1. - photometric aspects of image formation gray level images
point-wise operations
linear filtering
CS682, Jana Kosecka

2. Image
Brightness values
I(x,y)
CS682, Jana Kosecka

3. Analog intensity function Temporal/spatial sampled function
Image model
Mathematical tools
Analog intensity function
Temporal/spatial sampled function
Quantization of the gray levels
Point sets
Random fields
List of image features, regions
Analysis
Linear algebra
Numerical methods
Set theory, morphology
Stochastic methods
Geometry, AI, logic
CS682, Jana Kosecka

4. Basic Photometry
CS682, Jana Kosecka

5. Basic ingredients
Radiance – amount of energy emitted along certain direction
Iradiance – amount of energy received along certain direction
BRDF – bidirectional reflectance distribution
Lambertian surfaces – the appearance depends only on radiance, not
on the viewing direction
Image intensity for a Lambertian surface
CS682, Jana Kosecka

6. Challenges
CS682, Jana Kosecka

7. Images Images contain noise – sources sensor quality, light
fluctuations, quantization effects
CS682, Jana Kosecka

8. Image Noise Models Additive noise: Most commonly used
Multiplicative noise:
Impulsive noise (salt and pepper):
CS682, Jana Kosecka

9. Additive Noise Models Noise Amount: SNR = s/ n
Gaussian - Usually, zero-mean, uncorrelated
Uniform
CS682, Jana Kosecka

10. Example
CS682, Jana Kosecka

11. How can we reduce noise?
Image acquisition noise due to light fluctuations and sensor noise can be reduced by acquiring a sequence of images and averaging them.
Solution – filtering the image
CS682, Jana Kosecka

12. Image Processing 1D signal and its sampled version
f = { f(1), f(2), f(3), …, f(n)}
f = {0, 1, 2, 3, 4, 5, 5, 6, 10 }
CS682, Jana Kosecka

13. Discrete time system maps 1 discrete time signal to another
f[x]
g[x] h
Special class of systems – linear , time-invariant systems
Superposition principle
Shift (time) invariant – shift in input causes shift in output
Examples
CS682, Jana Kosecka

14. unit impluse – if x = 0 it’s 1 and zero everywhere else
Convolution sum:
unit impluse – if x = 0 it’s 1 and zero everywhere else
Every discrete time signal can be written as a sum of scaled
and shifted impulses
The output the linear system is related to the input and the transfer
function via convolution h f g
filter f g
Convolution sum:
Notation:
CS682, Jana Kosecka

15. Averaging filter
Original image
Smoothed image
CS682, Jana Kosecka

16. Center pixel weighted more
Averaging filter
Averaging box filter
Averaging filter
Center pixel weighted more
2D convolution - next
CS682, Jana Kosecka

17. Convolution in 2D X 10 10 11 9 1 2 99 f g h 1 1 1 1 1 1
1/9 1 1 1
1/9.(10x1 + 11x1 + 10x1 + 9x1 + 10x1 + 11x1 + 10x1 + 9x1 + 10x1) =
1/9.( 90) = 10
CS682, Jana Kosecka

18. Example: X X X X X X 10 11 10 1 10 7 4 1 X X 9 10 11 1 1 O X X 10 9 10 I 2 1 X X 11 10 9 9 11 10 1 F X X 9 10 11 9 99 11 X X X X X X 10 9 9 11 10 10
1/9
1/9.(10x1 + 0x1 + 0x1 + 11x1 + 1x1 + 0x1 + 10x1 + 0x1 + 2x1) =
1/9.( 34) =
CS682, Jana Kosecka

19. Example: X X X X X X 10 11 10 1 10 7 4 1 X X 9 10 11 1 1 O X X I 10 91 10 7 4 1 X X 9 10 11 1 1 O X X I 10 9 10 2 1 X X 11 10 9 9 11 10 1 F X 20 X 9 10 11 9 99 11 X X X X X X 10 9 9 11 10 10
1/9
1/9.(10x1 + 9x1 + 11x1 + 9x1 + 99x1 + 11x1 + 11x1 + 10x1 + 10x1) =
1/9.( 180) = 20
CS682, Jana Kosecka

20. Example: X X X X X X 10 11 10 1 10 7 4 1 X X 9 10 11 1 1 O X X 10 9 10 I 2 1 X 18 X 11 10 9 9 11 10 1 F X 20 X 9 10 11 9 99 11 X X X X X X 10 9 9 11 10 10
1/9
1/9.(10x1 + 0x1 + 2x1 + 9x1 + 10x1 + 9x1 + 11x1 + 9x1 + 99x1) =
1/9.( 159) =
CS682, Jana Kosecka

21. How big should the mask be?
The bigger the mask,
more neighbors contribute.
smaller noise variance of the output.
bigger noise spread.
more blurring.
more expensive to compute.
CS682, Jana Kosecka

22. Gaussian Filter A particular case of averaging
The coefficients are samples of a 1D Gaussian.
Gives more weight at the central pixel and less weights to the neighbors.
The further away the neighbors, the smaller the weight.
Sample from the continuous Gaussian
CS682, Jana Kosecka

23. How big should the mask be?
The std. dev of the Gaussian  determines the amount of smoothing.
The samples should adequately represent a Gaussian
For a 98.76% of the area, we need
m = 5
5.(1/)  2    0.796, m 5
5-tap filter
g[x] = [0.136, , 1.00, 0.606, 0.136]
CS682, Jana Kosecka

24. Image Smoothing Convolution with a 2D Gaussian filter
Gaussian filter is separable, convolution can be accomplished as two 1-D convolutions
CS682, Jana Kosecka

25. Non-linear Filtering
Replace each pixel with the MEDIAN value of all the pixels in the neighborhood.
Non-linear
Does not spread the noise
Can remove spike noise
Expensive to run
CS682, Jana Kosecka

26. Example: 10 11 9 1 2 99 I X 10 O median sort 10,11,10,9,10,11,10,9,1099 I X 10 O
median
sort
10,11,10,9,10,11,10,9,10
9,9,10,10,10,10,10,11,11
CS682, Jana Kosecka

27. Example: 10 11 10 1 X X X X X X 9 10 11 1 1 10 1 1 X 10 X O 10 9 10 I 2 1 X X 11 10 9 9 11 X X 10 9 10 11 9 99 11 X X 10 9 9 11 10 10 X X X X X X
median
sort
11,10,0,10,11,1,9,10,0
0,0,1,9,10,10,10,11,11
CS682, Jana Kosecka

28. Example: X X X X X X 10 11 10 1 10 X X 9 10 11 1 1 O X X 10 I 10 9 2 11 10 X X 9 10 11 1 1 O X X 10 I 10 9 2 1 X 9 X 11 10 9 9 11 10 X 10 X 9 10 11 9 99 11 X X X X X X 10 9 9 11 10 10
median
sort
10,9,11,9,99,11,11,10,10
9,9,10,10,10,11,11,11,99
CS682, Jana Kosecka

29. Pointwise Image Operations
Lookup table – match image intensity to the
displayed brightness values
Manipulation of the lookup table – different
Visual effects – mapping is often non-linear
CS682, Jana Kosecka

30. Contrast gamma
Contrast
CS682, Jana Kosecka
Brightness

31. Quantization Thresholding Histogram
Histogram – frequency gray-level -> empirical distribution
h[i] – number of pixels of intensity i
Histogram equalization – making histogram flat
CS682, Jana Kosecka