1. Image Restoration – Degradation Model and General Approaches

2. Image enhancement vs. restoration Degradation model
Image Restoration - 1
Outline
Image enhancement vs. restoration
Degradation model
Noise only
Linear, space-invariant
General approaches
Inverse filters
Wiener filters
Constrained least squares filtering

3. Image Restoration - 1
Image enhancement vs. restoration
Image enhancement : process image so that the result is more suitable for a specific application, is largely a subjective process.
Image restoration : recover image from distortions to its original image, is largely an objective process.

4. Apply the inverse process to recover the original image
Image Restoration - 1
General approaches
Model the degradation
Apply the inverse process to recover the original image
The degradation process is modeled as a degradation function with an additive noise term.

5. Degradation models : noise only
Image Restoration - 1
Degradation models : noise only
Noise models
Spatial characteristics (independent or dependent)
Intensity ( distribution, spectrum)
Uniform, Gaussian, Rayleigh, Gamma (Erlang), Exponential, impulse
Correlation with the image (additive, multiplicative)
De-noising
Spatial filtering
Frequency domain filtering
The principal sources of noise arise during image acquisition and/or transmission.

6. Image Restoration - 1
Noise models – examples

7. Image Restoration - 1
Noise models – examples

8. Degradation models : noise only
Image Restoration - 1
Degradation models : noise only
Noise simulation (Random number/signal generator)
盛文，焦晓丽. 雷达系统建模与仿真导论. 国防工业出版社，2006
The principal sources of noise arise during image acquisition and/or transmission.

9. Degradation models : noise only
Image Restoration - 1
Degradation models : noise only
高斯分布随机变量的近似产生方法
中心极限定理：n个均值为 、方差为 的随机变量的和服从均值为 、方差为 的近似正态分布
取n个[0，1]区间上的均匀分布随机变量，则以下随机变量服从均值为0、方差为1的近似标准正态分布。

10. Image Restoration - 1
Noise models – periodic noise

11. Estimation of noise parameters
Image Restoration - 1
De-noising
Estimation of noise parameters
By spectrum inspection: for periodic noise
By test image: mean, variance and histogram shape, if imaging system is available
By small patches, if only image is available
De-noising
Spatial filtering ( for additive noise)
Mean filters
Order-statistics filters
Adaptive filters
Frequency domain filtering (for periodic noise)

12. De-noising – Gaussian noise example
Image Restoration - 1
De-noising – Gaussian noise example
Arithmetic mean filter
Geometric mean filter

13. Order-statistics filters
Image Restoration - 1
De-noising
Order-statistics filters
Median filter
Good for impulse noise reduction with less blurring
Max filter
Find the brightest points
Min filter
Find the darkest points
Midpoint filter
Combines order statistics and averaging, works best for randomly distributed noise.

14. Image Restoration - 1
De-noising – Salt & Pepper noise example

15. Image Restoration - 1
De-noising – Salt & Pepper noise example

16. De-noising – Adaptive filters
Image Restoration - 1
De-noising – Adaptive filters
Adaptive local noise reduction filter
Adaptive median filter
Homework: read pp

17. Image Restoration - 1
De-noising – Adaptive local filter

18. Image Restoration - 1
De-noising – Periodic noise example

19. Image Restoration - 1
De-noising – evaluation
PSNR
Visual perception

20. De-noising – evaluation example
Image Restoration - 1
De-noising – evaluation example
ImageDenoising.m

21. Degradation models: linear vs. non-linear
Image Restoration - 1
Degradation models: linear vs. non-linear
Many types of degradation can be approximated by linear, space invariant processes
Can take advantages of the mature techniques developed for linear systems
Non-linear and space variant models are more accurate
Difficult to solve
Unsolvable

22. Linear, space-invariant degradations
Image Restoration - 1
Linear, space-invariant degradations
Sampling theorem -->
Linearity, additivity -->
Linearity, homogeneity -->
Space-invariant -->
(convolution integral)

23. Image Restoration - 1
Linear, space-invariant degradations (cont’)
Point Spread Function:
Linear, space-invariant degradation model:

24. Estimating degradation function - 1
Image Restoration - 1
Estimating degradation function - 1
Estimation by image observation
Degradation system H is completely characterized by its impulse response
Select a small section from the degraded image
Reconstruct an unblurred image of the same size
The degradation function can be estimated by
By ignoring the noise term, G(u,v) = F(u,v)H(u,v). If F(u,v) is the Fourier transform of point source (impulse), then G(u,v) is approximates H(u,v).

25. Estimating degradation function - 2
Image Restoration - 1
Estimating degradation function - 2
Estimation by experimentation
Point spread function (PSF)
Used in optics
The impulse becomes a point of light
The impulse response is commonly referred to as the PSF

26. Image Restoration - 1
Estimating degradation function - 3
Estimation by modeling – atmospheric turbulence

27. Image Restoration - 1
Estimating degradation function - 3
Estimation by modeling – linear motion blurring

28. The regularization theory
Image Restoration - 1
Different approaches
Classical approaches
Inverse filter
Weiner filter
Algebraic approaches
The regularization theory

29. Image Restoration - 1 Inverse filtering Degradation model
This expression tells us that even if we know the degradation function we cannot recover the undegraded image exactly because of the random noise, whose Fourier transform is not known. Image restoration is an ill-posed problem. When H(u,v) is very small, the noise term dominates the restoration result.

30. Inverse filtering - examples
Image Restoration - 1
Inverse filtering - examples
This expression tells us that even if we know the degradation function we cannot recover the undegraded image exactly because of the random noise, whose Fourier transform is not known. Image restoration is an ill-posed problem. When H(u,v) is very small, the noise term dominates the restoration result.

31. Image Restoration - 1 Wiener filtering
In most images, adjacent pixels are highly correlated, while the gray level of widely separated pixels are only loosely correlated.
Therefore, the autocorrelation function of typical images generally decreases away from the origin.
Power spectrum of an image is the Fourier transform of its autocorrelation function, therefore we can argue that the power spectrum of an image generally decreases with frequency.
Typical noise sources have either a flat power spectrum or one that decreases with frequency more slowly than typical image power spectrum.
Therefore, the expected situation is for the signal to dominate the spectrum at low frequencies, while the noise dominates the high frequencies.

32. Image Restoration - 1
Wiener filtering (cont’)
Degradation model
Wiener filter

33. Image Restoration - 1
Wiener filtering - example

34. Image Restoration - 1
Wiener filtering - example

35. Image Restoration - 1
Wiener filtering - problems
The power spectra of the undegraded image and noise must be known.
Weights all errors equally regardless of their location in the image, while the eye is considerably more tolerant of errors in the dark areas and high-gradient areas in the image.
In minimizing the mean square error, Wiener filter also smooth the image more than the eye would prefer.

36. Image Restoration - 1
Constrained Least Squares Filtering
Only the mean and variance of the noise is required
The degradation model in vector-matrix form
The objective function

37. Image Restoration - 1
Constrained Least Squares Filtering
The solution

38. Image Restoration - 1
Constrained Least Squares Filtering - example