Convolution Spatial Filtering Operations Example 3 x 3
- Slides: 20
Convolution
Spatial Filtering Operations Example 3 x 3 5 x 5 g(x, y) = 1/M S f(n, m) in S
Noise Cleaning 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
x derivative y derivative Gradient magnitude
Edge Detection Image Vertical edges Horizontal edges
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-x 0, y-yo)= (f*g) (x-x 0, y-yo)
The Convolution Theorem and similarly:
Examples What is the Fourier Transform of * ?
Image Domain Frequency Domain
The Sampling Theorem Nyquist frequency, Aliasing, etc… (on the board)
Multi-Resolution Image Representation • Gaussian pyramids • Laplacian Pyramids • Wavelet Pyramids Good for: - pattern matching - motion analysis - image compression - other applications
Image Pyramid Low resolution High resolution
Fast Pattern Matching search
The Gaussian Pyramid Low resolution down-sample blur down-samp le blur down blur do wn -sa blur High resolution mp le -sam ple
The Laplacian Pyramid Gaussian Pyramid expan - exp d and ex pa = - = nd
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) v F(u, v) u
Computerized Tomography Original (simulated) 2 D image 8 projections. Frequency Domain Reconstruction from 8 projections 120 projections. Frequency Domain Reconstruction from 120 projections
End of Lesson. . . Exercise#1 -- will be posted on the website. (Theoretical exercise: To be done and submitted individually)
- Ingress filtering vs egress filtering
- Convolution sum signals and systems
- Digital image processing
- Intensity transformation and spatial filtering
- Intensity transformation and spatial filtering
- Abbe imaging experiment
- Intensity transformations and spatial filtering
- Intensity transformations and spatial filtering
- Intensity transformation function
- Contra harmonic mean filter
- Spatial filtering
- Spatial filtering
- Spatial filtering
- Spatial filtering
- Restoration in the presence of noise only-spatial filtering
- Spatial filtering matlab
- Intensity transformation and spatial filtering
- Spatial data vs non spatial data
- Spatial operations in image processing
- Circular convolution meaning
- Convolution symbol