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Chapter 5

Chapter 5. Image Restoration. Preview. Goal: improve an image in some predefined sense. Image enhancement: subjective process Image restoration: objective process Restoration attempts to reconstruct an image that has been degraded by using a priori knowledge of the degradation process.

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Chapter 5

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  1. Chapter 5 Image Restoration

  2. Preview • Goal: improve an image in some predefined sense. • Image enhancement: subjective process • Image restoration: objective process • Restoration attempts to reconstruct an image that has been degraded by using a priori knowledge of the degradation process. • Modeling the degradation and applying the inverse process to recover the original image. • When degradation model is unknown  blind deconvolution (ICA)

  3. A Model of Degradation • or • Given g(x,y), some knowledge about H, and some knowledge about the noise term, obtain an estimate of the original image.

  4. Noise Models • Gaussian noise: electronic circuit sensor noise • Rayleigh noise: range imaging • Erlang (Gamma noise): laser imaging • Exponential noise: laser imaging • Uniform noise • Impulse (salt-and-pepper noise): faulty switching • Periodic noise

  5. Gaussian Noise • The PDF of a Gaussian random variable, z, is given by:

  6. Rayleigh Noise • The PDF of Rayleigh noise is given by: • Mean and variance are given by: • Useful for approximating skewed histograms.

  7. Erlang (Gamma) Noise • The PDF of Erlang noise is given by: • Mean and variance:

  8. Exponential Noise • The PDF of exponential noise is given by: where a >0 • Mean and variance:

  9. Uniform Noise • The PDF of uniform noise is given by: • Mean and variance:

  10. Impulse (Salt-and-Pepper) Noise • The PDF of (bipolar) impulse noise is given by:

  11. Periodic Noise • Arises typically from electrical or electromechanical interference during image acquisition. • The only type of spatially dependent noise considered in this chapter.

  12. Illustration (I)

  13. Illustration (II)

  14. Estimation of Noise Parameters • Periodic noises: from Fourier spectrum • Others: try to compute the mean and variance of a subimage S (containing only constant gray levels).

  15. Restoration in the Presence of Noise Only – Spatial Filtering • Mean filters: • Arithmetic mean filters • Geometric mean filter • Harmonic mean filter: • Contraharmonic mean filter:Q: the order of the filter. Q>0 eliminates pepper noise, Q <0 eliminates salt noise.

  16. Illustration (I)

  17. Illustration (II)

  18. Illustration (III)

  19. Order-Statistics Filters • Median filters • Max and min filters • Midpoint filter: • Alpha-trimmed mean filter: delete the d/2 lowest and d/2 highest gray-level values of g(s,t) in the neighborhood of Sxy , the average

  20. Illustration (I)

  21. Illustration (II)

  22. Illustration (III)

  23. Adaptive Filters • Filter’s behavior changes based on statistical characteristics of the image inside the filter region defined by the mxn window. • Adaptive, local noise reduction filter • Adaptive median filter

  24. Adaptive, local noise reduction filter • (a) g(x,y): the value of the noisy image at (x,y) • (b) The variance of the noise • (c) The mean of the pixels in Sxy • (d) Local variance of the pixels in Sxy • If (b) is zero, return g(x,y) • If (d) is high relative to (b), the filter should return a value close to g(x,y) • If the two variances are equal, return the arithmetic mean of the pixels in Sxy

  25. Illustration

  26. Adaptive Median Filter • Notation: • zmin: minimum gray level value in Sxy • zmax: maximum gray level value in Sxy • zmed: median of gray levels in Sxy • zxy: gray level value at (x,y) • Smax: maximum allowed size of Sxy • Level A: A1= zmed– zmin, A2= zmed– zmaxif A1> 0 and A2 <0, go to level BElse increase the window sizeIf window size <= Smax repeat level Aelse output zxy • Level B: B1= zxy– zmin, B2= zxy– zmaxif B1> 0 and B2 <0, output zxyElse output zmed

  27. Illustration

  28. Periodic Noise Reduction • By Fourier domain filtering: • Bandreject filters • Bandpass filters • Notch filters

  29. Illustration

  30. Ideal Notch Reject Filter • Ideal notch reject filter: where

  31. Butterworth Notch Reject Filter

  32. Gaussian Notch Reject Filter

  33. Notch Filters

  34. Linear, Position-Invariant Degradations • Estimating the degradation function • By image observation • By experimentation • By modeling

  35. Estimation by Image Observation • In the strong signal area, using sample gray levels of the object and background to construct an unblurred image • Then, • Use Hs(u,v) to estimate H(u,v)

  36. Estimation by Experimentation • Simulate an impulse by a (very) bright dot of light, the response G(u,v) is related to H(u,v) by:

  37. Figure 5.24

  38. Estimation by Modeling • Modeling atmospheric turbulence

  39. Atmospheric Turbulence

  40. Estimation by Modeling (cont’d) • Modeling effect of planar motion x0(t),y0(t): • If T is the duration of the exposure, then • It can be shown that:

  41. Motion Blur • If x0(t)=at/T and y0(t)=0, then

  42. Motion Blur Example

  43. Deconvolution • Inverse filtering • Minimum mean square error (Wiener) filtering • Constrained least squares filtering • Geometric mean filter • http://vision.cs.nccu.edu.tw/publications/CVPRIP2003_A.pdf

  44. Results (Inverse Filter)

  45. Results (Inverse and Wiener)

  46. Results (Motion Blurs)

  47. Results (Constrained LS Filter)

  48. Recent Developments • Blind deconvolution • http://cs.unc.edu/~lazebnik/research/fall08/lec05_deblurring.pdf

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