Convolution Convolution Properties Commutative fg gf Associative fgh

Convolution

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

Image Pyramid Low resolution High resolution

Fast Pattern Matching search Also good for: - motion analysis - image compression - other applications

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 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 on the website by tomorrow. (Theoretical exercise: To be done and submitted individually)