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)