Chapter 4 - Frequency Domain Image Enhancement

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1、§4-FrequencyDomainImageEnhancementDr.DirkSchniedersDepartmentofComputerScienceTheUniversityofHongKongFilteringintheFrequencyDomain★TheFouriertransformandthefrequencydomaincanbeusedforimagefiltering★Youcandevelopasolidunderstandingofthistopicwithouthavingtobecomeas

2、ignalprocessingexpert★Ifyoufindyourselfhavingtroublefollowingthedefinitionsinthischapter★Focusonthefundamentalsandtheirrelevancetodigitalimageprocessing★Identifyconnectionsbetweenimagecharacteristicsandthemathematicaltoolsusedtorepresentthem2013-2014COMP75022Preli

3、minaryConcepts★Sineandcosinearetrigonometricfunctionsofanangle★Foragivenangle,cosineandsinegivetherespectivex,ycoordinatesonaunitcircle2013-2014COMP75023PreliminaryConcepts★Acomplexnumbercanbedefinedas★RandIarerealnumbersandjisanimaginarynumberequalto★Theconjugate

4、ofacomplexnumberCisdefinedas★Complexnumberscanbeviewedgeometricallyaspointsinacomplexplane2013-2014COMP75024PreliminaryConcepts★Apointinthecomplexplanecanberepresentedusingpolarcoordinateswhereisthelengthofthevectorand✓istheanglebetweenthevectorandtherealaxis★Usin

5、gEuler’sformulagivesthefollowingrepresentation✓ofacomplexnumberwhereandrepresentamplitudeandphase2013-2014COMP75025FourierSeries★Anyperiodicfunctioncanbeexpressedasthesumofsinesand/orcosinesofdifferentfrequencies,eachmultipliedbyadifferentcoefficient,andthisiscall

6、edtheFourierseries★Thefunctionatthebottomisthesumofthefourfunctionsaboveit2013-2014COMP75026Example1.1(b)1.1(c)0.60.60.10.1−0.4−0.41.1(a)−0.9−0.90.6−1.4−1.4012345601234560.1(d)(e)1.11.1−0.40.60.6−0.90.10.1−1.40123456−0.4−0.4−0.9−0.9−1.4−1.4012345601234562013-2014C

7、OMP75027Example1.1(f)1.1(g)0.60.60.10.1−0.4−0.41.1(a)−0.9−0.90.6−1.4−1.4012345601234560.11.1(h)1.1(i)−0.40.60.6−0.90.10.1−1.40123456−0.4−0.4−0.9−0.9−1.4−1.4012345601234562013-2014COMP75028FourierTransform★Non-periodicfunctionwhoseareaunderthecurveisfinitecanbeexpr

8、essedastheintegralofsinesand/orcosinesmultipliedbyaweightingfunction,andthisiscalledtheFouriertransform★AfunctionexpressedineitherFourierseriesorFourier