DIGITAL IMAGE PROCESSING

DIGITAL IMAGE PROCESSING
Instructors: Dr J. Shanbehzadeh M.Gholizadeh

Chapter 5 - Image Restoration and Reconstruction
DIGITAL IMAGE PROCESSING Chapter 5 - Image Restoration and Reconstruction Instructors: Dr J. Shanbehzadeh M.Gholizadeh ( J.Shanbehzadeh M.Gholizadeh )

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

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

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

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

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

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

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 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 )

Bandreject Filters Degraded image DFT Bandreject filter Restored 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 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 )

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 Filters Bandpass Filters Notch Filters ( J.Shanbehzadeh M.Gholizadeh )

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

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 Filters Notch Filters Notch Filters ( J.Shanbehzadeh M.Gholizadeh )

Must appear in symmetric pairs about the origin.
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. Butterworth Gaussian ( J.Shanbehzadeh M.Gholizadeh )

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

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 )

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 Filters Notch Filters ( J.Shanbehzadeh M.Gholizadeh )

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

5.6 Estimating the degradation Function

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 Estimation by Image Observation Estimation by Experimentation Estimation by Modeling ( J.Shanbehzadeh M.Gholizadeh )

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 estimate h(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 )

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 )

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 Observation Estimation by Image Observation Estimation by Experimentation Estimation by Modeling ( J.Shanbehzadeh M.Gholizadeh )

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

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 Observation Estimation by Experimentation Estimation by Experimentation Estimation by Modeling ( J.Shanbehzadeh M.Gholizadeh )

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 )

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 )

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 Observation Estimation by Experimentation Estimation by Modeling Estimation by Modeling ( J.Shanbehzadeh M.Gholizadeh )

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 )

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. Original image Severe turbulence Example: Atmospheric Turbulence model k = Mild turbulence Low turbulence k = 0.001 k = ( J.Shanbehzadeh M.Gholizadeh )

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 )

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 )

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

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 )

Typical Blur Mask Coefficients

5.7 Inverse Filtering ( J.Shanbehzadeh M.Gholizadeh )

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 )

Inverse Filtering Based on properties of the Fourier transforms
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), we get An estimate Fˆ(u,v) of the transform of the original image ( J.Shanbehzadeh M.Gholizadeh )

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 )

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 )

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

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 )

Inverse Filtering - Example

5.8 Minimum Mean Square Error (Wiener) Filtering

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 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: ( J.Shanbehzadeh M.Gholizadeh )

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 Minimum Mean square Estimator ( J.Shanbehzadeh M.Gholizadeh )

Inverse & Wiener Filtering - Example

Inverse & Wiener Filtering -Example

5.9 Constrained Least Square Filtering

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 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 ( J.Shanbehzadeh M.Gholizadeh )

Constrained Least Squares Filtering

5.10 Geometric Mean filter ( J.Shanbehzadeh M.Gholizadeh )

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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