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

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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 FusterProf.dr.ir. Marcel Breeuwerdr. Anna Vilanova

Histogram equalization

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

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

rk

Questions?

Any questions so far?

Histogram equalization

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

Result: improved/increased contrast!☺

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

Monotonically increasing function T(r)

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. 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. case

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

Histogram equalization – cont. case

What is the resulting probability distribution?

From probability theory

Histogram equalization – cont. case

Uniform:

What does this mean?

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

Original image

Original histogram

Transformation function T(r)

“Equalized” image

“Equalized” histogram

Histogram Features

• Mean
• Variance

Mean: image mean intensity, measure of brightness

Variance: measure of contrast

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

And now we deserve a break!