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)