2d image fourier spectrum. image fourier spectrum fourier transform -- examples

2D Image Fourier Spectrum

Image Fourier spectrum

Fourier Transform -- Examples

3

Phase and Magnitude

• Curious factAll natural images have very similar magnitude transform.So why do they look different…?

• DemonstrationTake two pictures, swap the phase transforms, compute the inverse - what does the result look like?

Phase in images matters a lot (more than magnitude)

4

Slide: Freeman & Durand

5

Slide: Freeman & Durand

6

Reconstruction with zebra phase, cheetah magnitude

Slide: Freeman & Durand

7

Reconstruction with cheetah phase, zebra magnitude

Slide: Freeman & Durand

Convolution

1

0

1

0

1

0

),(),(

),(),(),(*

)()(

)()()(*

N N

N

yxgf

ddyxgfyxgf

xgf

dxgfxgf

:discrete) s,(continuou 2D

:discrete) s,(continuou 1D

Spatial Filtering Operations

h(x,y) = 1/9 S f(n,m)(n,m)

in the 3x3 neighborhood

of (x,y)

Example

3 x 3

111

111

111

9

1),( yxf

Salt & Pepper Noise

3 X 3 Average 5 X 5 Average

7 X 7 Average Median

Noise Cleaning

Salt & Pepper Noise

3 X 3 Average 5 X 5 Average

7 X 7 Average Median

Noise Cleaning

x derivativeGradient magnitude

y derivativex

f

y

f

22yx fff

11),( yxf

1

1),( yxf

),(),1( yxfyxf ),()1,( yxfyxf

Vertical edges Horizontal edges

Edge Detection

x

f

y

f

Image f

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

Salt & Pepper Noise 3 X 3 Average

5 X 5 Average 7 X 7 Average

Going back to the Noise Cleaning example…

111

111

111

9

1

Convolution with a rect Multiplication with a sinc in the Fourier domain

= LPF (Low-Pass Filter)

Wider rect Narrower sinc = Stronger LPF

What is the Fourier Transform of ?

Examples

rectrect

xf

*

{sinc}*{sinc}

sinc}{sinc)}({

*

2)()(xf

2sinc)( xf

Image Domain Frequency Domain

The Sampling Theorem

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

• Gaussian pyramids

• Laplacian Pyramids

• Wavelet Pyramids

Multi-Scale Image Representation

Good for:- pattern matching- motion analysis- image compression- other applications

Image Pyramid

High resolution

Low resolution

search

search

search

search

Fast Pattern Matching

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,()( uFuP

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)