Convolution
1
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1
0
),(),(
),(),(),(*
)()(
)()()(*
N N
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ddyxgfyxgf
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:discrete) s,(continuou 2D
:discrete) s,(continuou 1D
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)
2)()(xfWhat is the Fourier Transform of ?
Examples
rectrect
xf
*
{sinc}*{sinc}
sinc}{sinc)}({
*
2sinc)( xf
search
search
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Fast Pattern Matching
Also good for:- motion analysis- image compression- other applications
2)*( 23 gaussianGG
1G
The Gaussian Pyramid
High resolution
Low resolution
Image0G
2)*( 01 gaussianGG
2)*( 12 gaussianGG
2)*( 34 gaussianGG
blur
blur
blur
down-sample
down-sample
down-sampleblurdown-sample
expand
expand
expand
Gaussian Pyramid Laplacian Pyramid
The Laplacian Pyramid
0G
1G
2GnG
- =
0L
- =1L
- = 2Lnn GL
)expand( 1 iii GGL
)expand( 1 iii GLG
- =
Laplacian ~ Difference of Gaussians
DOG = Difference of Gaussians
More details on Gaussian and Laplacian pyramidscan be found in the paper by Burt and Adelson(link will appear on the website).
Computerized Tomography
Original (simulated) 2D image
8 projections-Frequency
Domain
120 projections-Frequency
Domain
Reconstruction from8 projections
Reconstruction from120 projections