1. The Chinese University of Hong Kong
Math 3360: Mathematical Imaging
Lecture 12:
Linear filtering
Prof. Ronald Lok Ming Lui Department of Mathematics,
The Chinese University of Hong Kong

2. Linear filter = Convolution
Linear filtering of a (2M+1)x(2N+1) image I (defined on
[-M,M]x[-N,N]) = CONVOLUTION OF I and H
H is called the filter.
Different filter can be used:
Mean filter
Gaussian filter
Laplcian filter
Variation of these filters (Non-linear)
Median filter
Edge preserving mean filter

3. Linear filter

4. Type of filter

5. In Photoshop

6. Mean filter

7. Mean filter
Impulse noise
After mean filter

8. Mean filter
Gaussian noise
After mean filter

9. Mean filter
Real image
After mean filter

10. Gaussian filter Define a function using Gaussian function
Definition of H

11. Gaussian filter
Real image
After mean filter

12. Gaussian filter
Real image
After mean filter

13. Gaussian filter
Real image
After Gaussian filter

14. Gaussian filter
Real image
After mean filter

15. Gaussian filter
Real image
After Gaussian filter

16. Laplace filter )
Laplace filter (High pass filter)

17. Laplace filter
Original
Laplace filter

18. Laplace filter
Original
Laplace filter

19. Laplace filter
Laplace filter
Original

20. Median filter Median Nonlinear filter
Take median within a local window

21. Median filter
Real image
After mean filter

22. Median filter
Salt & Pepper
Mean filter
Median filter

23. Median filter
Noisy image
Median filter

24. Median filter

25. Median filter
Noisy image
Median filter

26. Median filter

27. Median filter
Noisy image
Can you guess what it is?
Median filter

28. Median preserving filtering
Step 1: Consider all windows of fixed size around a pixel (not necessarily centered at that pixel)
Step 2: Find a window with the least variance
Step 3: Do a linear filter in that window.

29. Median preserving filtering

30. Median preserving filtering