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Slide1
Basis beeldverwerking (8D040)dr. Andrea FusterProf.dr. Bart ter Haar RomenyProf.dr.ir. Marcel Breeuwerdr. Anna Vilanova
Histogram equalization
Slide2Contact
d
r. 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
Slide3Today
Definition of histogram
Examples
Histogram features
Histogram equalization:
Continuous case
Discrete case
Examples
Slide4Histogram definition
Histogram is a discrete function
h
(
r
k
)
=
N(
r
k
)
,
where
r
k
is the
k-th
intensity value, and
N(
r
k
)
is the number of pixels with intensity
r
k
Histogram normalization by dividing
N(
r
k
)
by the number of pixels in the image (MN)
Normalization turns histogram into a
probability distribution function
Slide5r
k
Histogram
MN: total number of pixels (image of dimensions
MxN
)
Slide6What do the histograms of these images look like?
Slide7Bimodal histogram
Slide8Tri- (or more) modal histogram
Slide9Natural image histogram
Slide10Example histograms
Slide11More examples histograms
Slide12More examples histograms
Slide13MeanVariance
Histogram Features
Mean: image mean intensity, measure of brightness
Variance: measure of contrast
Slide14Questions?
Any questions so far?
Slide15Histogram processing
Slide16Histogram processing
Slide17Histogram equalization
Idea: spread
the intensity values to cover the whole gray scale
Result: improved/increased contrast!☺
Slide18Histogram equalization – cont. case
Assume r is the intensity in an image with L levels:Histogram equalisation is a mapping of the formwith r the input gray value and s the resulting or mapped value
Slide19Histogram equalization – cont. case
Assumptions / conditions:① is monotonically increasing function in ②Make sure output range equal to input range
Slide20Histogram equalization – cont. case
Monotonically increasing function T(r)
Slide21Histogram 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-negativeThis means the cumulative probability function is monotonically increasing, ① ok!
Slide22Histogram 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!
Slide23Histogram equalization – cont. case
What happens when we apply the transformation function T(r) to the intensity values? – how does the histogram change?
Slide24Histogram equalization – cont. case
What is the resulting probability distribution?From probability theory
Slide25Histogram equalization – cont. case
Uniform:
What does this mean?
Slide26Histogram equalization – disc. case
Spreads the intensity values to cover the whole gray scale (improved/increased contrast)Fully automatic method, very easy to implement:
Slide27Histogram
equalization – disc. case
Notice something??
Slide28Demo of equalization in Mathematica
Original image
Original histogram
Transformation function T(r)
“
Equalised” image
“
Equalised
” histogram
Slide29End of part 1
And now we deserve a break!
Slide30Slide31Slide32
Basis
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