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Computer Vision - Restoration

Computer Vision - Restoration. Hanyang University Jong-Il Park. Restoration vs. Enhancement. To improve an image in some predefined sense. Restoration. Enhancement. Objective process A priori knowledge on degradation model Modeling the degradation and applying the inverse process

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Computer Vision - Restoration

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  1. Computer Vision -Restoration Hanyang University Jong-Il Park

  2. Restoration vs. Enhancement To improve an image in some predefined sense Restoration Enhancement Objective process A priori knowledge on degradation model Modeling the degradation and applying the inverse process to recover the original Subjective process

  3. Restoration process

  4. Noise models Assume noise is independent of spatial coordinates and it is uncorrelated w.r.t. the image. • Gaussian: electronic circuit noise, • sensor noise • Rayleigh: range images • Exponential and gamma: • laser images • impulse(salt-and-pepper): • faulty switching

  5. Eg. Sample noisy images

  6. Eg. Sample noisy images(cont.)

  7. Periodic noise • Spatially dependent noise • Periodic noise can be reduced significantly via frequency domain filtering

  8. Estimation of noise parameters • PDF from small patches

  9. When the only degradation is noise • Periodic noise  subtraction gives a good result • Random noise  mean filter, order-statistics filter,…

  10. Mean filters • Arithmetic mean filters • For Gaussian or uniform noise • Geometric mean filters • For Gaussian or uniform noise • Harmonic mean filters • Work well for salt noise but fail for pepper noise • Contraharmonic mean filters • Suited for impulse noise but require identification(salt or pepper)

  11. Arithmetic & Geometric mean filter

  12. Contraharmonic filters • Q<0 : eliminates salt noise Q=-1 harmonic mean filter • Q=0 : arithmetic mean filter • Q>0: eliminates pepper noise

  13. Eg. Contraharmonic filters

  14. Wrong sign in contraharmonic filters Disaster!

  15. Order-Statistics filters • Median filter • Max filter • Min filter • Midpoint filter • Alpha-trimmed mean filter

  16. Median filters 3x3 median 3x3 median 3x3 median blurred

  17. Max and Min filter • Max filter • Min filter Removes salt noise Removes light pixels Makes dark objects larger Removes pepper noise Removes dark pixels

  18. Eg. Comparison (a) Additive uniform noise (b) (a)+additive S&P 5x5 arithmetic mean 5x5 geometric mean 5x5 median 5x5 alpha-trimmed Mean(d=5)

  19. Adaptive filters • Behavior changes locally based on statistical characteristics of local support • Simple adaptive filter based on mean and variance • If global_var is zero, then f(x,y)=g(x,y) • If local_var>global_var, then f(x,y)=g(x,y) (high local var  edge  should be preserved) • If local_var==global_var, then arithmetic mean filtering

  20. Eg. Adaptive filter

  21. Adaptive median filter • Cope with impulse noise with large probability • Preserve detail while smoothing non-impulse noise Algorithm Level A: A1=zmed-zmin A2=zmed-zmax If A1>0 AND A2<0, go to level B Else increase the window size If window size<=Smax repeat level A Else output zxy Level B: B1=zxy-zmin B2=zxy-zmax If B1>0 AND B2<0, output zxy Else output zmed

  22. Eg. Adaptive median filter median adaptive median

  23. Periodic noise reduction • By frequency domain filtering • Band reject filter

  24. Eg. Periodic noise reduction

  25. Noise extraction • By bandpass filter Help understanding noise pattern

  26. Notch filters

  27. Eg. Notch filtering • Removing sensor scan-line patterns

  28. Optimum notch filtering • First isolating the principal contributions of the interference pattern • Then subtracting weighted portion of the pattern from the corrupted image

  29. Eg. Periodic interference(1/3) • Noisy image

  30. Eg. Periodic interference(2/3) • Extraction of noise interference pattern

  31. Eg. Periodic interference(3/3) • Restored image by subtracting weighted portion of periodic interference (Refer to the derivation of weights in pp.250-252)

  32. Linear, Position-Invariant Degradation

  33. Degradation knowledge • Degradation knowledge about 1. A priori (known) 2. A posteriori (unknown)  • blind restoration • or blind deconvolution Restoration: determine the original image , given the observed image and knowledge about the degradation (H).

  34. Fundamental issue • Restoration problem  restoration is to find , such that but, 1. does not exist: singular 2. may exist, but not be unique: ill-conditioned 3. may exist and unique, but there exists , which can be made arbitrarily small, such that which is not negligible  Image restoration is ill-conditioned at best and singular at worst

  35. Estimation of degradation function • Approaches • Observation • Experimentation • Mathematical modeling

  36. Estimation by observation • Looking at a small section of the image containing simple structures and then obtaining degradation function Observed sub-image: Estimate of original image:

  37. Estimation by experimentation • Possible only if equipment similar to the equipment used to acquire the degraded images is available • Eg. Use an impulse

  38. Estimation by modeling • Based on either physical characteristics or basic principles Eg.1. Physical characteristics: atmospheric turbulence Eg.2. Math derivation: motion blur • Starting from • After some manipulation(p.259) • Setting the motion model, we obtain the degradation func.

  39. Eg.1. Physical model • Atmospheric turbulence

  40. Eg.2. Math modeling • Motion blur

  41. Restoration methods • Inverse filtering • Wiener filtering • Constrained least square filtering • Geometric mean filtering • Etc..

  42. Inverse filtering • Poor performance! • Very sensitive to noise Noise amplification when H(u,v) is small

  43. Eg. Inverse filtering

  44. Minimum mean-square error filter • Necessary to handle noise explicitly • Statistical characteristics of noise should be incorporated into the restoration process • MMSE filter • To find an estimate of the uncorrupted image such that the mean square error between them is minimized: • Assume: • the noise and the image are uncorrelated • The one or the other has zero mean • The gray levels in the estimate are a linear function of the levels in the degraded image • Derivation: Homework

  45. MMSE filter (cont.) • Frequency domain expression: • Approximation of the Wiener filter Wiener filter PS of noise PS of image f

  46. Eg. Wiener filtering • Using the approximation • K is chosen interactively

  47. Eg. Restoration by Wiener filtermotion blur Severe noise Moderate noise Negligible noise

  48. Constrained Least Square Filtering • Difficulty in Wiener filter • The power spectra of the undegraded image and noise must be known • Minimization in a statistical sense • The constrained LS filtering • requires knowledge of • Mean of the noise • Variance of the noise • Optimal result for each image

  49. Vector-matrix form of convolution • g: MN-vector (lexicographical order of an image) • f: MN-vector • H: MNxMN matrix

  50. Formulation: Constrained LS filter To find the minimum of a criterion function C defined as subject to the constraint where is the Euclidean vector norm

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