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Chapter 2
Image formation
2.1 Geometricprimitivesandtransformations . . . . . . . . . . . . . . . . . . . . 31
2.1.1 Geometricprimitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.2 2Dtransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.3 3Dtransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1.4 3Drotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.5 3Dto2Dprojections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.1.6 Lensdistortions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2 Photometricimageformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2.1 Lighting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2.2 Reflectanceandshading . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.3 Optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.3 Thedigitalcamera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3.1 Samplingandaliasing . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.3.2 Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.3.3 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.4 Additionalreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
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30
ComputerVision:AlgorithmsandApplications(September3,2010draft)
n
^
(a)
(b)
z
i
=102mm
f= 100mm
z
o
=5m
d
G
R
G
R
B
G
B
G
G
R
G
R
B
G
B
G
(c)
(d)
Figure2.1 Afewcomponentsoftheimageformationprocess: (a)perspectiveprojection;
(b)lightscatteringwhenhittingasurface;(c)lensoptics;(d)Bayercolorfilterarray.
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2.1Geometricprimitivesandtransformations
31
Beforewecanintelligentlyanalyzeandmanipulateimages,weneedtoestablishavocabulary
fordescribingthegeometry ofascene. Wealsoneedtounderstandtheimageformation
processthatproducedaparticularimagegivenasetoflightingconditions,scenegeometry,
surfaceproperties,andcameraoptics.Inthischapter,wepresentasimplifiedmodelofsuch
animageformationprocess.
Section2.1introducesthebasicgeometricprimitivesusedthroughoutthebook(points,
lines,andplanes)andthegeometrictransformationsthatprojectthese3Dquantitiesinto2D
imagefeatures(Figure2.1a). Section2.2describeshow lighting,surfaceproperties(Fig-
ure2.1b),andcameraoptics(Figure2.1c)interactinordertoproducethecolorvaluesthat
fallontotheimagesensor.Section2.3describeshowcontinuouscolorimagesareturnedinto
discretedigitalsamplesinsidetheimagesensor(Figure2.1d)andhowtoavoid(oratleast
characterize)samplingdeficiencies,suchasaliasing.
Thematerialcoveredinthischapterisbutabriefsummaryofaveryrichanddeepsetof
topics,traditionallycoveredinanumberofseparatefields.Amorethoroughintroductionto
thegeometryofpoints,lines,planes,andprojectionscanbefoundintextbooksonmulti-view
geometry(HartleyandZisserman2004;FaugerasandLuong2001)andcomputergraphics
(Foley,vanDam,Feineretal.1995).Theimageformation(synthesis)processistraditionally
taughtaspartofacomputergraphicscurriculum(Foley,vanDam,Feineretal.1995;Glass-
ner1995;Watt1995;Shirley2005)butitisalsostudiedinphysics-basedcomputervision
(Wolff,Shafer,andHealey1992a).Thebehaviorofcameralenssystemsisstudiedinoptics
(M¨oller1988;Hecht2001;Ray2002). Twogoodbooksoncolortheoryare(Wyszeckiand
Stiles2000;HealeyandShafer1992),with(Livingstone2008)providingamorefunandin-
formalintroductiontothetopicofcolorperception.Topicsrelatingtosamplingandaliasing
arecoveredintextbooksonsignalandimageprocessing(Crane1997;J¨ahne1997;Oppen-
heimandSchafer1996;Oppenheim,Schafer,andBuck1999;Pratt2007;Russ2007;Burger
andBurge2008;GonzalesandWoods2008).
Anotetostudents: Ifyouhavealreadystudiedcomputergraphics,youmaywant to
skimthematerialinSection2.1,althoughthesectionsonprojectivedepthandobject-centered
projectionneartheendofSection2.1.5maybenewtoyou. Similarly,physicsstudents(as
wellascomputergraphicsstudents)willmostlybefamiliarwithSection2.2.Finally,students
withagoodbackgroundinimageprocessingwillalreadybefamiliarwithsamplingissues
(Section2.3)aswellassomeofthematerialinChapter3.
2.1 Geometricprimitives andtransformations
Inthissection,weintroducethebasic2Dand3Dprimitivesusedinthistextbook,namely
points,lines,andplanes. Wealsodescribehow3Dfeaturesareprojectedinto2Dfeatures.
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32
ComputerVision:AlgorithmsandApplications(September3,2010draft)
Moredetaileddescriptionsofthesetopics(alongwithagentlerandmoreintuitiveintroduc-
tion)canbefoundintextbooksonmultiple-viewgeometry(HartleyandZisserman2004;
FaugerasandLuong2001).
2.1.1 Geometricprimitives
Geometricprimitivesformthebasicbuildingblocksusedtodescribethree-dimensionalshapes.
In thissection, weintroducepoints,lines,andplanes. Latersectionsofthebook discuss
curves(Sections5.1and11.2),surfaces(Section12.3),andvolumes(Section12.5).
2Dpoints. 2Dpoints(pixelcoordinatesinanimage)canbedenotedusingapairofvalues,
x=(x;y)2R
2
,oralternatively,
x=
"
x
y
#
:
(2.1)
(Asstatedintheintroduction,weusethe(x
1
;x
2
;:::)notationtodenotecolumnvectors.)
2Dpointscanalsoberepresentedusinghomogeneouscoordinates,~x=(~x;~y; w~)2P
2
,
wherevectorsthatdifferonlybyscaleareconsideredtobeequivalent.P
2
=R
3
(0;0;0)
iscalledthe2Dprojectivespace.
Ahomogeneous vector ~x can beconverted back intoan inhomogeneous vector x by
dividingthroughbythelastelement w~,i.e.,
~x=(~x;~y; w~)= ~w(x;y;1)= ~wx;
(2.2)
wherex=(x;y;1)istheaugmentedvector.Homogeneouspointswhoselastelementis w~=
0arecalledidealpointsorpointsatinfinityanddonothaveanequivalentinhomogeneous
representation.
2Dlines. 2Dlinescanalsoberepresentedusinghomogeneouscoordinates
~
l= (a;b;c).
Thecorrespondinglineequationis
x
~
l=ax+by+c=0:
(2.3)
Wecannormalizethelineequationvectorsothatl=(^n
x
;^n
y
;d)=(^n;d)withk^nk=1.In
thiscase,^nisthenormalvectorperpendiculartothelineanddisitsdistancetotheorigin
(Figure2.2). (Theoneexceptiontothisnormalizationisthelineat infinity
~
l = (0;0;1),
whichincludesall(ideal)pointsatinfinity.)
Wecanalsoexpress^nasafunctionofrotationangle, ^n =(^n
x
;^n
y
) =(cos;sin)
(Figure2.2a). Thisrepresentationiscommonly used intheHoughtransformline-finding
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2.1Geometricprimitivesandtransformations
33
y
x
d
θ
n
l
^
z
x
d
n
m
y
^
(a)
(b)
Figure2.2 (a)2Dlineequationand(b)3Dplaneequation,expressedintermsofthenormal
^nanddistancetotheorigind.
algorithm,which isdiscussed inSection4.3.2. Thecombination (;d) is also known as
polarcoordinates.
Whenusinghomogeneouscoordinates,wecancomputetheintersectionoftwolinesas
~x=
~
l
1
~
l
2
;
(2.4)
whereisthecrossproductoperator.Similarly,thelinejoiningtwopointscanbewrittenas
~
l=~x
1
~x
2
:
(2.5)
Whentryingtofit anintersectionpointtomultiplelinesor, conversely,alineto multiple
points,leastsquarestechniques(Section6.1.1andAppendixA.2)canbeused,asdiscussed
inExercise2.1.
2Dconics. Thereareotheralgebraiccurvesthatcanbeexpressedwithsimplepolynomial
homogeneousequations.Forexample,theconicsections(socalledbecausetheyariseasthe
intersectionofaplaneanda3Dcone)canbewrittenusingaquadricequation
~x
T
Q~x=0:
(2.6)
Quadricequationsplayusefulrolesinthestudyofmulti-viewgeometryandcameracalibra-
tion(HartleyandZisserman2004;FaugerasandLuong2001)butarenotusedextensivelyin
thisbook.
3Dpoints. Pointcoordinatesinthreedimensionscanbewrittenusinginhomogeneousco-
ordinatesx=(x;y;z)2R3orhomogeneouscoordinates~x=(~x;~y;~z;w~)2P3.Asbefore,
itissometimesusefultodenotea3Dpointusingtheaugmentedvectorx=(x;y;z;1)with
~x= w~x.
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34
ComputerVision:AlgorithmsandApplications(September3,2010draft)
z
x
λ
p
q
y
r=(1-
λ
)p+
λ
q
Figure2.3 3Dlineequation,r=(1 )p+q.
3Dplanes. 3Dplanescanalsoberepresentedashomogeneouscoordinates ~m=(a;b;c;d)
withacorrespondingplaneequation
x m~ ~ =ax+by+cz+d=0:
(2.7)
Wecanalsonormalizetheplaneequationasm=(^n
x
;^n
y
;^n
z
;d)=(^n;d)withk^nk=1.
In thiscase, ^nisthenormalvectorperpendicularto theplaneand disitsdistancetothe
origin(Figure2.2b). Aswiththecaseof2Dlines, theplaneatinfinity m~ = (0;0;0;1),
whichcontainsallthepointsatinfinity,cannotbenormalized(i.e.,itdoesnothaveaunique
normalorafinitedistance).
Wecanexpress^nasafunctionoftwoangles(;),
^n=(coscos;sincos;sin);
(2.8)
i.e.,usingsphericalcoordinates,butthesearelesscommonlyusedthanpolarcoordinates
sincetheydonotuniformlysamplethespaceofpossiblenormalvectors.
3Dlines. Linesin3Darelesselegantthaneitherlinesin2Dorplanesin3D.Onepossible
representationistousetwopointsontheline,(p;q). Anyotherpointonthelinecanbe
expressedasalinearcombinationofthesetwopoints
r=(1 )p+q;
(2.9)
asshowninFigure2.3.Ifwerestrict01,wegetthelinesegmentjoiningpandq.
Ifweusehomogeneouscoordinates,wecanwritethelineas
~r=~p+~q:
(2.10)
Aspecialcaseofthisiswhenthesecondpointisatinfinity,i.e.,~q=(
^
d
x
;
^
d
y
;
^
d
z
;0)=(
^
d;0).
Here,weseethat
^
disthedirectionoftheline.Wecanthenre-writetheinhomogeneous3D
lineequationas
r=p+
^
d:
(2.11)
2.1Geometricprimitivesandtransformations
35
Adisadvantageoftheendpointrepresentationfor3Dlinesisthatithastoomanydegrees
offreedom,i.e.,six(threeforeachendpoint)insteadofthefourdegreesthata3Dlinetruly
has. However,ifwefixthetwopointsonthelinetolieinspecificplanes,weobtainarep-
resentationwithfourdegreesoffreedom.Forexample,ifwearerepresentingnearlyvertical
lines,then z = 0andz = 1 formtwosuitableplanes,i.e., the(x;y)coordinatesinboth
planesprovidethefourcoordinatesdescribingtheline. Thiskindoftwo-planeparameteri-
zationisusedinthelightfieldandLumigraphimage-basedrenderingsystemsdescribedin
Chapter13torepresent thecollectionofraysseenbyacameraasit movesinfrontofan
object. Thetwo-endpointrepresentationisalsousefulforrepresentinglinesegments,even
whentheirexactendpointscannotbeseen(onlyguessedat).
Ifwewishtorepresentallpossiblelineswithoutbiastowardsanyparticularorientation,
wecanusePl¨uckercoordinates(HartleyandZisserman2004,Chapter2;FaugerasandLuong
2001,Chapter3).Thesecoordinatesarethesixindependentnon-zeroentriesinthe44skew
symmetricmatrix
L=~p~q
T
~q~p
T
;
(2.12)
where~pand ~qareanytwo(non-identical)pointsontheline. Thisrepresentationhasonly
fourdegreesoffreedom,sinceLishomogeneousandalsosatisfiesdet(L)=0,whichresults
inaquadraticconstraintonthePl¨uckercoordinates.
In practice,theminimalrepresentationisnotessential formostapplications. Anade-
quatemodelof3Dlinescanbeobtainedbyestimatingtheirdirection(whichmaybeknown
ahead of time, e.g.,for architecture)andsomepointwithinthevisibleportionoftheline
(seeSection7.5.1)orbyusingthetwoendpoints,sincelinesaremostoftenvisibleasfinite
linesegments. However,ifyou areinterested in moredetailsaboutthetopicof minimal
lineparameterizations,F¨orstner(2005)discussesvariouswaystoinferandmodel3Dlinesin
projectivegeometry,aswellashowtoestimatetheuncertaintyinsuchfittedmodels.
3Dquadrics. The3Danalogofaconicsectionisaquadricsurface
x
T
Qx=0
(2.13)
(HartleyandZisserman2004,Chapter2). Again, whilequadricsurfacesareusefulinthe
study ofmulti-viewgeometry and can alsoserve as usefulmodeling primitives(spheres,
ellipsoids,cylinders),wedonotstudythemingreatdetailinthisbook.
2.1.2 2Dtransformations
Havingdefinedourbasicprimitives,wecannowturnourattentiontohowtheycanbetrans-
formed.Thesimplesttransformationsoccurinthe2DplaneandareillustratedinFigure2.4.
36
ComputerVision:AlgorithmsandApplications(September3,2010draft)
y
x
similarity
Euclidean
affine
projective
translation
Figure2.4 Basicsetof2Dplanartransformations.
Translation. 2Dtranslationscanbewrittenasx
0
=x+tor
x
0
=
h
I t
i
x
(2.14)
whereIisthe(22)identitymatrixor
x
0
=
"
I
t
0
T
1
#
x
(2.15)
where0isthezerovector.Usinga23matrixresultsinamorecompactnotation,whereas
usingafull-rank33matrix(whichcanbeobtainedfromthe23matrixbyappendinga
[0
T
1]row)makesitpossibletochaintransformationsusingmatrixmultiplication.Notethat
inanyequationwhereanaugmentedvectorsuchasxappearsonbothsides,itcanalwaysbe
replacedwithafullhomogeneousvector~x.
Rotation+translation. Thistransformationisalsoknownas2Drigidbodymotionorthe
2DEuclideantransformation(sinceEuclideandistancesarepreserved). Itcanbewrittenas
x0=Rx+tor
x
0
=
h
R
t
i
x
(2.16)
where
R=
"
cos  sin
sin
cos
#
(2.17)
isanorthonormalrotationmatrixwithRR
T
=IandjRj=1.
Scaledrotation. Alsoknownasthesimilaritytransform, thistransformationcanbeex-
pressedasx
0
=sRx+twheresisanarbitraryscalefactor.Itcanalsobewrittenas
x
0
=
h
sR
t
i
x=
"
 b t
x
b
a
t
y
#
x;
(2.18)
whereweno longerrequirethata
2
+b
2
= 1. Thesimilaritytransformpreservesangles
betweenlines.
2.1Geometricprimitivesandtransformations
37
Affine. Theaffinetransformationiswrittenasx
0
= Ax,whereAisan arbitrary23
matrix,i.e.,
x
0
=
"
a
00
a
01
a
02
a
10
a
11
a
12
#
x:
(2.19)
Parallellinesremainparallelunderaffinetransformations.
Projective. Thistransformation, alsoknownasaperspectivetransformorhomography,
operatesonhomogeneouscoordinates,
~x
0
=
~
H~x;
(2.20)
where
~
Hisanarbitrary33matrix. Notethat
~
H ishomogeneous,i.e.,itisonlydefined
uptoascale,andthattwo
~
Hmatricesthatdifferonlybyscaleareequivalent.Theresulting
homogeneouscoordinate~x
0
mustbenormalizedinordertoobtainaninhomogeneousresult
x,i.e.,
x
0
=
h
00
x+h
01
y+h
02
h
20
x+h
21
y+h
22
and y
0
=
h
10
x+h
11
y+h
12
h
20
x+h
21
y+h
22
:
(2.21)
Perspectivetransformationspreservestraightlines(i.e.,theyremainstraightafterthetrans-
formation).
Hierarchy of2D transformations. Theprecedingsetoftransformationsareillustrated
inFigure2.4and summarizedin Table2.1. Theeasiest way to thinkofthem isasaset
of(potentiallyrestricted)33matricesoperatingon2Dhomogeneouscoordinatevectors.
HartleyandZisserman(2004)containsamoredetaileddescriptionofthehierarchyof2D
planartransformations.
Theabovetransformationsformanestedsetofgroups,i.e.,theyareclosedundercom-
positionandhaveaninversethatisamemberofthesamegroup. (Thiswillbeimportant
laterwhenapplyingthesetransformationstoimagesinSection3.6.)Each(simpler)groupis
asubsetofthemorecomplexgroupbelowit.
Co-vectors. Whiletheabovetransformationscanbeusedtotransformpointsina2Dplane,
cantheyalsobeuseddirectlytotransformalineequation?Considerthehomogeneousequa-
tion
~
l~x=0.Ifwetransformx
0
=
~
Hx,weobtain
~
l
0
~x
0
=
~
l
0T
~
H~x=(
~
H
T
~
l
0
)
T
~x=
~
l~x=0;
(2.22)
i.e.,
~
l
0
=
~
H
T
~
l.Thus,theactionofaprojectivetransformationonaco-vectorsuchasa2D
lineor3Dnormalcanberepresentedbythetransposedinverseofthematrix,whichisequiv-
alenttotheadjointof
~
H,sinceprojectivetransformationmatricesarehomogeneous. Jim
38
ComputerVision:AlgorithmsandApplications(September3,2010draft)
Transformation
Matrix
#DoF
Preserves
Icon
translation
h
I
t
i
23
2
orientation
rigid(Euclidean)
h
R
t
i
23
3
lengths
S
S
S
S
similarity
h
sR
t
i
23
4
angles
S
S
affine
h
A
i
23
6
parallelism
projective
h
~
H
i
33
8
straightlines
Table2.1 Hierarchyof2Dcoordinatetransformations. Eachtransformationalsopreserves
thepropertieslistedintherowsbelowit,i.e.,similaritypreservesnotonlyanglesbutalso
parallelismandstraightlines.The23matricesareextendedwithathird[0
T
1]rowtoform
afull33matrixforhomogeneouscoordinatetransformations.
Blinn(1998)describes(inChapters9and10)theinsandoutsofnotatingandmanipulating
co-vectors.
Whiletheabovetransformationsaretheonesweusemostextensively,anumberofaddi-
tionaltransformationsaresometimesused.
Stretch/squash. Thistransformationchangestheaspectratioofanimage,
x
0
=
s
x
x+t
x
y
0
=
s
y
y+t
y
;
andisarestrictedformofanaffinetransformation. Unfortunately,itdoesnotnestcleanly
withthegroupslistedinTable2.1.
Planarsurfaceflow. Thiseight-parametertransformation(Horn1986;Bergen,Anandan,
Hannaetal.1992;Girod,Greiner,andNiemann2000),
x
0
=
a
0
+a
1
x+a
2
y+a
6
x
2
+a
7
xy
y
0
=
a
3
+a
4
x+a
5
y+a
7
x
2
+a
6
xy;
ariseswhenaplanarsurfaceundergoesasmall3Dmotion. Itcanthusbethoughtofasa
smallmotionapproximationtoafullhomography.Itsmainattractionisthatitislinearinthe
motionparameters,a
k
,whichareoftenthequantitiesbeingestimated.
Documents you may be interested
Documents you may be interested