Convolution

1

0

1

0

1

0

),(),(

),(),(),(*

)()(

)()()(*

N N

N

yxgf

ddyxgfyxgf

xgf

dxgfxgf

: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)

The Convolution Theorem

xgxfxgxf

and similarly:

xgxfxgxf

2)()(xfWhat is the Fourier Transform of ?

Examples

rectrect

xf

*

{sinc}*{sinc}

sinc}{sinc)}({

*

2sinc)( xf

Image Domain Frequency Domain

The Sampling Theorem

Nyquist frequency, Aliasing, etc… (on the board)

• Gaussian pyramids

• Laplacian Pyramids

• Wavelet Pyramids

Multi-Resolution Image Representation

Image Pyramid

High resolution

Low resolution

search

search

search

search

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 (CT)

f(x,y)

)(1 xp )(2 xp

dyyxfxp ),()(

)()0,( uPuF

u

vF(u,v)

Computerized Tomography

Original (simulated) 2D image

8 projections-Frequency

Domain

120 projections-Frequency

Domain

Reconstruction from8 projections

Reconstruction from120 projections

End of Lesson...

Exercise#1 -- will be posted on the website.

(Theoretical exercise: To be done and submitted individually)