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Convolution
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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)
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The Convolution Theorem
and similarly:
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Examples What is the Fourier Transform of ? *
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Image Domain Frequency Domain
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The Sampling Theorem Nyquist frequency, Aliasing, etc… (on the board)
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Multi-Resolution Image Representation
Gaussian pyramids Laplacian Pyramids Wavelet Pyramids
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Image Pyramid Low resolution High resolution
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Fast Pattern Matching Also good for: - motion analysis
search search search search Also good for: - motion analysis - image compression - other applications
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The Gaussian Pyramid Low resolution down-sample blur down-sample blur
High resolution
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- = - = - = The Laplacian Pyramid Gaussian Pyramid Laplacian Pyramid
expand - = expand - = expand - =
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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).
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Computerized Tomography (CT)
f(x,y) u v F(u,v)
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Computerized Tomography
Original (simulated) 2D image 8 projections- Frequency Domain 120 projections- Frequency Domain Reconstruction from 8 projections Reconstruction from 120 projections
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End of Lesson... Exercise#1 -- will be posted on the website.
(Theoretical exercise: To be done and submitted individually)
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