Computer Vision

1. Lecture 10 Roger S. Gaborski Computer Vision Roger S. Gaborski

2. Algorithm Development: Edges • We would like to develop algorithms that detect important edges in images • The ‘quality’ of edges will depend on the image characteristics • Noise can result in false edges Roger S. Gaborski

3. Profiles of an Ideal Edge Roger S. Gaborski

4. Profile of Edge - Analysis • Same gray level input range [0,1] • Edge is located at ‘peak’ of first derivative • Cannot choose only one threshold value because range of first derivative depends on slope of gray level values • The edge is located where the second derivative goes through zero-independent of slope of gray level signal Roger S. Gaborski

5. Edge Detection is Not Simple Roger S. Gaborski

6. The derivative operation can be used to detect edgesHow can the derivative operator be approximated in a digital image?? Roger S. Gaborski

7. MATLAB Examples • Approximation to derivative – difference of adjacent data values • data = [ .10 .14 .13 .12 .11 .10 .11 .34 .33 .32 .35 .10 .11 .12 .12 .12]; • diff operator: second- first data value (.14-.10) • diff(data) = 0.04 -0.01 -0.01 -0.01 -0.01 0.01 0.23 -0.01 -0.01 0.03 -0.25 0.01 0.01 0 0 diff operator is approximation to first derivative Roger S. Gaborski

8. Estimate of Second Derivative • diff(diff( data)) • -0.05 0 0.0 0 0.02 0.22 -0.24 0 0.04 -0.28 0.26 0 -0.0100 0 Roger S. Gaborski

9. Simulated Data data diff(data) diff(diff(data)) Roger S. Gaborski

10. How would you implement in a filter? • diff: second value – first value [ -1 +1 ] (could also use [+1 -1]) • Could also represent as: [-1 0 +1] • Use in correlation filter – same results Roger S. Gaborski

11. Actual Image Data Roger S. Gaborski

12. Row 80 Roger S. Gaborski

13. Roger S. Gaborski

14. diff Operator Roger S. Gaborski

15. Smooth Image First Roger S. Gaborski

16. Roger S. Gaborski

17. Gradient of a 2D Function Gradient is defined by a vector (see pg 366): Δ f = gx = f / x gy f / y The magnitude of the vector: mag( f) =[ gx2 + gy2 ]1/2 Δ Roger S. Gaborski

18. Direction of Gradient • The gradient points in the direction of maximum change of intensity α(x,y) = tan-1 (gy / gx) Roger S. Gaborski

19. Partial Derivatives in Vertical and Horizontal Directions • Vertical (y direction) • Horizontal (x direction) Roger S. Gaborski

20. Partial Derivatives in Vertical and Horizontal Directions 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 .* = 6 VERTICAL GRADIENT Roger S. Gaborski

21. First Derivative Approximation y+1 y y-1 eq1 eq2 x-1 x x+1 Roger S. Gaborski

22. First Derivative Approximation • First Derivative in x direction: fx (x,y) = f(x,y) – f(x-1,y) Eq 1 = f(x+1,y) – f(x,y) Eq 2 • First Derivative in y direction: fy(x,y) = f(x,y) – f(x, y-1) Eq3 = f(x, y+1) - f(x,y) Eq4 Roger S. Gaborski

23. First Derivative Approximation y+1 y y-1 eq1 eq2 x-1 x x+1 Roger S. Gaborski

24. Edge Detection • An Edge is defined as a discontinuity within a local set of pixels. x-1 x x+1 y+1 y y-1 Roger S. Gaborski

25. fy = (28-15)/2+(27-17)/2+(26-16)/2= fy = -6.5- 5.0 - 5.0 = 16.5 fx = (26-28)/2+(31-30)/2+(16-15)/2= fx = -1 + .5 + .5 = 0 Divide by 2 because 2 pixels apart. • = tan-1 (fy/fx) = tan-1 (16.5/0) = 90 degrees mag(f ) = [ fy2 + fx2 ]1/2 = 16.5 x-1 x x+1 y+1 y y-1 Roger S. Gaborski Y –direction X -direction

26. fy = (38-12)/2+(66-15)/2+(65-42)/2= fy = 13.00 + 25.5+ 11.5 = 50 fx = (65-38)/2+(64-14)/2+(42-12)/2= fx = 13.5 0+ 25.00 + 15.00 = 53.5 Divide by 2 because 2 pixels apart. Ignore dividing by two simply introduces a scaling error, but results in faster calculation • = tan-1 (fy/fx) = tan-1 (50/53.5)= 43 degrees mag(f ) = [ fy2 + fx2 ]1/2 = 73  Roger S. Gaborski Y –direction X -direction

27. Second Derivative Approximation • Discrete version of 2nd Partial Derivation of f(x,y) in x direction is found by taking the difference of Eq1 and Eq2: • 2 f(x,y) /  x2 = Eq1-Eq2 = 2f(x,y)-f(x-1,y)-f(x+1,y) In y direction: • 2 f(x,y) /  y2 = Eq3-Eq4 = 2f(x,y)-f(x,y-1)-f(x,y+1) Roger S. Gaborski

28. Derivative Approximation for Laplacian y+1 y y-1 x-1 x x+1 Roger S. Gaborski

29. Laplacian • Independent of edge orientation • Combine 2 f(x,y)/ x2 and 2 f(x,y)/ y2 =4 f(x,y) - f(x-1,y) – f(x+1,y) – f(x,y-1) – f(x,y+1) Roger S. Gaborski

30. EXAMPLE: Using Sobel Edge Filter Roger S. Gaborski

31. Simple First Derivative Approximation Difference x = Smoothing Roger S. Gaborski

32. Rotate Filter Sensitive to edges at different orientations Roger S. Gaborski

33. Sobel Filter • Consider unequal weights for smoothing operation = Sy x = Roger S. Gaborski

34. Sobel Filter • Consider unequal weights for smoothing operation = Sx x = Roger S. Gaborski

35. Sobel Edge Filter >> fh=fspecial('sobel') fh = 1 2 1 0 0 0 -1 -2 -1 >> Ifh=imfilter(Ig,fh); >> fv=f' fv = 1 0 -1 2 0 -2 1 0 -1 >> Ifv=imfilter(Ig,fv); Ifh is an image containing the Horizontal edges Ifv is an image containing the Vertical edges Can we use the command: figure, imshow(Ifh) to display the horizontal edges?? Roger S. Gaborski

36. >> max(Ifh(:)) ans = 2.8849 >> min(Ifh(:)) ans = -2.9685 Roger S. Gaborski

37. Horizontal Edges Vertical Edges Roger S. Gaborski

38. Sum of Horizontal and Vertical Edges Roger S. Gaborski

39. Absolute Value of Sum of Horizontal and Vertical Edges Roger S. Gaborski

40. Threshold of Absolute Value of Sum of Horizontal and Vertical Edges T=1 T=.5 Roger S. Gaborski

41. Threshold of Absolute Value of Sum of Horizontal and Vertical Edges T=.25 T=.1 Roger S. Gaborski

42. build1.jpg Roger S. Gaborski

43. Edge image Roger S. Gaborski

44. ~ Operator Roger S. Gaborski

45. Box Image Roger S. Gaborski

46. Note missing Horizontal Edges Roger S. Gaborski

47. What are Edgels Edgels are defined as a rapid change in the image function over a small area Roger S. Gaborski

48. Edgels • Edgels have three properties: • Direction • Magnitude • Location • Edgels are not complete edges • They provide evidence for higher level structures, such as, boundary lines, contours Roger S. Gaborski