《数字图像处理》（第二版）英文版习题答案（全）

DigitalImageProcessing

SecondEdition

InstructorzsManual

RafaelC.Gonzalez

RichardE.Woods

PrenticeHall

UpperSaddleRiver,NJ07458

/gonzalezwoods

or

Revisionhistory

1098765432I

Copyright°c1992-

2002byRafaelC.

Gonzalezand

RichardE.Woods

Preface

ThismanualcontainsdetailedsolutionstoallproblemsinDigitalImageProcessing,2nd

Edition.Wealsoincludeasuggestedsetofguidelinesforusingthebook,anddiscuss

theuseofcomputerprojectsdesignedtopromoteadeeperunderstandingofthesubject

matter.Thenotationusedthroughoutthismanualcorrespondstothenotationusedin

thetext.

Thedecisionofwhatmaterialtocoverinacourserestswiththeinstructor,anditde­

pendsonthepurposeofthecourseandthebackgroundofthestudents.Wehavefound

thatthecourseoutlinessuggestedherecanbecoveredcomfortablyinthetimeframes

indicatedwhenthecourseisbeingtaughtinanelectricalengineeringorcomputersci­

encecurriculum.Ineachcase,nopriorexposuretoimageprocessingisassumed.We

givesuggestedguidelinesforone-semestercoursesattheseniorand®rst-yeargraduate

levels.Itispossibletocovermostofthebookinatwo-semestergraduatesequence.

Thebookwascompletelyrevisedinthisedition,withthepurposenotonlyofupdating

thematerial,butjustasimportant,makingthebookabetterteachingaid.Tothis

end,theinstructorwill®ndtheneworganizationtobemuchmore-exibleandbetter

illustrated.Althoughthebookisselfcontained,werecommenduseofthecompanion

website,wherethestudentwill®nddetailedsolutionstotheproblemsmarkedwitha

starinthetext,reviewmaterial,suggestedprojects,andimagesfromthebook.Oneof

theprincipalreasonsforcreatingthewebsitewastofreetheinstructorfromhavingto

preparematerialsandhandoutsbeyondwhatisrequiredtoteachfromthebook.

Computerprojectssuchasthosedescribedinthewebsiteareanimportantpartof

acourseonimageprocessing.Theseprojectsgivethestudenthands-onexperience

withalgorithmimplementationandreinforcethematerialcoveredintheclassroom.

Theprojectssuggestedatthewebsitecanbeimplementedonalmostanyreasonably-

equippedmulti-userorpersonalcomputerhavingahardcopyoutputdevice.

1Introduction

Thepurposeofthischapteristopresentsuggestedguidelinesforteachingmaterialfrom

thisbookattheseniorand@rst-yeargraduatelevel.Wealsodiscussuseofthebook

website.Althoughthebookistotallyself-contained,thewebsiteoffers,amongother

things,complementaryreviewmaterialandcomputerprojectsthatcanbeassignedin

conjunctionwithclassroomwork.Detailedsolutionstoallproblemsinthebookalso

areincludedintheremainingchaptersofthismanual.

TeachingFeaturesoftheBook

Undergraduateprogramsthatofferdigitalimageprocessingtypicallylimitcoverageto

onesemester.Graduateprogramsvary,andcanincludeoneortwosemestersofthema­

terial.Inthefollowingdiscussionwegivegeneralguidelinesforaone-semestersenior

course,aone-semestergraduatecourse,andafull-yearcourseofstudycoveringtwo

semesters.Weassumea15-weekprogrampersemesterwiththreelecturesperweek.

Inordertoprovide-exibiiityforexamsandreviewsessions,theguidelinesdiscussed

inthefollowingsectionsarebasedonforty,50-minutelecturespersemester.Theback­

groundassumedonthepartofthestudentissenior-levelpreparationinmathematical

analysis,matrixtheory,probability,andcomputerprogramming.

Thesuggestedteachingguidelinesarepresentedintermsofgeneralobjectives,andnot

astimeschedules.Thereissomuchvarietyinthewayimageprocessingmaterialis

taughtthatitmakeslittlesensetoattemptabreakdownofthematerialbyclassperiod.

Inparticular,theorganizationofthepresenteditionofthebookissuchthatitmakesit

mucheasierthanbeforetoadoptsigni@cantlydifferentteachingstrategies,depending

oncourseobjectivesandstudentbackground.Forexample,itispossiblewiththenew

organizationtoofferacoursethatemphasizesspatialtechniquesandcoverslittleorno

transformmaterial.Thisisnotsomethingwerecommend,butitisanoptionthatoften

isattractiveinprogramsthatplacelittleemphasisonthesignalprocessingaspectsofthe

®eldandprefertofocusmoreontheimplementationofspatialtechniques.

2Chapter1Introduction

Thecompanionwebsite

www:prenhall:com=gonzalezwoods

or

www:imageprocessingbook:com

isavaluableteachingaid,inthesensethatitincludesmaterialthatpreviouslywascov­

eredinclass.Inparticular,thereviewmaterialonprobability,matrices,vectors,and

linearsystems,waspreparedusingthesamenotationasinthebook,andisfocusedon

areasthataredirectlyrelevanttodiscussionsinthetext.Thisallowstheinstructorto

assignthematerialasindependentreading,andspendnomorethanonetotallecturepe­

riodreviewingthosesubjects.Anothermajorfeatureisthesetofsolutionstoproblems

markedwithastarinthebook.Thesesolutionsarequitedetailed,andwereprepared

withtheideaofusingthemasteachingsupport.Theon-lineavailabilityofprojects

anddigitalimagesfreestheinstructorfromhavingtoprepareexperiments,data,and

handoutsforstudents.Thefactthatmostoftheimagesinthebookareavailablefor

downloadingfurtherenhancesthevalueofthewebsiteasateachingresource.

OneSemesterSeniorCourse

Abasicstrategyinteachingaseniorcourseistofocusonaspectsofimageprocessingin

whichboththeinputsandoutputsofthoseprocessesareimages.Inthescopeofasenior

course,thisusuallymeansthematerialcontainedinChapters1through6.Depending

oninstructorpreferences,wavelets(Chapter7)usuallyarebeyondthescopeofcoverage

inatypicalseniorcurriculum).However,werecommendcoveringatleastsomematerial

onimagecompression(Chapter8)asoutlinedbelow.

Wehavefoundinmorethantwodecadesofteachingthismaterialtoseniorsinelectrical

engineering,computerscience,andothertechnicaldisciplines,thatoneofthekeysto

successistospendatleastonelectureonmotivationandtheequivalentofonelecture

onreviewofbackgroundmaterial,astheneedarises.Themotivationalmaterialis

providedinthenumerousapplicationareasdiscussedinChapterI.Thischapterwas

totallyrewrittenwiththisobjectiveinmind.Someofthismaterialcanbecoveredin

classandtherestassignedasindependentreading.Backgroundreviewshouldcover

probabilitytheory(ofonerandomvariable)beforehistogramprocessing(Section3.3).

Abriefreviewofvectorsandmatricesmayberequiredlater,dependingonthematerial

covered.Thereviewmaterialincludedinthebookwebsitewasdesignedforjustthis

purpose.

OneSemesterSeniorCourse3

Chapter2shouldbecoveredinitsentirety.Someofthematerial(suchaspartsof

Sections2.1and2.3)canbeassignedasindependentreading,butadetailedexplanation

ofSections2.4through2.6istimewellspent.

Chapter3servestwoprincipalpurposes.Itcoversimageenhancement(atopicofsignif­

icantappealtothebeginningstudent)anditintroducesahostofbasicspatialprocessing

toolsusedthroughoutthebook.Foraseniorcourse,werecommendcoverageofSec­

tions3.2.1through3.2.2uSection3.3.luSection3.4uSection3.5uSection3.6uSection

3.7.1,3.7.2(throughExample3.11),and3.7.3.Section3.8canbeassignedasindepen­

dentreading,dependingontime.

Chapter4alsodiscussesenhancement,butfromafrequency-domainpointofview.The

instructorhassigni®cant-exibilityhere.Asmentionedearlier,itispossibletoskip

thechapteraltogether,butthiswilltypicallyprecludemeaningfulcoverageofother

areasbasedontheFouriertransform(suchas©Iteringandrestoration).Thekeyin

coveringthefrequencydomainistogettotheconvolutiontheoremandthusdevelop

atiebetweenthefrequencyandspatialdomains.Allthismaterialispresentedinvery

readableforminSection4.2.ILight}coverageoffrequency-domainconceptscanbe

basedondiscussingallthematerialthroughthissectionandthenselectingafewsimple

©Iteringexamples(say,low-andhighpass©IteringusingButterworth®lters,asdiscussed

inSections4.3.2and4.4.2),Atthediscretionoftheinstructor,additionalmaterialcan

includefullcoverageofSections4.3and4.4.Itisseldompossibletogobeyondthis

pointinaseniorcourse.

Chapter5canbecoveredasacontinuationofChapter4.Section5.1makesthisaneasy

approach.Then,itispossiblegivethestudental-avor}ofwhatrestorationis(andstill

keepthediscussionbrief)bycoveringonlyGaussianandimpulsenoiseinSection5.2.1,

andacoupleofspatial©ItersinSection5.3.Thislattersectionisafrequentsourceof

confusiontothestudentwho,basedondiscussionsearlierinthechapter,isexpectingto

seeamoreobjectiveapproach.Itisworthwhiletoemphasizeatthispointthatspatial

enhancementandrestorationarethesamethingwhenitcomestonoisereductionby

spatial©Itering.Agoodwaytokeepitbriefandconcludecoverageofrestoration

istojumpatthispointtoinverse©Itering(whichfollowsdirectlyfromthemodelin

Section5.1)andshowtheproblemswiththisapproach.Then,withabriefexplanation

regardingthefactthatmuchofrestorationcentersaroundtheinstabilitiesinherentin

inverse©Itering,itispossibletointroducethelinteractive}formoftheWiener®lterin

Eq.(5.8-3)andconcludethechapterwithExamples5.12and5.13.

Chapter6oncolorimageprocessingisanewfeatureofthebook.Coverageofthis

4Chapter1Introduction

chapteralsocanbebriefattheseniorlevelbyfocusingonenoughmaterialtogivethe

studentafoundationonthephysicsofcolor(Section6.1),twobasiccolormodels(RGB

andCMY/CMYK),andthenconcludingwithabriefcoverageofpseudocolorprocessing

(Section6.3).

Wetypicallyconcludeaseniorcoursebycoveringsomeofthebasicaspectsofimage

compression(Chapter8).Interestonthistopichasincreasedsigni®cantlyasaresultof

theheavyuseofimagesandgraphicsovertheInternet,andstudentsusuallyareeasily

motivatedbythetopic.MinimumcoverageofthismaterialincludesSections8.1.1and

8.1.2,Section8.2,andSection8.4.1.Inthislimitedscope,itisworthwhilespending

one-halfofalectureperiod®llinginanygapsthatmayarisebyskippingearlierpartsof

thechapter.

OneSemesterGraduateCourse(NoBackgroundinDIP)

Themaindifferencebetweenasenioranda®rst-yeargraduatecourseinwhichneither

grouphasformalbackgroundinimageprocessingismostlyinthescopeofmaterial

covered,inthesensethatwesimplygofasterinagraduatecourse,andfeelmuchfreer

inassigningindependentreading.Inadditiontothematerialdiscussedintheprevious

section,weaddthefollowingmaterialinagraduatecourse.

Coverageofhistogrammatching(Section3.3.2)isadded.Sections4.3,4.4,and4.5

arecoveredinfull.Section4.6istoucheduponbrie-yregardingthefactthatimple­

mentationofdiscreteFouriertransformtechniquesrequiresnon-intuitiveconceptssuch

asfunctionpadding.TheseparabilityoftheFouriertransformshouldbecovered,and

mentionoftheadvantagesoftheFFTshouldbemade.InChapter5weaddSections5.5

through5.8.InChapter6weaddtheHSImodel(Section6.3.2),Section6.4,andSec­

tion6.6.Aniceintroductiontowavelets(Chapter7)canbeachievedbyacombination

ofclassroomdiscussionsandindependentreading.Theminimumnumberofsectionsin

thatchapterare7.1,7.2,7.3,and7.5,withappropriate(butbrief)mentionoftheexis­

tenceoffastwavelettransforms.Finally,inChapter8weaddcoverageofSections8.3,

8.4.2,8.5.1(throughExample8.16),Section8.5.2(throughExample8.20)andSection

8.5.3.

Ifadditionaltimeisavailable,anaturaltopictocovernextismorphologicalimage

processing(Chapter9).Thematerialinthischapterbeginsatransitionfrommethods

whoseinputsandoutputsareimagestomethodsinwhichtheinputsareimages,but

theoutputsareattributesaboutthoseimages,inthesensede®nedinSection1.1.We

OneSemesterGraduateCourse(withBackgroundinDIP)5

recommendcoverageofSections9.1through9.4,andsomeofthealgorithmsinSection

9.5.

OneSemesterGraduateCourse(withBackgroundinDIP)

Someprogramshaveanundergraduatecourseinimageprocessingasaprerequisiteto

agraduatecourseonthesubject.Inthiscase,itispossibletocovermaterialfromthe

@rstelevenchaptersofthebook.Usingtheundergraduateguidelinesdescribedabove,

weaddthefollowingmaterialtoformateachingoutlineforaonesemestergraduate

coursethathasthatundergraduatematerialasprerequisite.Giventhatstudentshavethe

appropriatebackgroundonthesubject,independentreadingassignmentscanbeusedto

controltheschedule.

Coverageofhistogrammatching(Section3.3.2)isadded.Sections4,3,4.4,4.5,and4.6

areadded.Thisstrengthensthestudentzsbackgroundinfrequency-domainconcepts.

AmoreextensivecoverageofChapter5ispossiblebyaddingsections5.2.3,5.3.3,

5.4.3,5.5,5.6,and5.8.InChapter6weaddfull-colorimageprocessing(Sections6.4

through6.7).Chapters7and8arecoveredasintheprevioussection.Asnotedinthe

previoussection.Chapter9beginsatransitionfrommethodswhoseinputsandoutputs

areimagestomethodsinwhichtheinputsareimages,buttheoutputsareattributes

aboutthoseimages.Asaminimum,werecommendcoverageofbinarymorphology:

Sections9.1through9.4,andsomeofthealgorithmsinSection9.5.Mentionshould

bemadeaboutpossibleextensionstogray-scaleimages,butcoverageofthismaterial

maynotbepossible,dependingontheschedule.InChapter10,werecommendSections

10.1,10.2.1and10.2.2,10.3.1through10.3.4,10.4,and10.5.InChapter11wetypically

coverSections11.1through11.4.

TwoSemesterGraduateCourse(NoBackgroundinDIP)

Afull-yeargraduatecourseconsistsofthematerialcoveredintheonesemesterunder­

graduatecourse,thematerialoutlinedintheprevioussection,andSections12.1,12.2,

12.3.1,and12.3.2.

Projects

Oneofthemostinterestingaspectsofacourseindigitalimageprocessingisthepictorial

6Chapter1Introduction

natureofthesubject.Ithasbeenourexperiencethatstudentstrulyenjoyandbene@t

fromjudicioususeofcomputerprojectstocomplementthematerialcoveredinclass.

Sincecomputerprojectsareinadditiontocourseworkandhomeworkassignments,we

trytokeeptheformalprojectreportingasbriefaspossible.Inordertofacilitategrading,

wetrytoachieveuniformityinthewayprojectreportsareprepared.Ausefulreport

formatisasfollows:

Page1：Coverpage.

Projecttitle

0Projectnumber

0Coursenumber

CStudentzsname

0Datedue

0Datehandedin

0Abstract(nottoexceed1/2page)

Page2:Onetotwopages(max)oftechnicaldiscussion.

Page3(or4)：Discussionofresults.Onetotwopages(max).

Results:Imageresults(printedtypicallyonalaserorinkjetprinter).Allimagesmust

containanumberandtitlereferredtointhediscussionofresults.

Appendix:Programlistings,focusedonanyoriginalcodepreparedbythestudent.For

brevity,functionsandroutinesprovidedtothestudentarereferredtobyname,butthe

codeisnotincluded.

Layout:Theentirereportmustbeonastandardsheetsize(e.g.,8:5£11inches),

stapledwiththreeormorestaplesontheleftmargintoformabooklet,orboundusing

clearplasticstandardbindingproducts.

Projectresourcesavailableinthebookwebsiteincludeasampleproject,alistofsug­

gestedprojectsfromwhichtheinstructorcanselect,bookandotherimages,andMAT­

LABfunctions.InstructorswhodonotwishtouseMATLABwill®ndadditionalsoft­

waresuggestionsintheSupport/Softwaresectionofthewebsite.

2ProblemSolutions

Problem2.1

Thediameter,x,oftheretinalimagecorrespondingtothedotisobtainedfromsimilar

triangles,asshowninFig.P2.1.Thatis,

(d=2)(x=2)

0:2=0:014~

whichgivesx=0:07d.FromthediscussioninSection2.1.1,andtakingsomeliberties

ofinterpretation,wecanthinkofthefoveaasasquaresensorarrayhavingontheorderof

337,000elements,whichtranslatesintoanarrayofsize580£580elements.Assuming

equalspacingbetweenelements,thisgives580elementsand579spacesonaline1.5

mmlong.Thesizeofeachelementandeachspaceisthens=[(l:5mm)=l;159]=

l:3£10j6m.Ifthesize(onthefovea)oftheimageddotislessthanthesizeofasingle

resolutionelement,weassumethatthedotwillbeinvisibletotheeye.Inotherwords,

theeyewillnotdetectadotifitsdiameter,d,issuchthat0:07(d)<1:3d0.6m,or

d<18:6£10i6m.

FigureP2.1

8Chapter2ProblemSolutions

Problem2.2

Brightnessadaptation.

Problem2.3

86

5=c=v=2:998£10(m/s)=60(l/s)=4:99£10m=5000Km.

Problem2.4

(a)FromthediscussionontheelectromagneticspectruminSection2.2,thesourceof

theilluminationrequiredtoseeanobjectmusthavewavelengththesamesizeorsmaller

thantheobject.Becauseinterestliesonlyontheboundaryshapeandnotonotherspec­

tralcharacteristicsofthespecimens,asingleilluminationsourceinthefarultraviolet

(wavelengthof.001micronsorless)willbeabletodetectallobjects.Afar-ultraviolet

camerasensorwouldbeneededtoimagethespecimens,(b)Noanswerrequiredsince

theanswerto(a)isaf®rmative.

Problem2.5

FromthegeometryofFig.2.3,7mm=35mm=z=500mm,orz=100mm.Sothetarget

sizeis100mmontheside.Wehaveatotalof1024elementsperline,sotheresolution

of1lineis1024=100=10elements/mm.Forlinepairswedivideby2,givingan

answerof5Ip/mm.

Problem2.6

Onepossiblesolutionistoequipamonochromecamerawithamechanicaldevicethat

sequentiallyplacesared,agreen,andabluepass@lterinfrontofthelens.Thestrongest

cameraresponsedeterminesthecolor.Ifallthreeresponsesareapproximatelyequal,

theobjectiswhite.Afastersystemwouldutilizethreedifferentcameras,eachequipped

withanindividual®lter.Theanalysiswouldbethenbasedonpollingtheresponseof

eachcamera.Thissystemwouldbealittlemoreexpensive,butitwouldbefasterand

morereliable.Notethatbothsolutionsassumethatthe®eldofviewofthecamera(s)is

suchthatitiscompletely@lledbyauniformcolor[i.e.,thecamera(s)is(are)focusedon

Problem2.79

apartofthevehiclewhereonlyitscolorisseen.Otherwisefurtheranalysiswouldbe

requiredtoisolatetheregionofuniformcolor,whichisallthatisofinterestinsolving

thisproblem].

Problem2.7

Theimageinquestionisgivenby

f(x;y)=i(x;y)r(x;y)

—255e；l(xjxO):+(yjyO):](j:0)

2

=255ei[(xixo/+(yiyo)]

AcrosssectionoftheimageisshowninFig.P2.7(a).Iftheintensityisquantizedusing

mbits,thenwehavethesituationshowninFig.P2.7(b),where4G=(255+1)=2m.

Sinceanabruptchangeof8graylevelsisassumedtobedetectablebytheeye,itfollows

that4G=8=256=2m,orm=5.Inotherwords,32,orfewer,graylevelswill

producevisiblefalsecontouring.

Intensity

㈤

FigureP2.7

Chapter2ProblemSolutions

Problem2.8

Theuseoftwobits(m=2)ofintensityresolutionproducesfourgraylevelsintherange

0to255.Onewaytosubdividethisrangeistoletalllevelsbetween0and63becoded

as63,alllevelsbetween64and127becodedas127,andsoon.Theimageresulting

fromthistypeofsubdivisionisshowninFig.P2.8.Ofcourse,thereareotherwaysto

subdividetherange[0;2551intofourbands.

FigureP2.8

Problem2.9

(a)Thetotalamountofdata(includingthestartandstopbit)inan8-bit,1024£1024

image,is(1024)-£[8+2]bits.Thetotaltimerequiredtotransmitthisimageovera

At56Kbaudlinkis(1024)2£[8+2]=56000=187:25secorabout3.1min.(b)At

750Kthistimegoesdowntoabout14sec.

Problem2.10

Thewidth-to-heightratiois16/9andtheresolutionintheverticaldirectionis1125lines

(or,whatisthesamething,1125pixelsintheverticaldirection).Itisgiventhatthe

Problem2.1111

resolutioninthehorizontaldirectionisinthe16/9proportion,sotheresolutioninthe

verticaldirectionis(1125)X16=9)=2000pixelsperline.ThesystemIpaints)afull

1125£2000.8-bitimageevery1/30secforeachofthered,green,andbluecomponent

images.Thereare7200secintwohours,sothetotaldigitaldatageneratedinthistime

intervalis(1125)(2000)(8)(30)(3)(7200)=1:166£1(Tbits,or1:458£1(Tbytes

(i.e.,about1.5terrabytes).These®guresshowwhyimagedatacompression(Chapter

8)issoimportant.

Problem2.11

LetpandqbeasshowninFig.P2.11.Then,(a)SiiUldS2arenot4-connectedbecause

qisnotinthesetN4(p)u(b)S)andS2are8-connectedbecauseqisinthesetNx(p)u

(c)S(andS;arem-connectedbecause(i)qisinND(p),and(ii)thesetN4(p)\N4(q)

isempty.

FigureP2.ll

Problem2.12

Thesolutiontothisproblemconsistsofde®ningallpossibleneighborhoodshapesto

gofromadiagonalsegmenttoacorresponding4-connectedsegment,asshowninFig.

P2.12.Thealgorithmthensimplylooksfortheappropriatematcheverytimeadiagonal

segmentisencounteredintheboundary.

Problem2.13

ThesolutiontothisproblemisthesameasforProblem2.12becauseconvertingfrom

an/^-connectedpathtoa4-connected