### Histogram equalization - BioMedical Image Analysis

```Basis beeldverwerking (8D040)
dr. Andrea Fuster
Prof.dr. Bart ter Haar Romeny
Prof.dr.ir. Marcel Breeuwer
dr. Anna Vilanova
Histogram equalization
Contact
• dr. Andrea Fuster – [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */
• Mathematical image analysis at W&I and Biomedical
image analysis at BMT
• HG 8.84 / GEM-Z 3.108
Today
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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
Histogram
MN: total number of pixels
(image of dimensions MxN)
rk
What do the histograms of these
images look like?
Bimodal histogram
Tri- (or more) modal histogram
Example histograms
More examples histograms
More examples histograms
Histogram Features
• Mean
• Variance
Mean: image mean intensity, measure of brightness
Variance: 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 ② satisfied?
• Cumulative distribution function (CDF)
• Probability density function (PDF) p is always nonnegative
• This means the cumulative probability function is
monotonically increasing, ① ok!
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!
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