2D Image Fourier Spectrum

Fourier Transform -- Examples Image Fourier spectrum

Phase and Magnitude Curious fact Demonstration 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)

Slide: Freeman & Durand

Slide: Freeman & Durand

Reconstruction with zebra phase, cheetah magnitude Slide: Freeman & Durand

Reconstruction with cheetah phase, zebra magnitude Slide: Freeman & Durand

Convolution

Spatial Filtering Operations Example 3 x 3 h(x,y) = 1/9 S f(n,m) (n,m) in the 3x3 neighborhood of (x,y)

Noise Cleaning Salt & Pepper Noise 3 X 3 Average 5 X 5 Average Median

Noise Cleaning Salt & Pepper Noise 3 X 3 Average 5 X 5 Average Median

Gradient magnitude x derivative y derivative

Vertical edges Horizontal edges Edge Detection Image

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 and similarly:

Going back to the Noise Cleaning example… 3 X 3 Average Salt & Pepper Noise Convolution with a rect  Multiplication with a sinc in the Fourier domain = LPF (Low-Pass Filter) 7 X 7 Average 5 X 5 Average Wider rect  Narrower sinc = Stronger LPF

Examples What is the Fourier Transform of ? *

Image Domain Frequency Domain

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

Multi-Scale Image Representation Gaussian pyramids Laplacian Pyramids Wavelet Pyramids Good for: - pattern matching - motion analysis - image compression - other applications

Image Pyramid High resolution Low resolution

Fast Pattern Matching search search search search

The Gaussian Pyramid Low resolution down-sample blur down-sample blur High resolution

- = - = - = The Laplacian Pyramid Gaussian Pyramid Laplacian Pyramid expand - = expand - = expand - =

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).

Computerized Tomography (CT) f(x,y) u v F(u,v)

Computerized Tomography Original (simulated) 2D image 8 projections- Frequency Domain 120 projections- Frequency Domain Reconstruction from 8 projections Reconstruction from 120 projections

End of Lesson... Exercise#1 -- will be posted on the website. (Theoretical exercise: To be done and submitted individually)