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

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**Edge-Detection and Wavelet Transform**Time Frequency Analysis and Wavelet Transform Midterm Presentation Kuang-Tsu Shih 2011.11.24**Outline**• Introduction to Edge Detection • Gradient-Based Methods • Canny Edge Detector • Wavelet Transform-Based Methods • The Lipschitz Exponent • Conclusion**Outline**• 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 Sobeloperator**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 inall 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) • 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)**Wavelet Transform**• More on the mother wavelet • Admissibility: • Regularity: “Wave” “Let” (vanishing moments) WHY? Decays fast as b is small Vanishes!**Wavelet Transform**We focus on this one Fig. Some common mother wavelets.**The Mexican Hat Function**• 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.**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**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 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**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**(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.**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-JuChang, “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

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