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

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

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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 )

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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 )

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

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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 Bandreject Filters Bandpass FiltersNotch Filters ( J.Shanbehzadeh M.Gholizadeh )

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

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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 Bandreject Filters Bandpass FiltersNotch Filters Bandreject Filters ( J.Shanbehzadeh M.Gholizadeh )

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Bandreject Filters -D(u,v) =distance from the origin of the centered freq. rectangle -W =width of the band -D 0 =Radial center of the band. Use to eliminate frequency components in some bands. Ideal Band-reject 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 )

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Bandreject filterRestored image Degraded image DFT Periodic noise can be reduced by setting frequency components corresponding to noise to zero. Bandreject 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 )

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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 Bandreject Filters Bandpass FiltersNotch FiltersBandpass Filters ( J.Shanbehzadeh M.Gholizadeh )

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Bandpass Filters Opposite operation of a band-reject filter: Periodic noise from the previous slide that is Filtered out. 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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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 Bandreject Filters Bandpass FiltersNotch Filters ( J.Shanbehzadeh M.Gholizadeh )

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Notch Filters A notch reject filter is used to eliminate some frequency components. Rejects (or passes) frequencies in predefined neighborhoods about a center frequency. Ideal Butterworth Gaussian Must appear in symmetri c pairs about the origin. 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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Notch reject Filter - Example Degraded image DFT Notch filter (freq. Domain) Restored imageNoise 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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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 )

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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 )

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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 Bandreject Filters Bandpass FiltersNotch Filters ( J.Shanbehzadeh M.Gholizadeh )

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Image Degraded by Periodic Noise Degraded image DFT (no shift) Restored image NoiseDFT of noise Several pairs of components are present more than just one sinusoidal component 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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5.6 Estimating the degradation Function ( J.Shanbehzadeh M.Gholizadeh )

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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 Estimation by Image ObservationEstimation by ExperimentationEstimation by Modeling ( J.Shanbehzadeh M.Gholizadeh )

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Estimating the Degradation Function Degradation model: Purpose: Methods: 1. Estimation by Image Observation 2. Estimation by Experiment 3. Estimation by Modeling or 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). To estimate h(x,y) or H(u,v) 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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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 : 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 ( J.Shanbehzadeh M.Gholizadeh )

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Restoration in the Presence of Noise Only - Spatial Filtering Estimation by Image ObservationEstimation by ExperimentationEstimation by ModelingEstimation 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 )

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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 )

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f(x,y) f(x,y)*h(x,y) g(x,y) Subimage Reconstructed Subimage DFT Restoration process by estimation Original image (unknown)Degraded image Estimated Transfer function Observation This case is used when we know only g(x,y) and cannot repeat the experiment! 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 )

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Restoration in the Presence of Noise Only - Spatial Filtering Estimation by Image ObservationEstimation by ExperimentationEstimation by ModelingEstimation by Experimentation 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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Estimation by Experiment 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 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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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 )

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o Used when we have the same equipment set up and can repeat the experiment. Input impulse image System H( ) Response image from the system DFT 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 )

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Restoration in the Presence of Noise Only - Spatial Filtering Estimation by Image ObservationEstimation by ExperimentationEstimation 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 )

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Estimation by Modeling 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 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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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 )

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Used when we know physical mechanism underlying the image formation process that can be expressed mathematically. Atmospheric Turbulence model Example: Original image Severe turbulence k = 0.00025k = 0.001 k = 0.0025 Low turbulenceMild turbulence 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 )

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Estimation by Modeling: Motion Blurring Assume that camera velocity is The blurred image is obtained by where T = exposure time. 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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Motion Blurring - Example For constant motion Original image Motion blurred image a = b = 0.1, T = 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 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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Blur Linear in one direction Horizontal Vertical Diagonal 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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PSF (Point Spread Function) 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 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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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 )

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5.7 Inverse Filtering ( J.Shanbehzadeh M.Gholizadeh )

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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 )

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Based on properties of the Fourier transforms Assume degradation can be expressed as convolution After applying the Fourier transform to Eq. (a), we get An estimate Fˆ(u,v) of the transform of the original image 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 )

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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 )

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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: 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 )

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Degradation function Cutting off values of the ratio outside a radius of 40, 70,85. 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 )

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Restoration Cut-off Frequency Limiting the restoration to a specific frequency about the origin Result: L ow-pass image B lurred R inging 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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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 )

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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 )

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

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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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The inverse filtration gives poor results, since the information about noise properties is not taken into account. Wiener filtration incorporates a priori knowledge about the noise properties. Restoration by the filter gives an estimate f of the original uncorrupted image f with minimal mean square error : Minimization is easy if the estimate f is a linear combination of the values in the image g; The estimate F of the Fourier transform F of the original image f can be expressed as: 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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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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Minimum Mean square Estimator 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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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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Inverse & Wiener 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 )

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Inverse & Wiener 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 )

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Inverse & Wiener 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 )

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5.9 Constrained Least Square Filtering ( J.Shanbehzadeh M.Gholizadeh )

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Constrained Least Squares Filtering Provides a filter that can eliminate some of the artifacts caused by other frequency domain filters Done by smoothing criterion in the filter derivation The result does not have undesirable oscillations 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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Constrained Least Squares 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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Constrained Least Squares 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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Constrained Least Squares 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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5.10 Geometric Mean filter ( J.Shanbehzadeh M.Gholizadeh )

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Geometric 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 ( J.Shanbehzadeh M.Gholizadeh )

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