Time Frequency Analysis and Wavelet Transform
Midterm Presentation
Edge-Detection and Wavelet Transform
2011.11.24
Kuang-Tsu Shih
Outline
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•
•
•
•
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Introduction to Edge Detection
Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion
Outline
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•
•
•
•
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Introduction to Edge Detection
Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion
Edge-Detection
• A fundamental element in image analysis
• Wide applications:
– Pattern recognition
– Image segmentation
– Scene analysis
– …etc.
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
Ambiguity in Edge Detection
Edge!
Edge?
Edge?
Edge?
Fig. The ambiguity of the locality of edges.
Outline
•
•
•
•
•
•
Introduction to Edge Detection
Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion
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
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
Outline
•
•
•
•
•
•
Introduction to Edge Detection
Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion
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.
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
Outline
•
•
•
•
•
•
Introduction to Edge Detection
Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion
Wavelet Transform
• Basic form of continuous wavelet transform (CWT)
: The mother wavelet
a: The dimension of translation (location axis)
b: The dimension of dilation (scale axis)

, that is,  f (t ) dt   . (finite energy)
• The functions generated by mother wavelet should be a basis
of the
space.
• f belongs to
2
Wavelet Transform
• More on the mother wavelet
– Admissibility:

 ( )

“Wave”
2

d     (t )dt  0
“Let”
n
M

x
– Regularity:
n
  ( x)dx  0
(vanishing moments)
WHY?
 tp
1
t 
( p)

W f (0, b) 
f (0)   ( ) dt

b 
b p
 p!
1

b
Decays fast as b is small

f (1) (0)
f ( 2) (0)
f ( n ) (0)
2
3
n 1
n2 
f
(
0
)
M
b

M
b

M
b

...

M
b

O
(
b
)
0
1
2
n

1!
2!
n!


Vanishes!
Wavelet Transform
We focus on this one
Fig. Some common mother wavelets.
The Mexican Hat Function
• The Mexican hat function


25 / 4
2 t 2
 (t ) 
1  2t e
3
• 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,
d p t 2
 (t )  p e
dt
the wavelet has exactly p vanish moment.
Wavelet Transform and Edge Detection
• Let f(x) be a function in
,  (x) be a smoothing
function. (impulse response of a low-pass filter)
•
1
x
Let  s ( x)   ( ) be the stretched version of  (x) .
s
s
• Let
and
Wavelet Transform and Edge Detection
KEY POINT!
Wavelet transform
Smooth + Differentiation
Wavelet transform
Smooth + Differentiation
Wavelet Transform and Edge Detection
Smooth
Differentiation
Differentiation
Wavelet Transform and Edge Detection
Fig. Edges can be detected by examine the wavelet transform of the signal.
Wavelet Transform and Edge Detection
• We can easily generalize this to 2D signals:
KEY POINT!
Smooth + Differentiation
Wavelet transform
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.
Original
Image
Filtered Image
s = 24

x

y
    
    
 x   y 
2
s = 21
s = 22
s = 23
s = 24
2
  / y 
tan1 

  / x 
Local
Maximum of
Modulus
Local
Maximum of
Modulus
after
thresholding
s = 21
s = 22
s = 23
s = 24
Outline
•
•
•
•
•
•
Introduction to Edge Detection
Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion
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.
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”
Lipschitz Exponent
h0  0, s.t. h  h0 it is true
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.
Lipschitz Exponent
Wavelet-Based Method with Lipschitz Exponent
Wavelet-Based Method with Lipschitz Exponent
Wavelet-Based Method with Lipschitz Exponent
Wavelet-Based Method with Lipschitz Exponent
Outline
•
•
•
•
•
•
Introduction to Edge Detection
Gradient-Based Methods
Canny Edge Detector
Wavelet Transform-Based Methods
The Lipschitz Exponent
Conclusion
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.
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