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Edge Detection

Explore various edge detection techniques, including differentiation, Hilbert Transform, and improved algorithms. Learn about first-order and second-order derivatives, Canny edge detection, and Short Response Hilbert Transform. Discover simulation results and conclusions.

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Edge Detection

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  1. Edge Detection Speaker: Che-Ming Hu Advisor: Jian-Jiun Ding Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC

  2. Outline • Introduction • Edge detection method • Differentiation • Hibert Transform • Short Response Hilbert Transform • Improved Harri’s Algorithm • Simulation results • Conclusion • Reference

  3. Introduction Fig.1. Edges are boundaries between different textures.Edge also can be defined as discontinuities inimage intensity from one pixel to another..[4]

  4. EDGE DETECTION METHOD • Fist-Order Derivative Edge Detection • Second-Order Derivative Edge Detection • Hilbert Transform for Edge Detection • Short Response Hilbert Transform for Edge Detection • Improved Harri’s Algorithm For Corner And Egde Detections

  5. Fist-Order Derivative Edge Detection • Introduction • The Roberts operators • The Prewitt operators • The Sobel operators • First-Order of Gausssian (FDOG)

  6. Fist-Order Derivative Edge Detection Definition: the gradient vector the magnitude of this vector the direction of the gradient vector

  7. Fist-Order Derivative Edge Detection Fig.2. Orthogonal gradient generation.[2] The gradient along the line normal to the edgeslope Thespatial gradient amplitude The gradient amplitude combination

  8. The Roberts operators

  9. The Prewitt operators

  10. The Sobel operators

  11. Operators Fig.3. Impulse response arrays for 3 ×3 orthogonal differential gradient edge operators.[2]

  12. First-Order of Gausssian It is hard to find the gradient by using the equation In order to simplify the computation M=|Mx|+|My|

  13. First-order derivative edge detection Fig.4. Using 1st-order differentiation to detect (a) the sharp edges, (c) the step edges with noise, and (e) the ramp edges. (b)(d)(e) are the results of differentiation of (a)(c)(e).[3]

  14. Canny Edge Detection • Good detection • Good localization • Single response • Many edge candidate • The accurate edge

  15. Simulation The original image FDOG The Sobel operators Fig.5. Comparison with FDOG and the Sobel operators

  16. Simulation (a) (c) (a) Original image. (b) Roberts operator. (b) (d) (c) Prewitt operator. (d) Sobel operator. Fig.6. Simulation of first-order of derivative edge detection[3]

  17. Second-order derivative edge detection The Laplacian of a 2-D function f (x, y) is a second-order derivative defined as:

  18. Second-order derivative edge detection

  19. Second-order derivative edge detection The 2-D Gaussian function: The Laplacian of Gaussian (LOG):

  20. Second-order derivative edge detection Fig.7. An 11×11 mask approximation to Laplacian of Gaussian (LOG).[3]

  21. Second-order derivative edge detection • Sensitive to noise • Gaussian function • Low-pass filter

  22. Second-order derivative edge detection (a) (c) (a) Original image. (b) Laplacian mask of Fig.(a) (b) (d) (c) Laplacian mask of Fig.(b) (d) LOG mask of Fig.(b) Fig.9. Simulation of second-order of derivative edge detection[3]

  23. Second-order derivative edge detection Fig.10. Using 1st-order differentiation to detect (a) the sharp edges, (c) the step edges with noise, and (e) the ramp edges. (b)(d)(e) are the results of differentiation of (a)(c)(e).[3]

  24. Second-order derivative edge detection The Drawbacks of the Differentiation Method for Edge Detection : • Sensitivity to noise • Not good for ramp edges • Make no difference between the significant edge and the detailed edge

  25. Hilbert Transform for Edge Detection H(f)= −jsgn(f), sgn(f)= 1 ,when f>0, sgn(f)=- 1 ,when f<0, sgn(0)= 0

  26. Hilbert Transform for Edge Detection Fig.11. Using HLTs to detect (a) the sharp edges, (c) the step edges with noise, and (e) the ramp edges. (b)(d)(e) are the results of the HLTs of (a)(c)(e). [3]

  27. Hilbert Transform for Edge Detection • Slove the problem of differentiation • Edge is too thick • SRHLT

  28. Short Response Hilbert Transform for Edge Detection From We obtain:

  29. Short Response Hilbert Transform for Edge Detection we can define the short response Hilbert transform (SRHLT) as: SRHLT(depend on the value of b) Large (Differentiation) b Small (HLT)

  30. Short Response Hilbert Transform for Edge Detection

  31. Short Response Hilbert Transform for Edge Detection Fig.13. Using SRHLTs to detect the sharp edges, the step edges with noise, and the ramp edges. Here we choose b = 1, 4, 12, and 30. [3]

  32. Short Response Hilbert Transform for Edge Detection Without noise: (a) Original image (b) Results of differentiation (c) Results of the HLT (d) Results of the SRHLT, b=8

  33. Short Response Hilbert Transform for Edge Detection With noise: (a) Lena + noise, SNR = 32 (b) Results of differentiation (c) Results of the HLT (d) Results of the SRHLT, b=8

  34. Short Response Hilbert Transform for Edge Detection (c) b = 8 for Lena image (d) b = 8 for Lena + noise

  35. Short Response Hilbert Transform for Edge Detection How to choose a edge detection filter? By Canny’s Theorem: • Higher distinction • Noise immunity

  36. Short Response Hilbert Transform for Edge Detection To have both the advantage, we have to satisfy: (Constraint 1) A1 < T < A2, (Constraint 2) (Constraint 3) if |x2| > |x1|  |x0|, (Constraint 4) h(x) = h(x)

  37. Short Response Hilbert Transform for Edge Detection The other alternative ways to define SRHLT:

  38. Improved Harri’s Algorithm For Corner And Edge Detections Comparision with SRHLT: • More effective to defect corner • More effective to defect edge • Worse noise immunity

  39. Improved Harri’s Algorithm For Corner And Edge Detections Instead of we use:

  40. Improved Harri’s Algorithm For Corner And Edge Detections Basis: • Harris’ Algorithm: x2, y2, xy • Improved Agorithm:x2, y2, xy, x, y, 1 More types of edge detection

  41. Improved Harri’s Algorithm For Corner And Edge Detections (step 1) Find the orthonormal polynomial set that isorthogonal respect to a weighting function w[m, n]. (step 2) We do the inner product for L1[m, n, x, y] and k[x, y], where k = 1, 2, 3, 4, 5,6:

  42. Improved Harri’s Algorithm For Corner And Egde Detections (step 3) Then we express the variation around [m, n] by:

  43. Improved Harri’s Algorithm For Corner And Egde Detections (step 4) The principal axes are the eigen vector of

  44. Improved Harri’s Algorithm For Corner And Egde Detections (step 5) we can observe the variation along the two principal axes.

  45. Improved Harri’s Algorithm For Corner And Egde Detections Fig.17.Variations of gray levels along four principal directions for the pixel at a corner, on an edge, at a peak, and on a ridge[1]

  46. Improved Harri’s Algorithm For Corner And Egde Detections Table 1 Case table.[1]

  47. Improved Harri’s Algorithm For Corner And Egde Detections (step 6) Choose the best one

  48. Improved Harri’s Algorithm For Corner And Egde Detections Using the proposed algorithm to do corner detection Using the proposed algorithm to do edge detection Using Harris’ algorithm to do corner detection Fig.18. Compare Harri’s Algorithm with the proposed algorithm.[1]

  49. Improved Harri’s Algorithm For Corner And Egde Detections Fig.19. (a) Image consists of three dots (upper-left), a valley (upper-right), a ridge (lower-left), and a noise-interfered region (lower-right). (b) (c) The results of corner detections.[1]

  50. Conlusion • Fist-Order Derivative Edge Detection: • the simplest method • Second-Order Derivative Edge Detection: • sensitive to noise • Hilbert Transform for Edge Detection • good for ramp edge, better noise immunity, but bad for accurate detection

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