadeh )

  42. Inverse 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 • Based on properties of the Fourier transforms • Assume degradation can be expressed as convolution • After applying the Fourier transform to Eq. (a), weget • An estimate Fˆ(u,v) of the transform of the original image ( J.Shanbehzadeh M.Gholizadeh )

  43. Inverse 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 ( J.Shanbehzadeh M.Gholizadeh )

  44. Inverse 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 • The degradation can be eliminated using the restoration filter with a transfer function that is inverse to the degradation h. • The Fourier transform of the inverse filter is then expressed as H-1(u,v) • We obtain the original undegraded image F from its degraded version G Example: ( J.Shanbehzadeh M.Gholizadeh )

  45. Inverse 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 Degradation function Cutting off values of the ratio outside a radius of 40, 70,85. ( J.Shanbehzadeh M.Gholizadeh )

  46. Restoration Cut-off Frequency • 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 • Limiting the restoration to a specific frequency about the origin Result: • Low-pass image • Blurred • Ringing ( J.Shanbehzadeh M.Gholizadeh )

  47. Inverse Filtering - 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 )

  48. Inverse Filtering - 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 )

  49. 5.8 Minimum Mean Square Error (Wiener) Filtering ( J.Shanbehzadeh M.Gholizadeh )

  50. Minimum Mean Square Error (Wiener) 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 ( J.Shanbehzadeh M.Gholizadeh )

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