Fourier Spectrum Image Fourier spectrum Fourier Transform Examples Phase and Magnitude Curious fact All natural images have very similar magnitude transform So why do they look different ID: 248690
Download Presentation The PPT/PDF document "2D Image" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
2D Image
Fourier SpectrumSlide2
Image
Fourier spectrum
Fourier Transform -- ExamplesSlide3
Phase and MagnitudeCurious fact
All natural images have very similar magnitude transform.So why do they look different…?
Demonstration
Take two pictures, swap the phase transforms, compute the inverse - what does the result look like?
Phase in images
matters a lot (more than magnitude)Slide4
Slide: Freeman & DurandSlide5
Slide: Freeman & DurandSlide6
Reconstruction with zebra phase, cheetah magnitude
Slide: Freeman & DurandSlide7
Reconstruction with cheetah phase, zebra magnitude
Slide: Freeman & DurandSlide8
ConvolutionSlide9
Spatial Filtering Operations
h(
x,y
) =
1/9 S f(n,m)
(
n,m
) in the 3x3 neighborhoodof (x,y
)
Example
3 x 3Slide10
Salt & Pepper Noise
3 X 3 Average
5 X 5 Average
7 X 7 Average
Median
Noise CleaningSlide11
Salt & Pepper Noise
3 X 3 Average
5 X 5 Average
7 X 7 Average
Median
Noise CleaningSlide12
x derivative
Gradient magnitude
y derivativeSlide13
Vertical edges
Horizontal edges
Edge Detection
ImageSlide14
Convolution Properties
Commutative:f*g = g*fAssociative:(f*g)*h = f*(g*h)Homogeneous:
f*(g)= f*g
Additive (Distributive): f*(g+h)= f*g+f*hShift-Invariantf*g(x-x0,y-yo)= (f*g) (x-x0,y-yo) Slide15
The Convolution Theorem
and similarly:Slide16
Salt & Pepper Noise
3 X 3 Average
5 X 5 Average
7 X 7 Average
Going back to the Noise Cleaning example…
Convolution with a
rect
Multiplication with a
sinc
in the Fourier domain
=
LPF
(Low-Pass Filter)
Wider
rect
Narrower
sinc
=
Stronger
LPFSlide17
What is the Fourier Transform of ?
Examples
*Slide18
Image Domain
Frequency DomainSlide19
The Sampling Theorem
(developed on the board)Nyquist frequency, Aliasing, etc…Slide20
Gaussian pyramids
Laplacian Pyramids Wavelet Pyramids
Multi-Scale Image Representation
Good for:
- pattern matching
- motion analysis
- image compression
- other applicationsSlide21
Image Pyramid
High resolution
Low resolutionSlide22
search
search
search
search
Fast
Pattern MatchingSlide23
The Gaussian Pyramid
High resolution
Low resolution
blur
blur
blur
down-sample
down-sample
down-sample
blur
down-sampleSlide24
expand
expand
expand
Gaussian Pyramid
Laplacian Pyramid
The Laplacian Pyramid
-
=
-
=
-
=Slide25
-
=
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).Slide26
Computerized Tomography (CT)
f(x,y)
u
v
F(u,v)Slide27
Computerized Tomography
Original
(simulated)
2D image
8 projections-
Frequency
Domain
120 projections-
Frequency
Domain
Reconstruction from
8 projections
Reconstruction from
120 projectionsSlide28
End of Lesson...
Exercise#1 -- will be posted on the website.(Theoretical exercise: To be done and submitted individually)