1. Edge Detection Speaker: Che-Ming Hu Advisor: Jian-Jiun Ding
Graduate Institute of Communication Engineering
National Taiwan University, Taipei, Taiwan, ROC
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2. Outline Introduction Edge detection method Simulation results
Differentiation
Hibert Transform
Short Response Hilbert Transform
Improved Harri’s Algorithm
Simulation results
Conclusion
Reference
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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]
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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
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5. Fist-Order Derivative Edge Detection
Introduction
The Roberts operators
The Prewitt operators
The Sobel operators
First-Order of Gausssian (FDOG)
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6. Fist-Order Derivative Edge Detection
Definition:
the gradient vector
the magnitude of this vector
the direction of the gradient vector
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7. Fist-Order Derivative Edge Detection
Fig.2. Orthogonal gradient generation.[2]
The gradient along the line normal to the edge slope
The spatial gradient amplitude
The gradient amplitude combination
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8. The Roberts operators
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9. The Prewitt operators
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10. The Sobel operators
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11. Operators Fig.3. Impulse response arrays for 3 ×3 orthogonal
differential gradient edge operators.[2]
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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|
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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]
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14. Canny Edge Detection Good detection Good localization Single response
Many edge candidate
The accurate edge
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15. Fig.5. Comparison with FDOG and the Sobel operators
Simulation
The original image
FDOG
The Sobel operators
Fig.5. Comparison with FDOG and the Sobel operators
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16. Fig.6. Simulation of first-order of derivative edge detection[3]
(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]
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17. Second-order derivative edge detection
The Laplacian of a 2-D function f (x, y) is a second-order derivative defined as:
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18. Second-order derivative edge detection
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19. Second-order derivative edge detection
The 2-D Gaussian function:
The Laplacian of Gaussian (LOG):
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20. Second-order derivative edge detection
Fig.7. An 11×11 mask approximation to Laplacian of Gaussian (LOG).[3]
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21. Second-order derivative edge detection
Sensitive to noise
Gaussian function
Low-pass filter
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22. Second-order derivative edge detection
(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]
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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]
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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
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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
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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]
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27. Hilbert Transform for Edge Detection
Slove the problem of differentiation
Edge is too thick
SRHLT
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28. Short Response Hilbert Transform for Edge Detection
From
We obtain:
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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)
Small (HLT) b
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30. Short Response Hilbert Transform for Edge Detection
Fig.12.Impulse responses and their FTs of the SRHLT for different b.
We can compare them with the impulse response of the
differential operation and the original HLT.[3]
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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]
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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
Fig.14. Experiments that use differentiation, the HLT, and the SRHLT
(b=8) to do edge detection for Lena image.[3]
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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
Fig.15. Experiments that use differentiation, the HLT, and the SRHLT (b=8) to
detect the edges of Lena image interfered by noise.[3]
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34. Short Response Hilbert Transform for Edge Detection
(a) b = 0.5 for Lena image (b) b = 0.5 for Lena + noise
(c) b = 8 for Lena image (d) b = 8 for Lena + noise
(e) b = 20 for Lena image (f) b = 20 for Lena + noise
Fig.16. Experiment of SRHLT with different b (0.5, 8, and 20)[3]
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35. Short Response Hilbert Transform for Edge Detection
How to choose a edge detection filter?
By Canny’s Theorem:
Higher distinction
Noise immunity
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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)
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37. Short Response Hilbert Transform for Edge Detection
The other alternative ways to define SRHLT:
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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
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39. Improved Harri’s Algorithm For Corner And Edge Detections
Instead of
we use:
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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
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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:
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42. Improved Harri’s Algorithm For Corner And Egde Detections
(step 3) Then we express the variation around [m, n] by:
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43. Improved Harri’s Algorithm For Corner And Egde Detections
(step 4) The principal axes are the eigen vector of
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44. Improved Harri’s Algorithm For Corner And Egde Detections
(step 5) we can observe the variation along the two principal axes.
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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]
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46. Improved Harri’s Algorithm For Corner And Egde Detections
Table 1 Case table.[1]
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47. Improved Harri’s Algorithm For Corner And Egde Detections
(step 6)
Choose the best one
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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]
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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]
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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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51. Conlusion Short Response Hilbert Transform for Edge Detection (SRHLT):
have both advantage of HLT and differentiation
Improved Harri’s Algorithm For Corner And Egde Detections:
detect more types (34=81)
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52. Conclusion
Application
Image segmentation
Data compression
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53. Reference
[1] Soo-Chang Pei, Jian-Jiun Ding, ” Imporved Harris’ Algotithm ForConer And Edge Detections”, vol 1, 2005.
[2] William K. Pratt , “Digital Image Processing_William K. Pratt 3rd ”,chapter 15.
[3] Jiun-De Huang, “Image Compression by Segmentation and Boundary Description”, chapter 2.
[4] J. Canny, “A Computational Approach to Edge Detection,” IEEE
Trans. Pattern Analysis and Machine Intelligence, PAMI-8, 6,November 1986, 679–698.
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54. END
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