1. Digital Image Processing Contrast Enhancement: Part II

2. Histogram Processing
Histogram : is the discrete function h(rk)=nk , where rk is the kth gray level in the range of [0, L-1] and nk is the number of pixels having gray level rk.
Normalized histogram : is p(rk)=nk/n, for k=0,1,…,L-1 and p(rk) can be considered to give an estimate of the probability of occurrence of ray level rk.

3. Histogram Equalization
Histogram equalization : is a method which increases the dynamic range of the gray-levels in a low-contrast image to cover full range of gray-levels.
How-to-Do: is achieved by having a transformation function which is the Cumulative Distribution Function (CDF) of a given PDF of gray-levels in a given image.

4. Histogram Equalization
Histogram equalization : the new intensity value of pixel x is calculated by:

5. Histogram Equalization
Histogram equalization : the probability function of the output levels is uniform.
Note : the transformation function is simply the CDF.

6. Histogram Equalization Histogram Equalization

7. Histogram Equalization
(a) A face image from the CALTECH face database, (b) its histogram, (c) the equalized face image using HE, (d) and its respective histogram.

8. Singular Value Equalization
Singular value decomposition : any matrix, A, can be written as multiplication of two orthogonal square matrices, U and V, and a matrix containing the sorted singular values on its main diagonal, Σ.
A=UΣVT

9. Singular Value Equalization
Note : as σ1 is much bigger than other σs then changing it will affect on the reconstructed image, i.e. changing σ1 will directly change the luminance of the image.

10. Singular Value Equalization
G(0.5, 1) : is a synthetic intensity matrix whose pixel values have Gaussian distribution with mean of 0.5 and variance of 1 with the same size of the original image.
ξ: is ratio of the largest singular value of the generated normalized matrix over a normalized image.

11. Singular Value Equalization

12. Singular Value Equalization Singular value equalization
Histogram equalization
Singular value equalization
Low contrast

13. Summary We have looked at:
How histogram equalization works.
What is SVD?
How SVE works
Next time we will continue our talk about image enhancement in spatial domain