Digital Image Processing & Pattern Analysis: Intensity Transformations

1. Digital Image Processing&Pattern Analysis(CSCE 563)Intensity Transformations Prof. Amr Goneid Department of Computer Science & Engineering The American University in Cairo

2. Intensity Transformations • Pixel Intensity Processing • Histogram Processing • Combining Images Prof. Amr Goneid, AUC

3. Pixel Intensity Processing • Point Processing • Gray Scale to Binary • Negative • Power Law Transformation • Contrast Stretching • Compression of Dynamic Range • Gray Level Slicing Prof. Amr Goneid, AUC

4. Point Processing • Local Processing: g(x,y) = T[f(x,y)] T = Operator on Locality (neighborhood) of (x,y) e.g. 3 x 3 T [ ] (8-neighborhood) • Point Processing: For 1 x 1 we have Point Processing S = T[r] = T[ ] S = New Intensity, r = Old Intensity p Prof. Amr Goneid, AUC

5. Gray scale to Binary s = T(r,a) = 0 for r < a, 1 otherwise Prof. Amr Goneid, AUC

6. Negative (Image Complement) s =T(r) = 1-r The MATLAB function: g = imcomplement(f) s 1 0 1 r Prof. Amr Goneid, AUC

7. Power Law Transformation • Has the form: • The MATLAB function g = imadjust (f, [low-in high-in], [low-out high-out], gamma) Prof. Amr Goneid, AUC

8. Power Law Transformation • Example: • g = imadjust (f, [0 1], [1 0]); equivalent to: g = imcomplement (f); • g = imadjust (f, [0.3 0.7], [0.2 1.0]); Prof. Amr Goneid, AUC

9. Contrast Stretching Low contrast images do not make full use of the dynamic range of grey levels. In this case the intensities are confined to a narrow range rmin < r < rmax. We can use a point transformation s = T(r) to stretch the range to smin < s < smax Prof. Amr Goneid, AUC

10. Contrast Stretching r = old intensity, s = new intensity Slope  =  s/ r > 1 produces contrast stretching.  = [(smax– smin)/(rmax– rmin)] Hence, s =T(r) =  (r – rmin) + smin s  s  r r Prof. Amr Goneid, AUC

11. Contrast Stretching Example rmin = 0.0 , rmax = 0.3 smin = 0.0 , smax = 1.0 Prof. Amr Goneid, AUC

12. Contrast Stretching Function • The function compresses the input • levels lower than m into a narrow • range of dark levels in the output image. • Similarly, it compresses intensities • above m into a narrow band of light • levels in the output. • The result is an image of higher contrast. Prof. Amr Goneid, AUC

13. Contrast Stretching Function Summer 2009 The lena image on the right has been obtained by: load lena; m = 0.5; E = 0.5; g = 1./(1+(m./(double(f)+eps)).^E); gs = im2uint8(mat2gray(g)); imshow(gs); Prof. Amr Goneid, AUC Prof. Amr Goneid, AUC 13

14. Contrast Stretching Function Summer 2009 In the limiting case it produces a binary image. Prof. Amr Goneid, AUC Prof. Amr Goneid, AUC 14

15. Compression of Dynamic Range Map r = {0,D} to s = {0,L-1} where D >> L-1 s = T(r) = c log(1 + r ) c = (L-1)/log(1+D) e.g. compression from 256 gray levels to 16 gray levels. Summer 2009 Prof. Amr Goneid, AUC Prof. Amr Goneid, AUC 15

16. Gray Level Slicing s Examples: • S = T(r) = {1 for A<r<B ; c otherwise} • S = T(r) = {1 for r>A; r otherwise} 1 c 0 A B 1 r s 1 A 0 1 r Prof. Amr Goneid, AUC

17. Gray Level Slicing Example S = T(r) = {1 for r>0.6; r otherwise} Prof. Amr Goneid, AUC

18. Histogram Processing Abad contrast image (F) can be enhanced to an image G by transforming every pixel intensity fij to another intensity gij using a method called Histogram Equalization Prof. Amr Goneid, AUC

19. Histogram Processing • PDF and CDF • Histogram Equalization (Theory) • Histogram Equalization (Practice) • Histogram Specification Prof. Amr Goneid, AUC

20. Probability Density Function (PDF) • Suppose the gray levels r are normalized such that 0 < r< 1.0 • pr (r) dr = probability of intensity taking a value between r and r+dr. • A plot of pr (r) against riscalled the PDF of the image. • The PDF is normalized: Prof. Amr Goneid, AUC

21. Cumulative Distribution Function (CDF) • The CDF = the probability of having values less than r • For a uniform PDF, ps (s) = 1 and CDF = s (i.e. CDF is Linear) Uniform cdf Prof. Amr Goneid, AUC

22. Histogram Equalization (Theory) Mapping r into s such that ps (s) is uniform P(s) P(r) Too Bright Too Dark Bad Contrast Uniform High Contrast r s Prof. Amr Goneid, AUC

23. How? • Gray level r {0,1} transformed to s = T(r) {0,1} • T(r) is single valued and monotonically increasing • To find T(r), force the two CDF’s to be the same: CDF(r) = CDF(s) = s • Then solve to find s in terms of r Prof. Amr Goneid, AUC

24. Example 2 0 1 Prof. Amr Goneid, AUC

25. Will s have a uniform distribution? Prof. Amr Goneid, AUC

26. Histogram Equalization (Practice) • In practice, gray levels are discrete (k = 0 .. L-1) • A histogram is an array whose element nk = no. of pixels with gray level k (k = 0 is black, k = L-1 is white). • A histogram is a plot of nk vs rke.g. for L = 256 • In MATLAB n =imhist(I,256) Prof. Amr Goneid, AUC

27. Histogram Equalization (Practice) • Algorithm to build histogram for an image with N rows and M columns (Ntot = NM pixels): Prof. Amr Goneid, AUC

28. Histogram EqualizationAlgorithm (1) The histogram is used in the following algorithm: Prof. Amr Goneid, AUC

29. Histogram EqualizationAlgorithm (2) A better algorithm first builds a Cumulative array: Prof. Amr Goneid, AUC

30. Histogram EqualizationSummary of Total Complexity Algorithm (1): Algorithm (2): Prof. Amr Goneid, AUC

31. Histogram EqualizationNumerical Example L = 256 gray levels Algorithm (1): Algorithm (2): Prof. Amr Goneid, AUC

32. Histogram Equalization (Practice) If N = total no. of pixels = numel (I), then the PDF of input image is Pr(rk) = n(k)/N Summer 2009 Prof. Amr Goneid, AUC Prof. Amr Goneid, AUC 32

33. Practical ExamplesFrom:http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip.html Original Contrast Stretched Histogram Equalized Prof. Amr Goneid, AUC

34. Negative Ramp Prof. Amr Goneid, AUC

35. Liver Tissue Image Prof. Amr Goneid, AUC

36. Girl Image Summer 2009 Prof. Amr Goneid, AUC Prof. Amr Goneid, AUC 36

37. Histogram Specification • How to Transform an image (A) with Pr(r) to an image (B) with specified Pz(z). • For the input image (A), the uniform variable s = T(r) = CDF(r) • For the output image (B), s = H(z) = CDF(z) • Hence z = H-1(s) = H-1(T(r)) • In MATLAB: g = histeq (f , HistB) Prof. Amr Goneid, AUC

38. Histogram Specification (Example) Image A has Pr(r) = 2(1-r) (-ve ramp over r {0,1}) Image Bhas Pz(z) = 2z (+ve ramp over z {0,1}) T(r) = CDF(r)= 1 – (1-r)2 H(z) = CDF(z) = z2 H-1(s) = sqrt(s) z = H-1(T(r)) = sqrt(T(r)) = sqrt {1 – (1-r)2} Summer 2009 Prof. Amr Goneid, AUC Prof. Amr Goneid, AUC 38

39. A Practical Example(-)ve To (+)ve Ramp Prof. Amr Goneid, AUC

40. Combining Images • Arithmetic Operations • Logical Operations Prof. Amr Goneid, AUC

41. Combining Images • By Arithmetic Operations: e.g. Addition & Subtraction: C = min (A + B , 1); C = max (A – B , 0); • By Logical operations(Binary Images): C = A & B Logical AND C = A | B Logical OR C = A xor B Logical Exclusive OR C = ~ A Logical NOT Prof. Amr Goneid, AUC

42. Example: Arithmetic • I=Image; RN = Random Noise; B=boundary I + RN + B RN - I Prof. Amr Goneid, AUC

43. Example: Logical Mask AND Image1 = Image2 Prof. Amr Goneid, AUC

44. Other ExamplesFrom: http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip.html Image (A) Image (B) ~ B A & (~ B) A & B A xor B A | B Prof. Amr Goneid, AUC