1. Intensity Transformations and Spatial Filtering

2. Basics of Intensity Transformation and Spatial Filtering
Spatial Domain Process
Neighborhood is rectangle, centered on (x,y), and much smaller in size than image.
Neighborhood size is 1x1, 3x3, 5x5, etc.

3. Intensity Transformation
T[f(x,y)] is Intensity Transformation, if the neighborhood size is 1x1.
Intensity Transformation can be written as follows
s = T[r],
where s = g(x,y), and r = f(x,y)

4. Image Negatives s = L-1 – r where intensity level is in the range

5. Log Transformations s = c Log(1+r)
Log Transformation is used to expand the value of the dark pixels while compressing the higher-level value.
It is used to compress the intensity of an image which has very large dynamic range.

6. Log Transformations of Fourier Spectrum
Original Image
Fourier Spectrum
Log Transform of Fourier Spectrum
We cannot see the Fourier spectrum, because its dynamic range is very large.

7. Power-Law (Gamma) Transformations
If <1, expand dark pixels, compress bright pixels.
If >1, compress dark pixels, expand bright pixels.

8. Examples of Gamma Transformations

9. Contrast Stretching If r<r1 then s = r*s1/r1
If r1<= r<=r2 then
s = (r-r1)*(s2-s1)/(r2-r1)+s1
If r>r2 then
s = (r-r2)*(255-s2)/(255-r2)+s2
If r1=r2 and s1=0,s2=255, the transform is called “Threshold Function”.

10. Examples of Contrast Stretching

11. Contrast Stretching in Medical Image
Window Width/Level(Center)
s1=0,s2=255
width (w)=r2-r1, level (c)=(r1+r2)/2

12. Histogram & PDF h(r) = nr
where nr is the number of pixels whose intensity is r.
The Probability Density Function (PDF)

13. Cumulative Distribution Function (CDF)
PDF CDF
Transfer Function s r

14. Example of Histogram and Cumulative Distribution Function (CDF)

15. Low Contrast Image
The image is highly concentrated on low intensity values.
The low contrast image is the image which is highly concentrated on a narrow histogram.
High
Concentrate
Low
Concentrate

16. Histogram Equalization
The Histogram Equalization is a method which makes the histogram of the image as smooth as possible

17. The PDF of the Transformed Variable
s = Transformed Variable.
= The PDF of r
= The PDF of s

18. Transformation Function of Histogram Equalization
The PDF of s

19. Histogram Equalization Example
Intensity
# pixels 20 1 5 2 25 3 10 4 15 6 7
Total
100
CDF of Pr
20/100 = 0.2
(20+5)/100 = 0.25
( )/100 = 0.5
( )/100 = 0.6
( )/100 = 0.75
( )/100 = 0.8
( )/100 = 0.9
( )/100 = 1.0
1.0

20. Histogram Equalization Example (cont.)
Intensity
(r)
No. of Pixels
(nj)
Acc Sum of Pr
Output value
Quantized Output (s) 20
0.2
0.2x7 = 1.4 1 5
0.25
0.25*7 = 1.75 2 25
0.5
0.5*7 = 3.5 3 10
0.6
0.6*7 = 4.2 4 15
0.75
0.75*7 = 5.25
0.8
0.8*7 = 5.6 6
0.9
0.9*7 = 6.3 7
1.0
1.0x7 = 7
Total
100

21. Histogram Matching
How to transform the variable r whose PDF is to the variable t whose PDF is
T( )
G-1( ) r s t