Chapter 9 Basic Signal Processing - cs.princeton.edu

1 Chapter9 BasicSignalProcessingMotivationManyaspec tsofcomputergraphicsandcomputerimagerydi fferfromaspectsofconventionalgraphicsand imagerybecausecomputerrepresentationsare digitalanddiscrete, previouslecturewediscussedtheimplication sofquantizingcontinuousorhighprecisionin tensityval-uesto continuousimageat a discretesetoflocations(usuallya regularlattice).Theimplicationsofthesamp lingprocessarequitesubtle,andtounderstan dthemfullyrequiresa framebufferholdsa saythatthediscreteimageisreconstructedto forma is oftenconvenienttothinkofeach2 Dpixelasa littlesquarethatabutsitsneighborstofillt heimageplane,thisviewofreconstructionis is bettertothinkofeachpixelasa surfacewhoseheightat a pointis singlesampleis thena spike.

2 Thespikeis locatedat thepositionofthesampleanditsheightis a setofspikes,andthecontinuousimageis a :A continuousimagereconstructedfroma discreteimagerepresentedasa ,theimageis drawnasa surfacewhoseheightis canmakea ,eachsamplerecordsthevalueoftheimageinte nsityat a CCDcamerarecordsimagevaluesbyturningligh tenergyintoelectricalenergy. Thelightsensitiveareaconsistofanarrayofs mallcells;eachcellproducesa singlevalue,andhence, theresultofallthelightfallingona singlecell,andcorrespondstoanintegralofa llthelightwithina smallsolidangle( ).Youreyeissimilar, eachsampleresultsfromtheactionofa singlephotoreceptor. However, justlikeCCDcells,photoreceptorcellsarepa ckedtogetherin yourretinaandintegrateovera mayseemlikethefactthatanindividualcellof aCCDcamera,orofyourretina,samplesoverana reais lessthanideal, producedbya rasterscanprocessinwhichthebeamsmovescon tinuouslyfromlefttoright, ,in television,theimageis continuousin reconstructionandsamplingleadsto aninterestingques-tion:Is it possibleto sampleanimageandthenreconstructit withoutanydistortion?

3 Jaggies,AliasingSimilarly, ,wecanconverta polygonto :A ,whena sampleis takenthelightis averagedovera pointis :forexample,inraytracing, , thesamplingprocessis illus-tratedwhena polygonorcheckerboardis polygonis notperfectlystraight,butinsteadisapproxi matedbya to samplea zoneplateasshownin seriesofconcentricrings;astheringsmoveou twardradiallyfromtheircenter, , wecandescribetheidealimageofa zoneplatebythesimpleformula: . If wesamplethezoneplate(tosampleanimagegive nbya formula at a pointis veryeasy;wesimplypluginthecoordinatesoft hepointintothefunction ), ratherthanseea singlesetofconcentricrings, strikingMoire :Whatcausesannoyingartifactssuchasjaggie sandmoirepatterns?Howcantheybeprevented? , :A raytracedimageofa shownat veryusefultoolin , wasacquired(forexample,if thecamerawasmoving)andit (orimages)canbecleverlycombinedintoa singlesignal, importantintelevision,wheredifferentcolo rimagesarecombinedtoforma singlesignalwhichis s beginwitha mathematicalfact:Anyperiodicfunction(exc eptvariousmon-strositiesthatwillnotconce rnus)canalwaysbewrittenasa :Samplingtheequation.

4 Ratherthana singlesetofringscenteredattheorigin,noti cethereareseveralsetsofsuperimposedrings beatingagainsteachothertoforma periodicfunctionis a functiondefinedinaninterval ! isperhapsthesimplestperiodicfunc-tionand hasanintervalequalto"$#. It is easyto seethatthesinefunctionis periodicsince % "$# & ' ( . Sinescanhaveotherfrequencies,forexample, thesinefunction ")# repeatsitself timesintheintervalfrom*to"$#. is thefre-quencyofthesinefunctionandis wecouldrepresenta periodicfunctionwitha sumofsinewaveseachofwhoseperiodswereharm onicsoftheperiodoftheoriginalfunction,th entheresultingsumwillalsobeperiodic(sinc eallthesinesareperiodic).Theabovemathema ticalfactsaysthatsucha thatwearefreetochoosethecoefficientsofea chsineofa differentfrequency, ,considera rathernastyfunction is discontinuousin valueandderivativeat squarepulseis thelimitas+.

5 Of/ 0 1 2 3 4" "#05687:9 8;4 6 < 9)= > ")?;4 A@ 4 4" "# =B> C@ E4F= > F@& G E4H= > H@I J LK K KM Wheretheangularfrequency(inradians)@L ")# . A plotofthisformulaforfourdifferentvalueso f+is +increases, , a non-periodicfunctioncanalsoberepresented asa sumofsin sandcos s, butnowwemustuseallfrequencies, replacedbyanintegral. 1 N O 4"$#PRQ<QTS U@V XW Y[Z]\_^]@whereWY`Z \ = > @& % bac @ 4 .S f@V arethecoefficientsofeachsineandcosine;S f@V is calledthespectrumofthefunction 1 2 .Thespectrumcanbecomputedfroma U@V O PRQ<Q 1 N XW<Y`Z \ ^c Unfortunately, wedonothavetimeto derivetheseformulas;thereaderwillhavetoa cceptthemas theirderivation,wereferyouto / :Fourapproximationstoa , theedgeofthepulse;thisis illustratethemathematicsoftheFouriertran sform,letuscalculatetheFouriertransformo fa squarepulseis describedmathematicallyas 8icj k]lXmc 2 O on4p prq9 *p prs9 TheFouriertransformofthisfunctionis <Q XiCjtk]l8mr 2 W<Y`Z \u^c Pwvx<vxW<Y`Z \y^c W<Y[Z]\;(aU@pvx<vx WYvxZ.

6 ZW<YvxZ"a9 @ 9 @9 @ = Hereweintroducethe =functiondefinedtobe = { # # Notethat # equalszeroforallintegervaluesof , except ,thesituationis , carefulanalysisshowsthat =* , = + O on4+ **+w| *A plotofthesincfunctionis shownbelow. Noticethattheamplitudeoftheoscilla-tiondecreasesas is > !@& is twospikes,oneat;V@andtheotherat functionis anex-pansionofthefunctionin ,expandingeithera singlesineora singlecosinein ,however, thattheFouriertransformofa cosineis twopositivespikes,whereasFouriertransfor mofa sineis anevenfunction(= > 2;V@O} =B> C@O}) whereasthesineis anoddfunction( ;~@O}O e; constantfunctionis a singlespikeat constantfunctiondoesnotvaryintimeorspace ,andhence, , ,if weknowthetransformfromthespacedomainto thefrequencydomain, ,theFouriertransformofa singlespikeat deltafunctionhasthepropertythatit is zeroeverywhereexceptat theorigin.)

7 2 * | *Thevalueofthedeltafunctionis notreallydefined, ,PRQ 2 ^c := > @& , @& , 1 N , tk] u 2 , andW<\ :AnimageanditsFouriertransformOneimagine sa deltafunctionto bea squarepulseofunitareain sequenceofspikesconsistof a sequenceof spikes(a sequenceofspikesis sometimesreferredtoastheshahfunction). alsoturnsoutthattheFouriertransformofa Gaussianis equaltoa , ,ordcterm,is ,highfrequencycomponentscontributetofine detail,sharpedges, thatofabandlimitedfunction. A functionis ban-dlimitedif itsspectrumhasnofrequenciesabovesomemaxi mumfrequency. Saidanotherway, thespectraofa bandlimitedfunctionoccupiesa finiteintervaloffre-quencies, summarize:thekeypointofthissectionis thata function canbeeasilyconvertedfromthespacedomain(t hatis,a functionof ) = == :Perfectlow-pass(top),high-pass(middle), andband-pass(bottom)fil-ters.

8 (thatis,a function,albeita differentfunction,of@), ,a func-tioncanbeinterpretedin , signaloranimageinthiswayis calledfiltering. Mathemat-ically, thepropertiesoffiltersareeasiesttodescri beinthefrequencydomain. U@! I S U@! V - U@V Here, is thespectrumofthefilteredfunction,Sis thespectrumoftheoriginalfunction,and is thespectrumofthefilter. Thesymbol multipliedbythecor-respondingfrequencyco mponentofthefilterfunctiontocomputetheva lueoftheoutputfunctionat ;ahigh-passfilterattenuateslowfrequencie srelativeto highfrequencies;aband-passfilterpreserve sa rangeoffrequenciesrela-tiveto :Applicationofa low-passfilterto blurry. Thisis frequency. Thus,inthefrequencydomain,a low-passfilteris a squarepulse( ).Simi-larly, a perfecthigh-passfiltercompletelyremovesa llfrequenciesbelowthecut-offfrequency, anda filtered,theeffectis blurryimage( ).

9 Removinglowfrequenciesenhancesthehighfre quenciesandcreatesa sharperimagecontainingmostlyedgesandothe rrapidlychangingtextures( ).Thecutoff frequencyforthehighpassandthelowpassfilt eris 1, ,if thepicturesintherighthandcolumnareaddedt ogether, theoriginalpictureinthelefthandcolumnis , itis ,filteringis achievedbysimplymultplyingspec-tra, ,filteringis achievedbya morecomplicatedoperationcalledconvolutio n. O R PRQ<Q 1 2 ;w 2 ^c :Applicationof a high-passfilterto theresultingimagethelowfrequencieshavebe enremovedandonlyplacesin theimagethatarechanging,suchasedges, :Forward:Slidethefilter alongtheaxis definingtheinputfunction . Ateachposition , multiplythefunction bythevalue ). Thescaledandtrans-latedfunction 2 1 J;b 2 is thenaccumulatedinto andtheprocesspro-ceedsbysliding , is sometimesreferredto asforwardconvolutionbecauseeachsingleinp utvalue 2 mapsforwardtoseveraloutputvalues w 2.

10 Backward:Slidethefilterg alongtheaxis definingtheoutputfunction . Now,foreachvalueof , multiplyit bythecorrespondingvalueof thenwrittenas .Inthisviewofconvolution, is sometimesreferredto asbackwardconvolutionbecauseeachsin-gleo utputvalue 2 is computedbymappingbackwardsintoseveralinp utvalues .To illustrateconvolution,supposetheinputfun ctionconsistsofa ,wecenterthefilterfunctionat theinputis a singlespikeat theorigin,thentheinputfunctionis zeroeverywhereexceptat ,thefilteris multipliedbyzeroeverywhereexceptattheori ginwhereit is ,theresultofconvolvingthefilterbya deltafunctionis , <Q 2 ;w N ^c sucha processis drivena deltafunction,orimpulse, , a veryimportantideasoletusconsideranothere xample ,theonecorrespondingtotheinputsignal,is ,representingthefilter, SquareSinc Triangle= Square Square = Cubic = :Theresultsofconvolvinga , ,beginstoincreaselinearlywhentheyfirstto uch,reachesa maximumwhentheyaresuperimposed,andthenbe ginsto decreaseuntiltheyarejusttouching,afterwh ichit a functionoranimagewitha squarepulseis ,noticethatthiscanbeinterpretedassetting theoutputtotheaverageoftheinputfunctiono vertheareawherethepulseis !