KuangTsu Shih Time Frequency Analysis and Wavelet Transform Midterm Presentation 20111124 Outline Introduction to Edge Detection GradientBased Methods Canny Edge Detector Wavelet TransformBased Methods ID: 181260
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Slide1
Edge-Detection and Wavelet Transform
Kuang-Tsu
Shih
Time Frequency Analysis and Wavelet Transform Midterm Presentation
2011.11.24Slide2
OutlineIntroduction to Edge DetectionGradient-Based Methods
Canny Edge DetectorWavelet Transform-Based MethodsThe Lipschitz ExponentConclusionSlide3
OutlineIntroduction to Edge Detection
Gradient-Based MethodsCanny Edge DetectorWavelet Transform-Based MethodsThe
Lipschitz ExponentConclusionSlide4
Edge-DetectionA fundamental element in image analysis
Wide applications:Pattern recognitionImage segmentationScene analysis…etc.Slide5
The Definition of An Edge
Definition:Neighboring pixels with large differences in value.Edges may be caused by various reasonsDiscontinuity 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 mapSlide6
Ambiguity in Edge Detection
Edge!
Edge?
Edge?
Edge?
Fig. The ambiguity of the
locality
of edges.Slide7
OutlineIntroduction to Edge Detection
Gradient-Based MethodsCanny Edge DetectorWavelet Transform-Based MethodsThe
Lipschitz ExponentConclusionSlide8
Gradient-Based MethodsThe gradient-based methods check the magnitude of image gradient.
The gradient map is generated by 2D convolution.Detects edges if the magnitude > threshold.Sobel operatorPrewitt operator
Robert’s cross operatorSlide9
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 operatorSlide10
OutlineIntroduction to Edge Detection
Gradient-Based MethodsCanny Edge DetectorWavelet Transform-Based MethodsThe
Lipschitz ExponentConclusionSlide11
Canny Edge DetectorFiltering
Pass to a low pass kernel (Gaussian) to raise SNR.Take gradient The angle of gradient is quantized into four bins. (米)Non-maximum suppressionDetermine local maximum of gradient according to the orientation of the gradient.
Hysteresis ThresholdTH and T
L, connectivity of edges.Slide12
Canny Edge DetectorAdvantage
Easy implementation, fast speed.Relatively robust and cost effect.DisadvantageThe result can still be affected by strong noise.Does
not examine edges in all scales.
Lena with noise
Canny resultSlide13
OutlineIntroduction to Edge Detection
Gradient-Based MethodsCanny Edge DetectorWavelet Transform-Based MethodsThe
Lipschitz ExponentConclusionSlide14
Wavelet TransformBasic 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)Slide15
Wavelet Transform
More on the mother wavelet
Admissibility:Regularity:
“Wave”
“Let”
WHY?
(vanishing moments)
Decays fast as b is small
Vanishes!Slide16
Wavelet Transform
Fig. Some common mother wavelets.
We focus on this oneSlide17
The Mexican
hat function
In
fact, it is the 2
nd
derivative of the Gaussian function (a “smoothing function”)
If we choose
the wavelet to be the p
th
derivative of Gaussian,
the wavelet has exactly p vanish moment.
The Mexican Hat FunctionSlide18
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)
Let andSlide19
Wavelet Transform and Edge Detection
KEY POINT!
Wavelet transform
Wavelet transform
Smooth + Differentiation
Smooth + DifferentiationSlide20
Wavelet Transform and Edge Detection
Smooth
Differentiation
DifferentiationSlide21
Wavelet Transform and Edge Detection
Fig. Edges can be detected by examine the wavelet transform of the signal.Slide22
We can easily generalize this to 2D signals:
Wavelet Transform and Edge Detection
KEY POINT!
Wavelet transform
Smooth + DifferentiationSlide23
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.Slide24
s = 2
1s = 22
s = 2
3
s = 24
Original Image
Filtered Image s = 24 Slide25
s = 2
1s = 22
s = 2
3
s = 24
Local Maximum of Modulus
Local Maximum of Modulus after thresholdingSlide26
OutlineIntroduction to Edge Detection
Gradient-Based MethodsCanny Edge DetectorWavelet Transform-Based MethodsThe Lipschitz
ExponentConclusionSlide27
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.Slide28
Wavelet-Based Method with
Lipschitz ExponentHow 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” Slide29
Lipschitz ExponentSlide30
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
.Slide31
Lipschitz ExponentSlide32
Wavelet-Based Method with Lipschitz ExponentSlide33
Wavelet-Based Method with Lipschitz ExponentSlide34
Wavelet-Based Method with Lipschitz ExponentSlide35
Wavelet-Based Method with Lipschitz ExponentSlide36
OutlineIntroduction to Edge Detection
Gradient-Based MethodsCanny Edge DetectorWavelet Transform-Based MethodsThe
Lipschitz ExponentConclusionSlide37
ConclusionWe 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.Slide38
ReferencesFeng-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=15Wikipedia Edge Detection http://en.wikipedia.org/wiki/Edge_detection Canny Edge Detector
http://en.wikipedia.org/wiki/Canny_edge_detectorhttp://140.115.11.235/~chen/course/vision/ch6/ch6.htm