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Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24
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Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Jan 29, 2016

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Page 1: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Edge-Detection and Wavelet Transform

Kuang-Tsu Shih

Time Frequency Analysis and Wavelet Transform Midterm Presentation

2011.11.24

Page 2: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Outline

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

Page 3: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Outline

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

Page 4: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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

Page 5: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

The Definition of An Edge• Definition:– 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

Page 6: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Ambiguity in Edge DetectionEdge!

Edge?Edge? Edge?

Fig. The ambiguity of the locality of edges.

Page 7: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Outline

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

Page 8: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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

Page 9: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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

Page 10: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Outline

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

Page 11: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Canny Edge Detector

• Filtering– 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.

Page 12: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Canny Edge Detector• Advantage– 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

Page 13: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Outline

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

Page 14: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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)

dttf

2)(

Page 15: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet Transform• More on the mother wavelet

– Admissibility:

– Regularity:

d2

)( 0)( dtt

“Wave”

0)( dxxxM nn

“Let”

dtb

t

p

tf

bbfW

p

p

p

)(

!)0(

1),0( )(

)(

!

)0(...

!2

)0(

!1

)0()0(

1 21)(

32

)2(2

1

)1(

0nn

n

n

bObMn

fbM

fbM

fbMf

b

WHY?

(vanishing moments)

Decays fast as b is small

Vanishes!

Page 16: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet Transform

Fig. Some common mother wavelets.

We focus on this one

Page 17: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

• 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.

The Mexican Hat Function

224/5

213

2)( tett

2

)( tp

p

edt

dt

Page 18: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

• Let be the stretched version of .

Wavelet Transform and Edge Detection• Let f(x) be a function in , be a smoothing

function. (impulse response of a low-pass filter)

)(x

)(1

)(s

x

sxs )(x

• Let and

Page 19: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet Transform and Edge Detection

KEY POINT!

Wavelet transform

Wavelet transform

Smooth + Differentiation

Smooth + Differentiation

Page 20: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet Transform and Edge Detection

Smooth

Differentiation

Differentiation

Page 21: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet Transform and Edge Detection

Fig. Edges can be detected by examine the wavelet transform of the signal.

Page 22: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

• We can easily generalize this to 2D signals:

Wavelet Transform and Edge Detection

KEY POINT!

Wavelet transformSmooth + Differentiation

Page 23: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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.

Page 24: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

s = 21 s = 22 s = 23 s = 24

Original Image

Filtered Image s = 24

x

y

22

yx

Page 25: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

s = 21 s = 22 s = 23 s = 24

x

y

/

/tan 1

Local Maximum of Modulus

Local Maximum of Modulus after thresholding

Page 26: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Outline

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

Page 27: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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.

Page 28: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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”

Page 29: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Lipschitz Exponent

trueisit s.t. ,0 00 hhh

Page 30: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Lipschitz Exponent

(Therefore, any differentiable point has L. E. greater than 1.)

KEY POINT

(The higher L. E., the smoother a function is, for that point.)

This important theorem relates the wavelet transform coefficients to L.E.The rates of change of coefficients across scales are different.

Page 31: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Lipschitz Exponent

Page 32: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet-Based Method with Lipschitz Exponent

Page 33: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet-Based Method with Lipschitz Exponent

Page 34: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet-Based Method with Lipschitz Exponent

Page 35: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Wavelet-Based Method with Lipschitz Exponent

Page 36: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

Outline

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

Page 37: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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.

Page 38: Edge-Detection and Wavelet Transform Kuang-Tsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 2011.11.24.

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,

www.polyvalens.com/blog/?page_id=15• Wikipedia Edge Detection http://en.wikipedia.org/wiki/Edge_detection Canny Edge Detector http://en.wikipedia.org/wiki/Canny_edge_detector• http://140.115.11.235/~chen/course/vision/ch6/ch6.htm