Histogram equalization

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Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr . Bart ter Haar Romeny Prof.dr.ir . Marcel Breeuwer dr. Anna Vilanova. Histogram equalization. Contact. d r. Andrea Fuster – A.Fuster@tue.nl Mathematical image analysis at W&I and Biomedical image analysis at BMT
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Basis beeldverwerking (8D040)dr. Andrea FusterProf.dr. Bart terHaarRomenyProf.dr.ir. Marcel Breeuwerdr. Anna VilanovaHistogram equalizationContactdr. Andrea Fuster – A.Fuster@tue.nlMathematical image analysis at W&I and Biomedical image analysis at BMT HG 8.84 / GEM-Z 3.108Today
  • 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
  • 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
  • HistogramMN: total number of pixels (image of dimensions MxN) rkWhat do the histograms of these images look like?Bimodal histogramTri- (or more) modal histogramExample histogramsMore examples histogramsMore examples histogramsHistogram Features
  • Mean
  • Variance
  • Mean: image mean intensity, measure of brightnessVariance: measure of contrastQuestions?Any questions so far?Histogram processing Histogram processing Histogram equalizationIdea: spread the intensity values to cover the whole gray scale Result: improved/increased contrast!☺Histogram equalization – cont. caseAssume ris 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 valueHistogram equalization – cont. case
  • Assumptions / conditions:
  • ① is monotonically increasing function in
  • Make sure output range equal to input range
  • Histogram equalization – cont. caseMonotonically 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 probability function is monotonically increasing, ① ok!
  • Histogram equalization – cont. caseSo 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. caseWhat happens when we apply the transformation function T(r) to the intensity values? – how does the histogram change? Histogram equalization – cont. caseWhat is the resulting probability distribution?From probability theoryHistogram equalization – cont. caseUniform:What does this mean?Histogram equalization – disc. caseSpreads the intensity values to cover the whole gray scale (improved/increased contrast)Fully automatic method, very easy to implement:Histogram equalization – disc. caseNotice something??Demo of equalization in MathematicaOriginal imageOriginal histogramTransformation function T(r)“Equalised” image“Equalised” histogramEnd of part 1 And now we deserve a break!
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