Edge Preserving Smoothing Techniques

1. Edge Preserving Smoothing Techniques Prof. PhD. Vasile Gui Polytechnic University of Timisoara

2. Content • Introduction • Brief review of linear operators • Linear image smoothing techniques • Nonlinear image smoothing techniques

3. Introduction Why do we need image smoothing? What is “image” and what is “noise”? • Frequency spectrum • Statistical properties

4. Brief Review of Linear Operators[Pratt 1991] • Generalized 2D linear operator • Separable linear operator: • Space invariant operator: Convolution sum

5. h22 h20 h21 L-1 h12 h11 h10 2 L-1 h02 h01 h00 h22 h20 h21 H 2 h12 h11 h10 h02 h01 h00 L-1 h22 G h20 h21 2 L-1 h12 h11 h10 2 h02 h01 h00 N-L+1 F N N+L-1 h22 h20 h21 h12 h11 h10 h22 h20 h21 h02 h12 h01 h00 h11 h10 h02 h01 h00 Brief Review of Linear Operators • Geometrical interpretation of 2D convolution

6. Brief Review of Linear Operators • Circular convolution theorem • Periodicity of f,h and g with N are assumed

7. Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

8. m,n  + L pixels m,n+1 Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator • Separable • Can be computed recursively, resulting in roughly 4 operations per pixel

9. Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator Optimality properties • Signal and additive white noise • Noise variance is reduced N times

10. Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator • Unknown constant signal plus noise • Minimize MSE of the estimation g:

11. Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator • i.i.d. Gaussian signal with unknown mean. • Given the observed samples, maximize • Optimal solution: arithmetic mean

12. Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator • Frequency response: Non-monotonically decreasing with frequency

13. Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator • An example

14. Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator • Image smoothed with 33, 55, 99 and 11 11 box filters

15. Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator Lena image filtered with 5x5 box filter Original Lena image

16. Linear Image smoothing techniquesBinomial filters [Jahne 1995] • Computes a weighted average of pixels in the window • Less blurring, less noise cleaning for the same size • The family of binomial filters can be defined recursively • The coefficients can be found from (1+x)n

17. Linear Image smoothing techniquesBinomial filters. 1D versions As size increases, the shape of the filter is closer to a Gaussian one

18. Linear Image smoothing techniquesBinomial filters. 2D versions

19. Linear Image smoothing techniquesBinomial filters. Frequency response Monotonically decreasing with frequency

20. Linear Image smoothing techniquesBinomial filters. Example Original Lena image Lena image filtered with binomial 5x5 kernel Lena image filtered with box filter 5x5

21. Step edge Result of size L box filter Size L binomial L Linear Image smoothing techniquesBinomial and box filters. Edge blurring comparison • Linear filters have to compromise smoothing with edge blurring

22. f(1) f1 f(2) f2 . . . f(m) . . . Order samples select f(N) fN median Nonlinear image smoothingThe median filter [Pratt 1991] • Block diagram N is odd

23. 3 7 8 2 3 7 4 6 7 2, 3, 3, 4, 6, 7, 7, 7, 8 6 Nonlinear image smoothingThe median filter • Numerical example

24. Nonlinear image smoothingThe median filter • Nonlinearity median{f1 + f2}  median{ f1} + median{ f2}. However: median{ c f} = c median{ f }, median{ c + f} = c + median{ f}. • The filter selects a sample from the window, does not average • Edges are better preserved than with liner filters • Best suited for “salt and pepper” noise

25. Nonlinear image smoothingThe median filter Noisy image 5x5 median filtered 5x5 box filter

26. Nonlinear image smoothingThe median filter • Optimality • Grey level plateau plus noise. Minimize sum of absolute differences: • Result: • If 51% of samples are correct and 49% outliers, the median still finds the right level!

27. Test images Results of 33 pixel median filter Nonlinear image smoothingThe median filter • Caution: points, thin lines and corners are erased by the median filter

28. Results of the 9 pixel cross shaped window median filter Nonlinear image smoothingThe median filter • Cross shaped window can correct some of the problems

29. Nonlinear image smoothingThe median filter • Implementing the median filter • Sorting needs O(N2) comparisons • Bubble sort • Quick sort • Huang algorithm (based on histogram) • VLSI median

30. x1 x(1) x2 x(2) x3 x(3) a Min(a,b) x4 x(4) b Max(a,b) x5 x(5) x6 x(6) x7 x(7) Nonlinear image smoothingThe median filter • VLSI median block diagram

31. Nonlinear image smoothingThe median filter • Color median filter • There is no natural ordering in 3D (RGB) color space • Separate filtering on R,G and B components does not guarantee that the median selects a true sample from the input window • Vector median filter, defined as the sample minimizing the sum of absolute deviations from all the samples • Computing the vector median is very time consuming, although several fast algorithms exist

32. Nonlinear image smoothingThe median filter • Example of color median filtering • 5x5 pixels window • Up: original image • Down: filtered image

33. Nonlinear image smoothingThe weighted median filter • The basic idea is to give higher weight to some samples, according to their position with respect to the center of the window • Each sample is given a weight according to its spatial position in the window. • Weights are defined by a weighting mask • Weighted samples are ordered as usually • The weighted median is the sample in the ordered array such that neither all smaller samples nor all higher samples can cumulate more than 50% of weights. • If weights are integers, they specify how many times a sample is replicated in the ordered array

34. 1 2 1 2 2 3 1 2 1 3 7 8 2 3 7 6 4 6 7 2, 2, 3, 3, 3, 3, 4, 6, 6, 7, 7, 7, 7, 7, 8 Nonlinear image smoothingThe weighted median filter • Numerical example for the weighted median filter

35. weights f(1) f1 a1 f(2) f2 a2 y . . . . . . ordering sum f(N) fN aN y = a1f(1) + a2f(2)+...+ aNf(N) Nonlinear image smoothingRank-order filters (L filters) • Block diagram

36. a1=1 . . . Min aN=1 . . . Max am=1 . . . . . . Median a1=.5 aN=.5 . . . Mid range Nonlinear image smoothingRank-order filters (L filters) • Some examples Percentile filters (rank selection): 0%, 50% 100% Particular cases of grey level morphological filters

37. Nonlinear image smoothingRank-order filters (L filters) • Optimality Minkovski distance Case r = 1: best estimator ismedian. Caser = 2, best estimator is arithmetic mean. Caser , best estimator is mid-range.

38. Nonlinear image smoothingRank-order filters (L filters) • Alpha-trimmed mean filter •  = (2Q + 1) / N, defines de degree of averaging • = 1 corresponds to arithmetic mean  = 1 / N corresponds to the median filter Properties: in between mean and median

39. Nonlinear image smoothingRank-order filters (L filters) Median of absolute differences trimmed mean • Better smoothing than the median filter and good edge preservation M2 is a robust estimator of the variance

40. Nonlinear image smoothingRank-order filters (L filters) • k nearest neighbour (kNN) median filter • Median of the k grey values nearest by rank to the central pixel • Aim: same as above

41. reg 2 reg 1 reg 4 reg 3 reg 6 reg 5 reg 8 reg 7 Nonlinear image smoothingSelected area filtering [Nagao 1980] • Kuwahara type filtering • Form regions Ri • Compute mean, mi and variance, Si for each region. • Result = mi : mi < mj for all j different from i, i,e. themean of the most homogeneous region • Matsujama & Nagao

42. Nonlinear image smoothingConditional mean • Pixels in a neighbourhood are averaged only if they differ from the central pixel by less than a given threshold: L is a spacescale parameter and th is a range scale parameter

43. Nonlinear image smoothingConditional mean • Example with L=3, th=32

44. Nonlinear image smoothingBilateral filter [Tomasi 1998] • Space and range are treated in a similar way • Space and range similarity is required for the averaged pixels • Tomasi and Manduchi [1998] introduced soft weights to penalize the space and range dissimilarity. s() and r() are space and range similarity functions (Gaussian functions of the Euclidian distance between their arguments).

45. Nonlinear image smoothingBilateral filter • The filter can be seen as weighted averaging in the joint space-range space (3D for monochromatic images and 5D – x,y,R,G,B - for colour images) • The vector components are supposed to be properly normalized (divide by variance for example) • The weights are given by:

46. Nonlinear image smoothingBilateral filter • Example of Bilateral filtering • Low contrast texture has been removed • Yet edges are well preserved

47. Nonlinear image smoothingMean shift filtering [Comaniciu 1999, 2002] • Mean shift filtering replaces each pixel’s value with the most probable local value, found by a nonparametric probability density estimation method. • The multivariate kernel density estimate obtained in the point x with the kernel K(x) and window radius r is: • For the Epanechnikov kernel, the estimated normalized density gradient is proportional to the mean shift: S is a sphere of radius r, centered on x and nx is the number of samples inside the sphere

48. Nonlinear image smoothingMean shift filtering • The mean shift procedure is a gradient ascent method to find local modes (maxima) of the probability density and is guaranteed to converge. • Step1: computation of the mean shift vector Mr(x). • Step2: translation of the window Sr(x) by Mr(x). • Iterations start for each pixel (5D point) and tipically converge in 2-3 steps.

49. Nonlinear image smoothingMean shift filtering • Example1.

50. Nonlinear image smoothingMean shift filtering Detail of a 24x40 window from the cameraman image a) Original data b) Mean shift paths for some points c) Filtered data d) Segmented data