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Convolution

Explore the commutative, associative, homogeneous, and additive properties of convolution in signal processing. Learn about shift-invariance and the Convolution Theorem, alongside examples and Fourier Transform in the Image Domain and Frequency Domain. Discover Multi-Resolution Image Representation through Gaussian, Laplacian, and Wavelet Pyramids, useful for fast pattern matching, motion analysis, image compression, and other applications.

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Convolution

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  1. Convolution

  2. 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-x0,y-yo)= (f*g) (x-x0,y-yo)

  3. The Convolution Theorem and similarly:

  4. * Examples What is the Fourier Transform of ?

  5. Image Domain Frequency Domain

  6. The Sampling Theorem Nyquist frequency, Aliasing, etc… (on the board)

  7. Multi-Resolution Image Representation • Gaussian pyramids • Laplacian Pyramids • Wavelet Pyramids

  8. Image Pyramid Low resolution High resolution

  9. search search search search FastPattern Matching Also good for: - motion analysis - image compression - other applications

  10. down-sample blur down-sample blur down-sample blur down-sample blur The Gaussian Pyramid Low resolution High resolution

  11. The Laplacian Pyramid expand - = expand - = expand - = Laplacian Pyramid Gaussian Pyramid

  12. 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).

  13. v F(u,v) u Computerized Tomography f(x,y)

  14. Computerized Tomography Original (simulated) 2D image 8 projections- Frequency Domain 120 projections- Frequency Domain Reconstruction from 120 projections Reconstruction from 8 projections

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