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Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr.ir . Marcel Breeuwer dr. Anna Vilanova

Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr.ir . Marcel Breeuwer dr. Anna Vilanova. Histogram equalization. Today. Definition of histogram Examples Histogram equalization: Continuous case Discrete case Examples Histogram features. Histogram definition.

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Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr.ir . Marcel Breeuwer dr. Anna Vilanova

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

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

  3. 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

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

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

  6. Bimodal histogram

  7. Tri- (or more) modal histogram

  8. Example histograms

  9. Questions? Any questions so far?

  10. Histogram processing

  11. Histogram processing

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

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

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

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

  16. 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 distributionfunction is monotonically increasing, ① ok!

  17. 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)

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

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

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

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

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

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

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

  25. Histogram features • Mean and variance can be used for local histogram processing… (see example 3.12 in Gonzalez and Woods)

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

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