1. Linear filtering

2. Overview: Linear filtering
Linear filters
Definition and properties
Examples
Gaussian smoothing
Separability
Applications
Denoising
Non-linear filters: median filter
Sharpening
Subsampling

3. Images as functions
demo with scanalyze on laptop
Source: S. Seitz

4. Images as functions
We can think of an image as a function, f, from R2 to R:
f( x, y ) gives the intensity at position ( x, y )
Realistically, we expect the image only to be defined over a rectangle, with a finite range:
f: [a,b] x [c,d]  [0,1]
A color image is just three functions pasted together. We can write this as a “vector-valued” function:
As opposed to [0..255]
Source: S. Seitz

5. What is a digital image?
In computer vision we usually operate on digital (discrete) images:
Sample the 2D space on a regular grid
Quantize each sample (round to nearest integer)
The image can now be represented as a matrix of integer values
Source: S. Seitz

6. Motivation 1: Noise reduction
Given a camera and a still scene, how can you reduce noise?
Answer: take lots of images, average them
Take lots of images and average them!
What’s the next best thing?
Source: S. Seitz

7. Motivation 2: Image half-sizing
This image is too big to fit on the screen. How can we reduce it by a factor of two?
How about taking every second pixel?
Source: S. Seitz

8. First attempt at a solution
Let’s replace each pixel with an average of all the values in its neighborhood
ys(i) =1/(2N+1)*(y(i + N) + y(i + N – 1) y(i – N))
ys(1) = y(1)
ys(2) = (y(1)+y(2)+y(3))/3
ys(3) = (y(1)+y(2)+y(3)+y(4)+y(5))/5
ys(4) = (y(2)+y(3)+y(4)+y(5)+y(6))/5
Source: S. Marschner

9. First attempt at a solution
Let’s replace each pixel with an average of all the values in its neighborhood
Moving average in 1D:
Source: S. Marschner

10. Weighted Moving Average
Can add weights to our moving average
Weights [1, 1, 1, 1, 1] / 5
Source: S. Marschner

11. Weighted Moving Average
Non-uniform weights [1, 4, 6, 4, 1] / 16
Source: S. Marschner

12. Moving Average In 2D 90 90
ones, divide by 9
Source: S. Seitz

13. Moving Average In 2D 90 90 10
ones, divide by 9
Source: S. Seitz

14. Moving Average In 2D 90 90 10 20
ones, divide by 9
Source: S. Seitz

15. Moving Average In 2D 90 90 10 20 30
ones, divide by 9
Source: S. Seitz

16. Moving Average In 2D 90 10 20 30
ones, divide by 9
Source: S. Seitz

17. Moving Average In 2D Source: S. Seitz ones, divide by 9 90 10 20 30 4090 10 20 30 40 60 90 50 80
ones, divide by 9
Source: S. Seitz

18. Generalization of moving average
Let’s replace each pixel with a weighted average of its neighborhood
The weights are called the filter kernel
What are the weights for a 3x3 moving average? 1
“box filter”
Source: D. Lowe

19. Practice with linear filters1 ?
Original
Source: D. Lowe

20. Practice with linear filters1
Original
Filtered
(no change)
Source: D. Lowe

21. Practice with linear filters1 ?
Original
Source: D. Lowe

22. Practice with linear filters1
Original
Shifted left
By 1 pixel
Source: D. Lowe

23. Practice with linear filters? 1
Original
Source: D. Lowe

24. Practice with linear filters1
Original
Blur (with a
box filter)
Source: D. Lowe

25. Practice with linear filters- 2 1 ?
(Note that filter sums to 1)
Original
Source: D. Lowe

26. Practice with linear filters- 2 1
Original
Sharpening filter
Accentuates differences with local average
Source: D. Lowe

27. Sharpening
Source: D. Lowe

28. Gaussian Kernel
5 x 5,  = 1
Constant factor at front makes volume sum to 1 (can be ignored, as we should re-normalize weights to sum to 1 in any case)
Source: C. Rasmussen

29. Choosing kernel width
Gaussian filters have infinite support, but discrete filters use finite kernels
Source: K. Grauman

30. Choosing kernel width
Rule of thumb: set filter half-width to about 3 σ

31. Gaussian noise Mathematical model: sum of many independent factors
Good for small standard deviations
Assumption: independent, zero-mean noise
Source: K. Grauman

32. Reducing Gaussian noise
Smoothing with larger standard deviations suppresses noise, but also blurs the image

33. Reducing salt-and-pepper noise
3x3
5x5
7x7
What’s wrong with the results?

34. Salt-and-pepper noise
Median filter
Salt-and-pepper noise
Median filtered
MATLAB: medfilt2(image, [h w])
Source: K. Grauman

35. Median vs. Gaussian filtering
3x3
5x5
7x7
Gaussian
Median

36. Sharpening revisited What does blurring take away? Let’s add it back:
original
smoothed (5x5)
detail = –
Let’s add it back:
original
detail
+ α
sharpened =

37. unit impulse (identity)
Unsharp mask filter
Gaussian
unit impulse
Laplacian of Gaussian
image
blurred image
unit impulse (identity)
f + a(f - f * g) = (1+a)f-af*g = f*((1+a)e-g)

38. - Recall 2 1 Original Sharpening filter2 1
Original
Sharpening filter
Accentuates differences with local average

39. Image half-sizing
This image is too big to fit on the screen. How can we reduce it by a factor of two?
How about taking every second pixel?
Source: S. Seitz

40. Image sub-sampling
1/8
1/4
Throw away every other row and column to create a 1/2 size image
- called image sub-sampling
Slide by Steve Seitz

41. Gaussian (low-pass) pre-filtering
Solution: filter the image, then subsample
Filter size should double for each ½ size reduction
Source: Steve Seitz

42. Subsampling with Gaussian pre-filtering
1/2
1/4 (2x zoom)
1/8 (4x zoom)
Source: Steve Seitz

43. Without prefiltering:
1/2
1/4 (2x zoom)
1/8 (4x zoom)
Source: Steve Seitz

44. Zmiany klimatyczne – El Nino

45. ENSO = EN + SO

46. Fourier series make use of the orthogonality relationships of
the sine and cosine functions

47. natural
fake

48. Anti-aliasing in 3D graphic display