DIGITAL IMAGE PROCESSING

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. Preview • Goal of Restoration: Improve Image Quality Example Degraded Image Develop Degradation Model Develop Inverse Degradation Process Knowledge Of Image Creation Process Input Image d (r,c ) Output Image I(r,c ) Apply Inverse Degradation Process ( J.Shanbehzadeh M.Gholizadeh )

6. Preview • Restoration is an objective process compared to image enhancement: • Image restoration is to restore a degraded image back to the original image. • Image Enhancement is to manipulate the image so that it is suitable for a specific application. • Contrast stretching is an enhancement technique while debluring function is considered a restoration. • Only consider in this chapter a degraded digital image. • Restoration can be categorized as two groups: • Deterministic methods are applicable to images with little noise and a known degradation • Stochastic methods try to find the best restoration according to a particular stochastic criterion, e.g., a least square method ( J.Shanbehzadeh M.Gholizadeh )

7. 5.1 A Model of the Image Degradation/Restoration Process ( J.Shanbehzadeh M.Gholizadeh )

8. A Model of the Image Degradation/Restoration Process • 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. A Model of the Image Degradation/Restoration Process • 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 • Spatial domain: additive noise • The degraded image in Spatial domain is • whereh(x,y)is a system thatcauses image distortion and h(x,y) is noise. • Frequency domain : blurring • The degraded image in Frequency domain is • Where the terms in capital letters are Fourier transforms. • Objective: obtain an estimate of ( J.Shanbehzadeh M.Gholizadeh )

10. A Model of the Image Degradation/Restoration Process • 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 • Three types of degradation that can be easily expressed mathematically • Relative motion of the camera and object • Wrong lens focus • Atmospheric turbulence ( J.Shanbehzadeh M.Gholizadeh )

11. Spatial and Frequency Properties of Noise Noise Models • Some Important Noise Probability Density Functions • Periodic Noise Estimation of Noise Parameters • 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. The Principal Source of Noise • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Noise arise … • During Image Acquisition • Environment conditions • Quality of sensing elements • For x. Two factors for CCD: light level and sensor temperature • Image Transmission ( J.Shanbehzadeh M.Gholizadeh )

13. Spatial and Frequency Properties of Noise • Spatial and Frequency Properties of Noise Noise Models • Some Important Noise Probability Density Functions • Periodic Noise Estimation of Noise Parameters • 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. Spatial and Frequency Properties of Noise • 5.1- A Model of the Image Degradation/Restoration Process • 5.2- Noise Models • 5.3- Restoration in the Presence of Noise Only-Spatial Filtering • 5.4- Periodic Noise Reduction by Frequency Domain Filtering • 5.5 - Linear, Position-Invariant Degradations • 5.6- Estimating the degradation Function • 5.7- Inverse Filtering • 5.8- Minimum Mean Square Error (Wiener) Filtering • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • White noise: • The Fourier spectrum of noise is constant. • This terminology is a carryover from the physical properties of white light, which contains nearly all frequencies in the visible spectrum in equal properties. • We assume in this chapter: Noise is independent of spatial coordinates. ( J.Shanbehzadeh M.Gholizadeh )

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

16. Noise Probability Density Functions • 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 • Noise cannot be predicted but can be approximately described in statistical way using the probability density function (PDF). • The statistical properties of the gray level of spatial noise can be considered random variables characterized by a PDF. ( J.Shanbehzadeh M.Gholizadeh )

17. Most Common PDFs of Noises • 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 • Gaussian noise • Are used frequently in practice • The PDF of a Gaussian random variable, Z, is given by: • Rayleigh noise • The PDF of Rayleigh noise: • Erlang (Gamma) noise • The PDF of Erlang noise : ( J.Shanbehzadeh M.Gholizadeh )

18. Most Common PDFs of Noises • 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 • Exponential noise • The PDF of exponential noise : • Uniform noise • The PDF of uniform noise is given by: • Impulse noise (Salt and pepper) • The PDF of impulse noise is given by: • If b>a gray level b will appear as a light dot; • If either Pa or Pbis zero, the impulse is called unipolar • If neither probability is zero (bipolar), and especially if they are approximately equal: salt and pepper noise ( J.Shanbehzadeh M.Gholizadeh )

19. Most Common PDFs of Noises • 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 PDF tells how much each z value occurs. ( J.Shanbehzadeh M.Gholizadeh )

20. Noise Factors • 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 • Gaussian noise: electronic circuit noise and sensors noise due to poor illumination and /or temperature • Rayleigh noise: helpful in characterizing noise phenomena in rang imaging • Exponential and gamma noise: application in laser imaging • Impulse noise: found in quick transient such as faulty-switching ; is the only one that is visually indicative • Uniform noise: basis for random number generator • Difficult to differentiate visually between the five image (Fig 5.4(a) ~Fig5.4(b)) ( J.Shanbehzadeh M.Gholizadeh )

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

22. Image Degradation with Additive Noise Original 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 images Histogram ( J.Shanbehzadeh M.Gholizadeh )