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Histogram equalization

Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr . Bart ter Haar Romeny Prof.dr.ir . Marcel Breeuwer dr. Anna Vilanova. Histogram equalization. Contact. d r. Andrea Fuster – A.Fuster@tue.nl Mathematical image analysis at W&I and Biomedical image analysis at BMT

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Histogram equalization

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  1. Basis beeldverwerking (8D040)dr. Andrea FusterProf.dr. Bart terHaarRomenyProf.dr.ir. Marcel Breeuwerdr. Anna Vilanova Histogram equalization

  2. Contact dr. Andrea Fuster – A.Fuster@tue.nl Mathematical image analysis at W&I and Biomedical image analysis at BMT HG 8.84 / GEM-Z 3.108

  3. Today • Definition of histogram • Examples • Histogram features • Histogram equalization: • Continuous case • Discrete case • Examples

  4. Histogram definition • Histogram is a discrete function h(rk) = N(rk), where • rkis the k-th intensity value, and • N(rk)is the number of pixels with intensity rk • Histogram normalization by dividing N(rk)by the number of pixels in the image (MN) • Normalization turns histogram into a probability distribution function

  5. Histogram MN: total number of pixels (image of dimensions MxN) rk

  6. What do the histograms of these images look like?

  7. Bimodal histogram

  8. Tri- (or more) modal histogram

  9. Example histograms

  10. More examples histograms

  11. More examples histograms

  12. Histogram Features • Mean • Variance Mean: image mean intensity, measure of brightness Variance: measure of contrast

  13. Questions? Any questions so far?

  14. Histogram processing

  15. Histogram processing

  16. Histogram equalization Idea: spread the intensity values to cover the whole gray scale Result: improved/increased contrast!☺

  17. Histogram equalization – cont. case Assume ris the intensity in an image with L levels: Histogram equalisation is a mapping of the form with r the input gray value and s the resulting or mapped value

  18. Histogram equalization – cont. case • Assumptions / conditions: • ① is monotonically increasing function in • ② • Make sure output range equal to input range

  19. Histogram equalization – cont. case Monotonically increasing function T(r)

  20. Histogram equalization – cont. case • Consider a candidate function for T(r) – conditions ① and ② satisfied? • Cumulative distribution function (CDF) • Probability density function (PDF) p is always non-negative • This means the cumulative probability function is monotonically increasing, ① ok!

  21. Histogram equalization – cont. case So ② ok! Does the CDF fit the second assumption? To have the same intensity range as the input image, scale with (L-1)

  22. Histogram equalization – cont. case What happens when we apply the transformation function T(r) to the intensity values? – how does the histogram change?

  23. Histogram equalization – cont. case What is the resulting probability distribution? From probability theory

  24. Histogram equalization – cont. case Uniform: What does this mean?

  25. Histogram equalization – disc. case Spreads the intensity values to cover the whole gray scale (improved/increased contrast) Fully automatic method, very easy to implement:

  26. Histogram equalization – disc. case Notice something??

  27. Demo of equalization in Mathematica Original image Original histogram Transformation function T(r) “Equalised” image “Equalised” histogram

  28. End of part 1 And now we deserve a break!

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