Convolution. Spatial Filtering Operations g(x,y) = 1/M  f(n,m) (n,m) in  S Example 3 x 3 5 x 5.

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Presentation transcript:

Convolution

Spatial Filtering Operations g(x,y) = 1/M  f(n,m) (n,m) in  S Example 3 x 3 5 x 5

Salt & Pepper Noise 3 X 3 Average5 X 5 Average 7 X 7 AverageMedian Noise Cleaning

Salt & Pepper Noise 3 X 3 Average5 X 5 Average 7 X 7 AverageMedian Noise Cleaning

x derivative Gradient magnitude 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:

What is the Fourier Transform of ? Examples *

Image DomainFrequency Domain

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

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

Image Pyramid High resolution Low resolution

search Fast Pattern Matching

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

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

- = 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)

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