Let be a given image represented as a by matrix of integer pixel intensities ranging from 0 to 1 is the number of possible intensity values often 256 Let denote the normalized histogram of with a bin for each possible intensity So number of pixels w ID: 22043
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HistogramEqualizationHistogramequalizationisatechniqueforadjustingimageintensitiestoenhancecontrast.Letfbeagivenimagerepresentedasamrbymcmatrixofintegerpixelintensitiesrangingfrom0toL 1.Listhenumberofpossibleintensityvalues,often256.Letpdenotethenormalizedhistogramoffwithabinforeachpossibleintensity.Sopn=numberofpixelswithintensityn totalnumberofpixelsn=0;1;:::;L 1:Thehistogramequalizedimagegwillbede nedbygi;j=\roor((L 1)fi;jXn=0pn);(1)where\roor()roundsdowntothenearestinteger.Thisisequivalenttotransformingthepixelintensities,k,offbythefunctionT(k)=\roor((L 1)kXn=0pn):ThemotivationforthistransformationcomesfromthinkingoftheintensitiesoffandgascontinuousrandomvariablesX,Yon[0;L 1]withYde nedbyY=T(X)=(L 1)ZX0pX(x)dx;(2)wherepXistheprobabilitydensityfunctionoff.TisthecumulativedistributivefunctionofXmultipliedby(L 1).AssumeforsimplicitythatTisdi erentiableandinvertible.ItcanthenbeshownthatYde nedbyT(X)isuniformlydistributedon[0;L 1],namelythatpY(y)=1 L 1.Zy0pY(z)dz=probabilitythat0Yy=probabilitythat0XT 1(y)=ZT 1(y)0pX(w)dwd dyZy0pY(z)dz=pY(y)=pX(T 1(y))d dy(T 1(y)):1 original image 0 100 200 0 0.1 0.2 0.3 original histogram transformed image 0 100 200 0 0.1 0.2 0.3 transformed histogram Figure1:HistogramequalizationappliedtolowcontrastimageNotethatd dyT(T 1(y))=d dyy=1,sodT dxjx=T 1(y)d dy(T 1(y))=(L 1)pX(T 1(y))d dy(T 1(y))=1;whichmeanspY(y)=1 L 1.OurdiscretehistogramisanapproximationofpX(x)andthetransformationinEquation1approximatestheoneinEquation2.Whilethediscreteversionwon'tresultinexactly\rathistograms,itwill\rattenthemandindoingsoenhancethecontrastintheimage.TheresultofapplyingEquation1totheelvis low contrast.bmptestimageisshowninFigure1.MATLAB:Totesttheaccompanyingcode,hist eq.m,typeg=hist_eq('elvis_low_contrast.bmp');HistogramequalizationisalsobuiltintoMATLAB.Type2 helphisteqtoseehowitworks.Question:WhathappensifEquation1isappliedtwice?Reference:R.C.GonzalezandR.E.Woods,DigitalImageProcessing,ThirdEdition,2008.3