1.FourierTransformproblem.Foranimagegivenbythefunctionf(x,y)=(x+y)3wherex,yarecontinuousvaribales;evaluatef(x,y)δ(x-1,y-2)andf(x,y)*δ(x-1,y-2),whereδistheDoracDeltafunction.Fortheopticalimagingsystemshoenbelow,consistingofanimagescalingandtwoforwardFouriertransformsshowthattheoutputimageisascaleandinvertedreplicaoftheoriginalimagef(x,y).f(x,y)Scalingf(ax,by)FFg(x,y)_3)threebinaryimages(withvalue1onblackareasandvalue0elsewhere)areshownbelow.Sketchthecontinuous2DFToftheseimages(don’tdothismathematically,trytouseinsteadtheconvolutiontheoremandknowledgeofFTsofcommonfunctions)2.TheratedistortionfunctionofazeromemoryGaussiansourceofarbitarymeanandvarianceσ2withrespecttothemean-squareerrorcriterionisPlotthisfunctionWhatisDmaxIfadistortionofnomorthan75%ofthesource’svarianceisallowed,whatisthemaximumcompressionthatcanbeachieved?3.ThePDFofanimageisgivenbyPr(r)asshownbelow.Findthetransformtoconverttheimage'sPDFtoPr(z).Assumecontinuity,andfindthetransformintermsofrandz.Explainthetransformation.4.Acertaininspectionapplicationgathersblack&whiteimagesofpartsastheytravelalongaconveyorbelt.Itisnecessarytosortthepartsintotwocategories:partswithholesandpartswith-outholes.Anexampleofanimagethatmightbetakenbytheinspectioncameraisshownattheright.Proposeamethodtoidentifyandlocatetheobjectsofeachcategoryintheimagesothattheycanbepickedupbyaroboticsystemandplacedindifferentbins.Assumethattheimagingsystemknowswhereeachimagepixelislocatedontheconveyorbeltateverypointintime.Provideanannotatedflowchartofthealgorithmyoupropose.5.Inagivenapplication,anaveragingmaskisappliedtoinputimagestoreducenoiseandthenaLaplacianmaskisappliedtoenhancesmalldetails.Wouldmathematicspredictthattheresultshouldbethesameiftheorderoftheoperationswerereversed?Whatpracticalissueswouldbeencounteredincomputerimplementation?DigitalImageProcessingExamination1.Apreprocessingstepinanapplicationofmicroscopyisconcernedwiththeissueofisolatingindividualroundparticlesfromsimilarparticlesthatoverlapingroupsoft