 # Convolution Spatial Filtering Operations Example 3 x 3

• Slides: 20 Convolution Spatial Filtering Operations Example 3 x 3 5 x 5 g(x, y) = 1/M S f(n, m) in S Noise Cleaning Salt & Pepper Noise 3 X 3 Average 5 X 5 Average 7 X 7 Average Median Noise Cleaning Salt & Pepper Noise 3 X 3 Average 5 X 5 Average 7 X 7 Average Median x derivative y derivative Gradient magnitude Edge Detection Image Vertical edges Horizontal edges Convolution Properties • Commutative: f*g = g*f • Associative: (f*g)*h = f*(g*h) • Homogeneous: f*( g)= f*g • Additive (Distributive): f*(g+h)= f*g+f*h • Shift-Invariant f*g(x-x 0, y-yo)= (f*g) (x-x 0, y-yo) The Convolution Theorem and similarly: Examples What is the Fourier Transform of * ? Image Domain Frequency Domain The Sampling Theorem Nyquist frequency, Aliasing, etc… (on the board) Multi-Resolution Image Representation • Gaussian pyramids • Laplacian Pyramids • Wavelet Pyramids Good for: - pattern matching - motion analysis - image compression - other applications Image Pyramid Low resolution High resolution Fast Pattern Matching search The Gaussian Pyramid Low resolution down-sample blur down-samp le blur down blur do wn -sa blur High resolution mp le -sam ple The Laplacian Pyramid Gaussian Pyramid expan - exp d and ex pa = - = nd Laplacian ~ Difference of Gaussians - = DOG = Difference of Gaussians More details on Gaussian and Laplacian pyramids can be found in the paper by Burt and Adelson (link will appear on the website). Computerized Tomography (CT) f(x, y) v F(u, v) u Computerized Tomography Original (simulated) 2 D image 8 projections. Frequency Domain Reconstruction from 8 projections 120 projections. Frequency Domain Reconstruction from 120 projections End of Lesson. . . Exercise#1 -- will be posted on the website. (Theoretical exercise: To be done and submitted individually)