edge-detection and wavelet transform kuang-tsu shih time frequency analysis and wavelet transform...
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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](https://reader035.vdocuments.net/reader035/viewer/2022062309/5697c0111a28abf838ccbd1e/html5/thumbnails/1.jpg)
Edge-Detection and Wavelet Transform
Kuang-Tsu Shih
Time Frequency Analysis and Wavelet Transform Midterm Presentation
2011.11.24
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Outline
• Introduction to Edge Detection• Gradient-Based Methods• Canny Edge Detector• Wavelet Transform-Based Methods• The Lipschitz Exponent• Conclusion
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Outline
• Introduction to Edge Detection• Gradient-Based Methods• Canny Edge Detector• Wavelet Transform-Based Methods• The Lipschitz Exponent• Conclusion
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Edge-Detection• A fundamental element in image analysis• Wide applications:– Pattern recognition– Image segmentation– Scene analysis– …etc.
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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
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Ambiguity in Edge DetectionEdge!
Edge?Edge? Edge?
Fig. The ambiguity of the locality of edges.
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Outline
• Introduction to Edge Detection• Gradient-Based Methods• Canny Edge Detector• Wavelet Transform-Based Methods• The Lipschitz Exponent• Conclusion
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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
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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
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Outline
• Introduction to Edge Detection• Gradient-Based Methods• Canny Edge Detector• Wavelet Transform-Based Methods• The Lipschitz Exponent• Conclusion
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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.
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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
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Outline
• Introduction to Edge Detection• Gradient-Based Methods• Canny Edge Detector• Wavelet Transform-Based Methods• The Lipschitz Exponent• Conclusion
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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)(
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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!
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Wavelet Transform
Fig. Some common mother wavelets.
We focus on this one
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• 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
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• 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
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Wavelet Transform and Edge Detection
KEY POINT!
Wavelet transform
Wavelet transform
Smooth + Differentiation
Smooth + Differentiation
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Wavelet Transform and Edge Detection
Smooth
Differentiation
Differentiation
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Wavelet Transform and Edge Detection
Fig. Edges can be detected by examine the wavelet transform of the signal.
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• We can easily generalize this to 2D signals:
Wavelet Transform and Edge Detection
KEY POINT!
Wavelet transformSmooth + Differentiation
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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.
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s = 21 s = 22 s = 23 s = 24
Original Image
Filtered Image s = 24
x
y
22
yx
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s = 21 s = 22 s = 23 s = 24
x
y
/
/tan 1
Local Maximum of Modulus
Local Maximum of Modulus after thresholding
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Outline
• Introduction to Edge Detection• Gradient-Based Methods• Canny Edge Detector• Wavelet Transform-Based Methods• The Lipschitz Exponent• Conclusion
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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.
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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”
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Lipschitz Exponent
trueisit s.t. ,0 00 hhh
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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.
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Lipschitz Exponent
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Wavelet-Based Method with Lipschitz Exponent
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Wavelet-Based Method with Lipschitz Exponent
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Wavelet-Based Method with Lipschitz Exponent
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Wavelet-Based Method with Lipschitz Exponent
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Outline
• Introduction to Edge Detection• Gradient-Based Methods• Canny Edge Detector• Wavelet Transform-Based Methods• The Lipschitz Exponent• Conclusion
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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.
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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