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