Chapter 16 Image Restoration

2. Chapter 16 Image Restoration • 16.1 Introduction • Image restoration means the removal of or reduction of degradations that were incurred while the digital image was obtained. • The degradations include the blurring that can be introduced by optical systems, image motion, and noise et.al. • Lenses, film, digitizer etc. may cause image degradations.

3. Chapter 16 Image Restoration • 16.1.1 Approaches and models • Estimation. When little is known about the image, one can model the source of degradations • Detection. When a great deal of prior knowledge of the image is available, a mathematical model of the original image may be developed. • Continuous and discrete; spatial and frequency domains.

4. 16.2 Classical restoration filters • Continuous image restoration model • Image degradation can be modeled as a linear system with impulse response

6. In the following, we first assume that H is the identity operator, and we deal only with degradation due to noise.

12. PDF of noises

32. Periodic Noise reduction by frequency domain filtering

34. When convolution is taken into consideration in image degradation,if the degradation is model as a linear position invariant system and assume noise is zero, the the restoration process is merely deconvolution (inverse filtering).

35. EXAMPLE OF CONVOLUTION = *  3% Poisson noise 10% Poisson noise

36. DECONVOLUTION PROBLEM Deconvolution problem: given D, P and N, find I(i.e. compensate for noise and the PSF of the imaging system) Blind deconvolution: P is also unknown. Equation (*) is mathematically ill-posed, i.e. its solution may not exist, may not be unique and may be unstable with respect to small perturbations of "input" data D, P and N. This is easy to see in the Fourier representation of eq.(*) 1) Non-existence: 2) Non-uniqueness: 3) Instability:

37. A SOLUTION OF THE DECONVOLUTION PROBLEM Convolution: Deconvolution: We assume that (otherwise there is a genuine loss of information and the problem cannot be solved). Then eq.(!) provides a nice solution at least in the noise-free case (as in reality the noise cannot be subtracted exactly). (*)-1 =

38. EFFECT OF NOISE In the presence of noise, the ill-posedness of deconvolution leads to artefacts in deconvolved images: The problem can be alleviated with the help of regularization without regularization = (*)-1 3% noise in the experimental data with regularization

39. EFFECT OF NOISE. II In the presence of stronger noise, regularization may not be able to deliver satisfactory results, as the loss of high frequency information becomes very significant. Pre-filtering (denoising) before deconvolution can potentially be of much assistance. without regularization = (*)-1 10% noise in the experimental data with regularization

40. 16.2 Classical restoration filters • 16.2.1 Deconvolution • The transfer function of a deconvolution filter usually increases with frequencies. It may take on extremely large magnitude at higher frequencies. • Nathan’s approach. Limit the deconvolution transfer function to some maximum value.

41. Deconvolution • Deconvolution is a standard technique for image restoration. Theoretical response Actual response Reverse response Response after calibration

42. 16.2 Classical Restoration filters • 16.2.2 Wiener deconvolution • One-dimensional( See Chapter 11 )

43. 16.2 Classical Restoration filters • For uncorrelated signal and noise, And thus

44. Wiener deconvolution • The transfer function of the 2-D Wiener deconvolution filter

45. Limitation of Wiener deconvolution filter • The MSE optimal criteria of Wiener deconvolution filter is not the best one for observation of a restored image by human eyes. • The classical Wiener deconvolution can not deal with spatially variable point spread functions. • The classical Wiener deconvolution cannot be used to nonstationary signal and noise.

46. 16.2.3 Power Spectrum Equalization • The power spectrum equalization (PSE) filter that restores the power spectrum of the degraded image to its original amplitude is • The PSE filter is phaseless, applicable for phaseless blurring of images.

47. Power spectrum equalization • Both PSE filter and Wiener deconvolution filter reduce to straight deconvolution in the absence of noise • Both cut off completely in the absence of signal. • The PSE filter does not cut off at zeros in the blurring transfer function. • The PSE filter may be preferable to Wiener deconvolution filter in some cases.

48. Geometric mean filters • Transfer function is given by • If , the filter reduces to a deconvolution filter. • If , it reduces to the PSE filter. • If , the filter is a geometric mean between ordinary deconvolution and Wiener deconvolution • If , the filter becomes a parametric Wiener filter.

49. 16.2.4 Geometric mean filters • Under conditions of slight blurring and moderate noise, straight decovolution is least desirable, and Wiener deconvolution produces lowpass filtering more severe than the human eye desires. • The parametric Wiener filter and the geometric filter with less than unity produce more pleasing results

50. Examples of Wiener deconvolution