Download - Histogram equalization
Basis beeldverwerking (8D040)
dr. Andrea FusterProf.dr. Bart ter Haar RomenyProf.dr.ir. Marcel Breeuwerdr. Anna Vilanova
Histogram equalization
Contact
• dr. Andrea Fuster – [email protected]• Mathematical image analysis at W&I and Biomedical
image analysis at BMT • HG 8.84 / GEM-Z 3.108
Today
• Definition of histogram • Examples • Histogram features• Histogram equalization:
• Continuous case• Discrete case
• Examples
Histogram definition
• Histogram is a discrete function h(rk) = N(rk) , where
• rk is 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
rk
Histogram
MN: total number of pixels (image of dimensions MxN)
What do the histograms of these images look like?
Bimodal histogram
Tri- (or more) modal histogram
Example histograms
More examples histograms
More examples histograms
• Mean
• Variance
Histogram Features
Mean: image mean intensity, measure of brightnessVariance: measure of contrast
Questions?
• Any questions so far?
Histogram processing
Histogram processing
Histogram equalization
• Idea: spread the intensity values to cover the whole gray scale
• Result: improved/increased contrast!☺
Histogram equalization – cont. case
• Assume r is 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
Histogram equalization – cont. case
• Assumptions / conditions:• ① is monotonically increasing function in • ②
• 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 satisfied?②
• Cumulative distribution function (CDF)• Probability density function (PDF) p is always non-
negative• This means the cumulative probability function is
monotonically increasing, ok!①
Histogram equalization – cont. case
• Does the CDF fit the second assumption?
•
• To have the same intensity range as the input image, scale with (L-1)
So ② ok!
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:
Histogram equalization – disc. case
Notice something??
Demo of equalization in Mathematica
Original image
Original histogram
Transformation function T(r)
“Equalised” image
“Equalised” histogram
End of part 1
• And now we deserve a break!