Image Restoration – Degradation Model and General Approaches

1. Image Restoration – Degradation Model and General Approaches

2. 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. Image Restoration - 1 General approaches • Model the degradation • Apply the inverse process to recover the original image

5. 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

6. Image Restoration - 1 Noise models – examples

7. Image Restoration - 1 Noise models – examples

8. 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

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

13. 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. Image Restoration - 1 De-noising – Adaptive filters • Adaptive local noise reduction filter • Adaptive median filter • Homework: read pp.241-243

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. Image Restoration - 1 De-noising – evaluation example

21. 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. 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. 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

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

29. Image Restoration - 1 Inverse filtering • Degradation model • Inverse filter

30. Image Restoration - 1 Inverse filtering - examples

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