Image Enhancement

1. Image Enhancement 

2. Image enhancement 1. simplifying interpretation 2. more pleasing outlook 3. normalization for further processing 

3. Three types of image enhancement 1. Noise suppression 2. Image de-blurring 3. Contrast enhancement Original Image Noise Blur Bad Contrast

4. Overview • Preliminaries • Reminders from previous lecture • Fourier power spectra of images • Noise suppression • Convolutional (Linear) filters • Non-linear filters • Image de-blurring • Unsharp masking • Inverse Filtering • Wiener Filters • Contrast enhancement • Histogram Equalization

5. Overview • Preliminaries • Reminders from previous lecture • Fourier power spectra of images • Noise suppression • Convolutional (Linear) filters • Non-linear filters • Image de-blurring • Unsharp masking • Inverse Filtering • Wiener Filters • Contrast enhancement • Histogram Equalization

6. Reminders from previous lecture: Fourier Transform Linear decomposition of functions in the new basis Scaling factor for basis function (u,v)  The Fourier transform Reconstruction of the original function in the spatial domain: weighted sum of the basis functions  The inverse Fourier transform

7. v v f(x,y) u u phase F(u,v) |F(u,v)|

8. Reminders from previous lecture: Convolution Theorem Space convolution = frequency multiplication Space multiplication = frequency convolution

9. modulation transfer function (MTF) Reminders from previous lecture: Modulation Transfer Function For any Linear Shift Invariant Operator For any Convolutional Operator

10. Fourier power spectra of images ɸii = |I(u,v)|2 i(x,y) Amount of signal at each frequency pair Images are mostly composed of homogeneous areas Most nearby object pixels have similar intensity Most of the signal lies in low frequencies! High frequency contains the edge information!

11. Fourier power spectra of noise ɸnn = |N(u,v)|2 n(x,y) -Pure noise has a uniform power spectra -Similar components in high and low frequencies.

12. Fourier power spectra of noisy image ɸff = |F(u,v)|2 f(x,y) Power spectra is a combination of image and noise

13. Signal to Noise Ratio Low SNR Low SNR High SNR Low SNR Low SNR ɸii(u,v) / ɸnn(u,v)

14. Only retaining the low frequencies Low signal/noise ratio at high frequencies  eliminate these Smoother image but we lost details! 

15. High frequencies contain noisebut also Edges! We cannot simply discard the higher frequencies They are also introduces by edges ; example : 

16. Overview • Preliminaries • Reminders from previous course • Fourier power spectra of images • Noise suppression • Convolutional (Linear) filters • Non-linear filters • Image de-blurring • Unsharp masking • Inverse Filtering • Wiener Filters • Contrast enhancement • Histogram Equalization

17. Noise Suppression NoisyObservation Original Image

18. Noise suppression • specific methods for specific types of noise • we only consider 2 general options : • 1. Convolutional linear filters low-pass convolution filters • 2. Non-linear filters edge-preserving filters • a. median • b. anisotropic diffusion 

19. Low-pass filters: principle Goal: remove low-signal/noise part of the spectrum Approach 1: Multiply the Fourier domain by a mask Such spectrum filters yield “rippling” due to ripples of the spatial filter and convolution 

20. Illustration of rippling 

21. Approach 2: Low-pass convolution filters generate low-pass filters that do not cause rippling Idea: Model convolutional filters in the spatial domain to approximate low-pass filtering in thefrequency domain Frequencymask Convolutional filter 

22. Averaging One of the most straight forwardconvolution filters: averaging filters 1/25 1/9 Separable: 1/9 1/3 1/3 * = 

23. Example for box averaging Noise is gone. Result is blurred! 

24. 3 x 3 (separable!) (1+2cos(2u)) (1+2cos(2v)) 5 x 5 (separable) (1+2cos(2u)+2cos(4 u))(1+2cos(2v)+2cos(4 v)) MTFs for averaging not even low-pass! 

25. So far • Masking frequency domain with window type low-pass filter yields sinc-type of spatial filter and ripples -> disturbing effect • box filters are not exactly low-pass, ripples in the frequency domain at higher freq. no ripples in either domain required! 

26. only odd filters : (1,2,1), (1,4,6,4,1) 2D : MTF : (2+2cos(2 u))(2+2cos(2 v)) Solution: Binomial filters iterative convolutions of (1,1) Also separable 

27. Result of binomial filter 

28. f : Limit of iterative binomial filtering Gaussian

29. Gaussian smoothing Gaussian is limit case of binomial filters noise gone, no ripples, but still blurred… Actually linear filters cannot solve this problem 

30. Some implementation issues separable filters can be implemented efficiently large filters through multiplication in the frequency domain integer mask coefficients increase efficiency powers of 2 can be generated using shift operations 

31. Noise suppression • specific methods for specific types of noise • we only consider 2 general options : • 1. Convolutional linear filters low-pass convolution filters • 2. Non-linear filters edge-preserving filters fighting blurring! • a. median • b. anisotropic diffusion 

32. Median filters : principle non-linear filter method : • 1. rank-order neighbourhood intensities • 2. take middle value no new grey levels emerge... 

33. Median filters : odd-man-out advantage of this type of filter is its “odd-man-out” effect e.g. 1,1,1,7,1,1,1,1  ?,1,1,1.1,1,1,? 

34. Median filters : example filters have width 5 : 

35. Median filters : analysis median completely discards the spike, linear filter always responds to all aspects median filter preserves discontinuities, linear filter produces rounding-off effects DON’T become all too optimistic 

36. Median filter : results 3 x 3 median filter : sharpens edges, destroys edge cusps and protrusions 

37. Median filters : results Comparison with Gaussian : e.g. upper lip smoother, eye better preserved 

38. Example of median 10 times 3 X 3 median patchy effect important details lost (e.g. ear-ring) 

39. Anisotropic diffusion : principle non-linear filter method : • 1. Gaussian smoothing across homogeneous intensity areas • 2. No smoothing across edges 

40. The Gaussian filter revisited The diffusion equation Initial/Boundary conditions If in1D: Solution is a convolution!

41. Diffusion as Gaussian lowpass filter Gaussian filter with time dependent standard deviation: Nonlinear version can change the width of the filter locally Specifically dependening on the edge information through gradients

42. Selection of diffusion coefficient or 𝛋 controls the contrast to be preserved by smooting actually edge sharpening happens

43. Dependence on contrast c

44. with Take with and yields with Let’s see what really happens in 1D

45. Anisotropic diffusion result : diffusion with gradient dependent sign : Anisotropic diffusion

46. Anisotropic diffusion: Numerical solutions When c is not a constant solution is found through solving the equation Partial differential equation Numerical solutions through discretizing the differential operators and integrating Finite differences in space and integration in time

47. 1 1 -2 1 -2 1 + = 1 = 1 -4 1 1 = 1 -1 1 -1 + -1 1 -1 1 Finite difference approximationof the divergence operator

48. 1 = 1 -1 -1 + -1 1 -1 1 Divergence in the presence of c Coefficients depend on derivatives of c

49. Anisotropic diffusion: Output End state is homogeneous

50. Restraining the diffusion adding restraining force :