Histograms show the distribution of pixel intensities in an image by counting the number of pixels for each intensity value. Normalized histograms provide an estimate of the probability of each intensity occurring. Histogram equalization transforms the pixel intensity distribution of an image to a uniform distribution in order to increase contrast. It does this by using the cumulative distribution function to map intensities to new output values. Local histogram equalization performs this on neighborhoods within an image to enhance local details. Arithmetic and logical operations can also be used for image enhancement, such as AND, OR, and subtraction between images on a pixel-by-pixel basis.

1. Histogram Processing : 1
HistogramsHistograms
Histogram
Discrete function h(rk) showing the number of occurrences nk for
the kth
gray level rk
Normalized Histogram
Gives estimate of probability of occurrence of gray level rk
(probability distribution function)
( )k kh r n=
1
0
( ) / , ( ) 1
L
k k kp r n n p r
−
= =∑

2. Histogram Processing : 2
Histograms: ExampleHistograms: Example

3. Histogram Processing : 3
Histogram EqualizationHistogram Equalization
High-contrast images are often desirable from a visual perspective
High-contrast images have histograms where the components
cover a wide dynamic range
Distribution of pixels are close to a uniform distribution
Intuitively, low-contrast images can be enhanced by transforming
its pixel distribution into a uniform distribution to achieve high
contrast

4. Histogram Processing : 4
Histogram EqualizationHistogram Equalization
Let p(r) and p(s) denote the probability density functions of r and s
For s=T(r), p(s) can be expressed as,
Since we want to transform the input image such that the pixel
distribution is uniform,
( ) ( )
dr
p s p r
ds
=
( ) 1 , 0 1p s s= ≤ ≤

5. Histogram Processing : 5
Histogram EqualizationHistogram Equalization
What transformation will give you a uniform distribution for s?
Cumulative distribution function (CDF)
Why?
0
( ) ( )
r
s T r p w dw= = ∫
0
( )
( )
r
d p w dw
ds
p r
dr dr
 
  = =
∫
1
( ) ( ) 1
( )
p s p r
p r
= =

6. Histogram Processing : 6
Histogram EqualizationHistogram Equalization

7. Histogram Processing : 7
Histogram EqualizationHistogram Equalization
For discrete case,
1
0
( ) / , ( ) 1
L
k k k
k
p r n n p r
−
=
= =∑
( ) ( )
k
k k j
j o
s T r p r
=
= = ∑

8. Histogram Processing : 8
Histogram Equalization: ExampleHistogram Equalization: Example

9. Histogram Processing : 9
Local Histogram EqualizationLocal Histogram Equalization
Global approach good for overall contrast enhancement
However, there may be cases where it is necessary to enhance
details over small areas in image
Solution: perform histogram equalization over a small neighborhood

10. Histogram Processing : 10
Local Histogram Equalization:Local Histogram Equalization:
ExampleExample

11. Histogram Processing : 11
Enhancement Using Arithmetic/LogicEnhancement Using Arithmetic/Logic
OperationsOperations
Performed on a pixel-by-pixel basis between two or more images
( , ) ( , ) ( , )g x y f x y h x y= ⊕

12. Histogram Processing : 12
Enhancement Using AND Operation:Enhancement Using AND Operation:
ExampleExample

13. Histogram Processing : 13
Enhancement Using OR Operation:Enhancement Using OR Operation:
ExampleExample

14. Histogram Processing : 14
Enhancement Using SubtractionEnhancement Using Subtraction
Operation: ExampleOperation: Example

15. Histogram Processing : 15
Enhancement Using SubtractionEnhancement Using Subtraction
Operation: ExampleOperation: Example

16. Histogram Processing : 16
Noise reduction by Image AveragingNoise reduction by Image Averaging