lec13 - Fourier Tansform Edge and Corner Detection Discrete...

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1 CS252A, Fall 2010 Computer Vision I Edge and Corner Detection Computer Vision I CSE252A Lecture 13 CS252A, Fall 2010 Computer Vision I Fourier Tansform Discrete Fourier Transform (DFT) of I[x,y] Inverse DFT x,y: spatial domain u,v: frequence domain N by N image Implemented via the “Fast Fourier Transform” algorithm (FFT) CS252A, Fall 2010 Computer Vision I The Fourier Transform and Convolution If H and G are images, and F(.) represents Fourier transform, then Or H*G = F -1 ( F(H)F(G)) This is referred to as the Convolution Theorem Fast Fourier Transform: complexity O(n log n) -> complexity of convolution is O( n logn) not O(n 2 ). Thus, one way of thinking about the properties of a convolution is by thinking of how it modifies the frequencies of the image to which it is applied. In particular, if we look at the power spectrum, then we see that convolving image H by G attenuates frequencies where G has low power, and amplifies those which have high power. F(H*G) = F(H)F(G) CS252A, Fall 2010 Computer Vision I Median filters : example filters have width 5 : CS252A, Fall 2010 Computer Vision I On segmentation CS252A, Fall 2010 Computer Vision I Edges
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2 CS252A, Fall 2010 Computer Vision I Physical causes of edges 1. Object boundaries 2. Surface normal discontinuities 3. Reflectance (albedo) discontinuities 4. Lighting discontinuities CS252A, Fall 2010 Computer Vision I Noisy Step Edge • Derivative is high everywhere. • Must smooth before taking gradient. CS252A, Fall 2010 Computer Vision I Edge is Where Change Occurs: 1-D • Change is measured by derivative in 1D Smoothed Edge First Derivative Second Derivative Ideal Edge • Biggest change, derivative has maximum magnitude • Or 2nd derivative is zero. CS252A, Fall 2010 Computer Vision I Numerical Derivatives f(x) x X 0 X 0 +h X 0 -h Take Taylor series expansion of f(x) about x 0 f(x) = f(x 0 )+f’(x 0 )(x-x 0 ) + ½ f’’(x 0 )(x-x 0 ) 2 + … Consider Samples taken at increments of h and first two terms, we have f(x 0 +h) = f(x 0 )+f’(x 0 )h+ ½ f’’(x 0 )h 2 f(x 0 -h) = f(x 0 )-f’(x 0 )h+ ½ f’’(x 0 )h 2 Subtracting and adding f(x 0 +h) and f(x 0 -h) respectively yields Convolve with First Derivative: [-1 0 1] Second Derivative: [1 -2 1] CS252A, Fall 2010 Computer Vision I Implementing 1-D Edge Detection 1. Filter out noise: convolve with Gaussian 2. Take a derivative: convolve with [-1 0 1] We can combine 1 and 2.
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This note was uploaded on 12/08/2010 for the course CSE 252a taught by Professor Staff during the Fall '08 term at UCSD.

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lec13 - Fourier Tansform Edge and Corner Detection Discrete...

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