1. Intensity Transformations (Chapter 3)
CS474/674 – Prof. Bebis

2. Spatial Domain Methods
f(x,y)
g(x,y)
Point
Processing
g(x,y)
f(x,y)
Area/Mask
Processing

3. Point Processing Transformations
Convert a given pixel value to a new pixel value based on some predefined function.

4. Identity Transformation

5. Negative Image
O(r,c) = 255-I(r,c)

6. Contrast Stretching or Compression
Stretch gray-level ranges where
we desire more information
(slope > 1).
Compress gray-level ranges that
are of little interest
(0 < slope < 1).

7. Thresholding
Special case of contrast compression

8. Intensity Level Slicing
Highlight specific ranges of gray-levels only.
Same as double
thresholding!

9. Bit-level Slicing
Highlighting the contribution made by a specific bit.
For pgm images, each pixel is represented by 8 bits.
Each bit-plane is a binary image

11. Logarithmic transformation
Enhance details in the darker regions of an image at the expense of detail in brighter regions.
compress
stretch

12. Exponential transformation
Reverse effect of that obtained using logarithmic mapping.
stretch
compress

13. Histogram Equalization
A fully automatic gray-level stretching technique.
Need to talk about image histograms first ...

14. Image Histograms
An image histogram is a plot of the gray-level frequencies (i.e., the number of pixels in the image that have that gray level).

15. Image Histograms (cont’d)
Divide frequencies by total number of pixels to represent as probabilities.

16. Image Histograms (cont’d)

17. Properties of Image Histograms
Histograms clustered at the low end correspond to dark images.
Histograms clustered at the high end correspond to bright images.

18. Properties of Image Histograms (cont’d)
Histograms with small spread correspond to low contrast images (i.e., mostly dark, mostly bright, or mostly gray).
Histograms with wide spread correspond to high contrast images.

19. Properties of Image Histograms (cont’d)
Low contrast
High contrast

20. Histogram Equalization
The main idea is to redistribute the gray-level values uniformly.

21. Histogram Equalization (cont’d)
In practice, the equalized histogram might not be completely flat.

22. Probability - Definitions
Random experiment: an experiment whose result is not certain in advance (e.g., throwing a die)
Outcome: the result of a random experiment
Sample space: the set of all possible outcomes (e.g., {1,2,3,4,5,6})
Event: a subset of the sample space (e.g., obtain an odd number in the experiment of throwing a die = {1,3,5})

23. Random Variables - Review
A function that assigns a real number to random experiment outcomes (i.e., helps to reduce space of possible outcomes)
X(j)
X: # of heads

24. Random Variables - Example
Consider the experiment of throwing a pair of dice
Define the r.v. X=“sum of dice”
X=x corresponds to the event

25. Probability density function
The probability density function (pdf) is a real-valued function fX(x) describing the density of probability at each point in the sample space.
In the discrete case, this is just a histogram!
Gaussian

26. Probability distribution function
The integral of fX(x) defines the probability distribution function FX(x) (i.e., cumulative probability)
In the discrete case, simply take the sum:
fX(x)
FX(x)
Gaussian
fX(a)da
non-decreasing

27. Probability distribution function (cont’d)

28. Uniform Distribution
fX(x)
FX(x)

29. Random Variable Transformations
Suppose Y=T(X)
e.g., Y=X+1
If we know fX(x), can we find fY(y)?
Yes - it can be shown that:

30. Transformations of r.v. - Example
μ=0, σ=1
μ=1, σ=1

31. Special transformation!
Proof:

32. Histogram Equalization (cont’d)
The intensity levels can be viewed as a random variable in [0,1]
pr(r)
ps(s)

33. Histogram Equalization (cont’d)
For PGM images:
L=256 (graylevels)
k=0,1,2, …, L-1 (possible graylevels)
rk=k/(L-1) (normalized graylevel in [0, 1])

34. Histogram Equalization (cont’d)
then, de-normalize:
sk x (L-1)

35. Histogram Equalization Example
3 bit
64 x 64 image
input histogram
equalized histogram

36. Histogram Equalization Examples
original images and histograms
equalized images and histograms

37. Histogram Specification (Matching)
Histogram equalization yields a uniform pdf only.
What if we want to obtain a histogram other than uniform?
fR(r)
fR(a)da
fZ(a)da
fZ(z)
only
desired
histogram
is known
so, Q(r)=G-1(T(r))

38. Histogram Specification (cont’d)
fS(s) and fV(v) are uniform
G(z) can be computed by specifying fZ(z) but I2 and I’2 are unknown!
z=G-1(v) requires that v is a r.v. with uniform pdf
IDEA: use z=G-1(s) instead of z=G-1(v)
s is a r.v. with uniform pdf
The desired transformation is
z=G-1(T(r))
fR(a)da
fz(a)da

39. Histogram Specification (cont’d)
Comments
We do not need to apply T( ) and G-1( ) separately!
Combine them: Q=G-1T, thus, z=Q(r)
Histogram specification assumes that we know G-1 (not always easy to find).
G( ) must be in [0,1] and must be non-decreasing.
z=G-1(T(r))=Q(r)

40. Histogram Specification Example
3 bit
64 x 64 image
input histogram
specified histogram
actual histogram

41. Histogram Specification (cont’d)
Histogram specification might yield superior results than histogram equalization.
results of histogram equalization

42. Histogram Specification (cont’d)
results of histogram specification
G(z)
G-1(z)

43. Local Histogram Processing
Histogram equalization/specification are global methods.
The intensity transformation is computed using pixels from the entire image.
Global transformations are not appropriate for enhancing little details in an image.
The number of pixels in these areas might be very small, contributing very little to the computation of the transformation.

44. Local Histogram Processing
Idea:
Define a transformation function based on the intensity distribution in a neighborhood of every pixel in the image!

45. Local Histogram Processing (cont’d)
1. Define a neighborhood and move its center from pixel to pixel.
2. At each location, the histogram of the points in the neighborhood is computed. Obtain histogram equalization or histogram specification transformation.
3. Map the intensity of the pixel centered in the neighborhood.
4. Move to the next location and repeat the procedure.

46. Local Histogram Processing: Example
equalization
3 x 3 neighborhood
global histogram
equalization

47. Histogram Statistics Mean (average intensity) n-th moment around mean
Variance
(2nd moment)

48. Example: Comparison of Standard Deviation Values
σ is useful for estimating image contrast!

49. Local Histogram Statistics

50. Using Histogram Statistics for Image Enhancement
Useful when parts of the image might contain hidden features.
Task: enhance dark
areas without changing
bright areas.
Idea: Find dark, low contrast
areas using local statistics.

51. Using Histogram Statistics for Image Enhancement: Example

52. Question
Intensity operations can yield pixel values outside of the range [0 – 255].
You should convert values back to the range [0 – 255] to ensure that the image is displayed properly.
How would you find the following mapping?
[fmin – fmax]  [ 0 – 255]