1. Intensity Transformations
Sections
Digital Image Processing
Gonzales and Woods
Irina Rabaev

2. Representing digital image
value f(x,y) at each x, y is called intensity level or gray level

4. Intensity Transformations and Filters
g(x,y)=T[f(x,y)]
f(x,y) – input image,
g(x,y) – output image
T is an operator on f defined over a neighborhood of point (x,y)

5. Intensity Transformation
1 x 1 is the smallest possible neighborhood.
In this case g depends only on value of f at a single point (x,y) and we call T an intensity (gray-level mapping) transformation and write
s = T(r)
where r and s denotes respectively the intensity of g and f at any point (x, y).

6. Some Intensity Transformation Functions

7. Image Negatives Denote [0, L-1] intensity levels of the image.
Image negative is obtained by s= L-1-r

8. Log Transformations s = clog(1+r), c – const, r ≥ 0
Maps a narrow range of low intensity values in the input into a wider range of
output levels. The opposite is true for higher values of input levels.

9. Power–Law (Gamma) transformation
s = crγ, c,γ –positive constants
curve the grayscale components either to brighten the intensity (when γ < 1)
or darken the intensity (when γ > 1).

10. Power –Law (Gamma) transformation

11. Power –Law (Gamma) transformation

12. Contrast stretching
Contrast stretching is a process that expands the range of intensity levels in a image so that it spans the full intensity range of the recording medium or display device. Contrast-stretching transformations increase the contrast between the darks and the lights

13. Thresholding function

14. Intensity-level slicing
Highlighting a specific range of gray levels in an image

15. Histogram processing
The histogram of a digital image with gray levels in the range [0, L-1] is a discrete function h(rk)=nk , where rk is the kth gray level and nk is the number of pixels in the image having gray level rk. It is common practice to normalize a histogram by dividing each of its values by the total number of pixels in the image, denoted by the product MN. Thus, a normalized histogram is given by h(rk)=nk/MN The sum of all components of a normalized histogram is equal to 1.

16. Histogram Equalization
Histogram equalization can be used to improve the visual appearance of an image.
Histogram equalization automatically determines a transformation function that produce and output image that has a near uniform histogram

18. Histogram Equalization
Let rk, k[0..L-1] be intensity levels and let p(rk) be its normalized histogram function.
The intensity transformation function for histogram equalization is

19. Histogram Equalization - Example
Let f be an image with size 64x64 pixels and L=8 and let f has the intensity distribution as shown in the table
p r(rk )=nk/MN nk rk
0.19
790
0.25
1023 2
0.21
850 1
0.16
656 3
0.08
329 4
0.06
245 5
0.03
122 6
0.02 81 7
round the values to the nearest integer

20. Local histogram Processing
Define a neighborhood and move its center from pixel to pixel. At each location, the histogram of the points in the neighborhood is computed and histogram equalization transformation is obtained.

21. Using Histogram Statistics for Image Enhancement
Denote:
ri – intencity value in the range [0, L-1],
p(i) - histogram component corresponding to value ri .
The intensity variance:

22. Using Histogram Statistics for Image Enhancement
Let (x, y) be the coordinates of a pixel in an image, and let Sxy denote a
neighborhood (subimage) of specified size, centered at (x, y).
The mean value of the pixels in this neighborhood is given by
where is the histogram of the pixels in region Sxy.
The variance of the pixels in the neighborhood is given by

23. Using Histogram Statistics for Image Enhancement
Tungsten filament

24. Using Histogram Statistics for Image Enhancement

25. Using Histogram Statistics for Image Enhancement