1. Edge-Detection and Wavelet Transform
Time Frequency Analysis and Wavelet Transform Midterm Presentation
Edge-Detection and Wavelet Transform
Kuang-Tsu Shih

2. Outline Introduction to Edge Detection Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion

3. Outline Introduction to Edge Detection Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion

4. Edge-Detection A fundamental element in image analysis
Wide applications:
Pattern recognition
Image segmentation
Scene analysis
…etc.

5. The Definition of An Edge
Neighboring pixels with large differences in value.
Edges may be caused by various reasons
Discontinuity in depth (Silhouettes)
Discontinuity in reflectance (texture)
Discontinuity in lighting (shade)
We do not distinguish them in this report.
Edge Detector
original image
a binary edge map

6. Ambiguity in Edge Detection
Fig. The ambiguity of the locality of edges.

7. Outline Introduction to Edge Detection Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion

8. Gradient-Based Methods
The gradient-based methods check the magnitude of image gradient.
The gradient map is generated by 2D convolution.
Detects edges if the magnitude > threshold.
Sobel operator
Prewitt operator
Robert’s cross operator

9. Gradient-Based Methods
Advantage:
Very simple, very fast.
Disadvantage:
Very susceptible to noise. (main drawback)
Not capable of detecting edges in different scales.
Parameter tuning.
Lena image with noise
The result by Sobel operator

10. Outline Introduction to Edge Detection Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion

11. Canny Edge Detector Filtering Take gradient Non-maximum suppression
Pass to a low pass kernel (Gaussian) to raise SNR.
Take gradient
The angle of gradient is quantized into four bins. (米)
Non-maximum suppression
Determine local maximum of gradient according to the orientation of the gradient.
Hysteresis Threshold
TH and TL, connectivity of edges.

12. Canny Edge Detector Advantage Disadvantage
Easy implementation, fast speed.
Relatively robust and cost effect.
Disadvantage
The result can still be affected by strong noise.
Does not examine edges in all scales.
Lena with noise
Canny result

13. Outline Introduction to Edge Detection Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion

14. Wavelet Transform Basic form of continuous wavelet transform (CWT)
f belongs to , that is, (finite energy)
The functions generated by mother wavelet should be a basis of the space.
: The mother wavelet
a: The dimension of translation (location axis)
b: The dimension of dilation (scale axis)

15. Wavelet Transform More on the mother wavelet Admissibility:
Regularity:
“Wave”
“Let”
(vanishing moments)
WHY?
Decays fast as b is small
Vanishes!

16. Wavelet Transform We focus on this one
Fig. Some common mother wavelets.

17. The Mexican Hat Function
In fact, it is the 2nd derivative of the Gaussian function (a “smoothing function”)
If we choose the wavelet to be the pth derivative of Gaussian,
the wavelet has exactly p vanish moment.

18. Wavelet Transform and Edge Detection
Let f(x) be a function in , be a smoothing function. (impulse response of a low-pass filter)
Let be the stretched version of
Let and

19. Wavelet Transform and Edge Detection
KEY POINT!
Wavelet transform
Smooth + Differentiation
Wavelet transform
Smooth + Differentiation

20. Wavelet Transform and Edge Detection
Smooth
Differentiation
Differentiation

21. Wavelet Transform and Edge Detection
Fig. Edges can be detected by examine the wavelet transform of the signal.

22. Wavelet Transform and Edge Detection
We can easily generalize this to 2D signals:
KEY POINT!
Smooth + Differentiation
Wavelet transform

23. Wavelet Transform and Edge Detection
The modulus of the wavelet transform at scale s:
A point is a multi-scale edge point at scale s if the magnitude of the gradient attains a local maximum.

24. Original Image
Filtered Image s = 24
s = 21
s = 22
s = 23
s = 24

25. s = 21 s = 22 s = 23 s = 24 Local Maximum of Modulus
Local Maximum of Modulus after thresholding
s = 21
s = 22
s = 23
s = 24

26. Outline Introduction to Edge Detection Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion

27. Wavelet-Based Method with Lipschitz Exponent
In fact, the wavelet-based method with dyadic (2k) scale alone is NOT optimally adapt to noise.
IDEA: We deal with sharp edges in big-scale (lower frequency) and not-so-sharp edges in small-scale (higher frequency).
Equivalently, we use kernels with larger support for sharp edges to better eliminate noise, and vice versa for weak edges.
Spatially variant kernel, none linear filtering.

28. Wavelet-Based Method with Lipschitz Exponent
How do we measure the “singularity” of a function?
Intuitively, an edge is a singular point of the function and the degree of singularity corresponds to the sharpness of an edge.
Note that the functions we care are not necessarily differentiable.
Solution: “The Lipschitz Exponent”

29. Lipschitz Exponent

30. Lipschitz Exponent
(Therefore, any differentiable point has L. E. greater than 1.)
(The higher L. E., the smoother a function is, for that point.)
KEY POINT
This important theorem relates the wavelet transform coefficients to L.E.
The rates of change of coefficients across scales are different.

31. Lipschitz Exponent

32. Wavelet-Based Method with Lipschitz Exponent

33. Wavelet-Based Method with Lipschitz Exponent

34. Wavelet-Based Method with Lipschitz Exponent

35. Wavelet-Based Method with Lipschitz Exponent

36. Outline Introduction to Edge Detection Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion

37. Conclusion
We reviewed several conventional edge detectors and their advantage and disadvantage.
We briefly introduced the concept of wavelet transform.
We proved the relationship between wavelet transform and low-pass filtering + gradient.
We introduced the concept of Lipschitz exponent and its application in edge detection.

38. References Feng-Ju Chang, “Wavelet for edge detection.”
J. C. Goswami, A. K. Chan, 1999, “Fundamentals of wavelets: theory, algorithms, and applications," John Wiley & Sons, Inc.
G. X. Ritter, J. N. Wilson, 1996, “Handbook of computer vision algorithms in image algebra," CRC Press, Inc.
謝豪駿, 小波分析於梁構件損傷檢測之應用
A really friendly guild to wavelet transform,
Wikipedia
Edge Detection
Canny Edge Detector