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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

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

Histogram equalization

today
Today
  • Definition of histogram
  • Examples
  • Histogram equalization:
    • Continuous case
    • Discrete case
  • Examples
  • Histogram features
histogram definition
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
histogram
Histogram

MN: total number of pixels (image of dimensions MxN)

rk

questions
Questions?

Any questions so far?

histogram equalization
Histogram equalization

Idea: spread the intensity values to cover the whole gray scale

Result: improved/increased contrast!☺

histogram equalization cont case
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

histogram equalization cont case1
Histogram equalization – cont. case
  • Assumptions / conditions:
    • ① is monotonically increasing function in
    • Make sure output range equal to input range
histogram equalization cont case2
Histogram equalization – cont. case

Monotonically increasing function T(r)

histogram equalization cont case3
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!
histogram equalization cont case4
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)

histogram equalization cont case5
Histogram equalization – cont. case

What happens when we apply the transformation function T(r) to the intensity values? – how does the histogram change?

histogram equalization cont case6
Histogram equalization – cont. case

What is the resulting probability distribution?

From probability theory

histogram equalization cont case7
Histogram equalization – cont. case

Uniform:

What does this mean?

histogram equalization disc case
Histogram equalization – disc. case

Spreads the intensity values to cover the whole gray scale (improved/increased contrast)

Fully automatic method, very easy to implement:

demo of equalization in mathematica
Demo of equalization in Mathematica

Original image

Original histogram

Transformation function T(r)

“Equalized” image

“Equalized” histogram

slide24

Histogram Features

  • Mean
  • Variance

Mean: image mean intensity, measure of brightness

Variance: measure of contrast

histogram features
Histogram features
  • Mean and variance can be used for local histogram processing… (see example 3.12 in Gonzalez and Woods)
end of part 1
End of part 1

And now we deserve a break!