The multidegree of the multi-image variety

0.25 MB·

by  Laura Escobar

The multidegree of the multi-image variety
LauraEscobarandAllenKnutson
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AbstractThemulti-imagevarietyisasubvarietyofGr(1,P3)n thatmodelstaking
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1 pictureswithnrationalcameras.Wecomputeitscohomologyclassinthecohomol-
ogyofGr(1,P3)n,andfromthereitsmultidegreeasasubvarietyof(P5)nunderthe

] Plu¨ckerembedding.
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1 Introduction
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Multi-viewgeometrystudiestheconstraintsimposedonathree-dimensionalscene
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from various two-dimensional images of the scene. Each image is produced by a

1 camera. Algebraic vision is a recent field of mathematics in which the techniques
v ofalgebraicgeometryandoptimizationareusedtoformulateandsolveproblemsin
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computervision.Oneofthemainobjectsstudiedbythisfieldisamulti-viewvariety.
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Roughly speaking, a multi-view variety parametrizes all the possible images that
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3 can be taken by a fixed collection of cameras. See [1, 11, 14] for more details on
0 multi-viewvarieties.See[13]forasurveyonvariouscameramodels.
. Thework[11]presentsanewpointofviewtostudythemulti-viewvarieties.A
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0 photographiccameramapsapointinthescenetoapointintheimage.[11]defines
7 ageometriccamera,whichmapsapointinthescenenottoapoint,buttoaviewing
1 ray.Moreprecisely,aphotographiccameraisamapP3(cid:57)(cid:57)(cid:75)P2 orP3(cid:57)(cid:57)(cid:75)P1×P1,
v: whereasageometriccameraisamapP3(cid:57)(cid:57)(cid:75)Gr(1,P3)suchthattheimageofeach
i pointisalinecontainingthepoint.Theviewingraycorrespondingtoapoint p∈P3
X
is the line in Gr(1,P3) this point gets mapped to. An important assumption is that
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LauraEscobar
UniversityofIllinoisatUrbana-Champaign,DepartmentofMathematics,Urbana,IL,61801,USA
e-mail:[email protected]
AllenKnutson
CornellUniversity,DepartmentofMathematics,Ithaca,NY,14850,USA,
e-mail:[email protected]
1
2 LauraEscobarandAllenKnutson
lighttravelsalongrayswhichmeansthattheimageofageometriccameraistwo-
dimensional.
We now illustrate these definitions using the example of pinhole cameras or
cameræ obscuræ, see Figure 1. As a geometric camera, a pinhole camera maps a
point p∈P3 tothelineφ(p)connecting ptothefocalpoint.Thismapisrational
since it is undefined at the focal point. As a photographic camera, a pinhole cam-
era maps a point p∈P3 to the intersection of φ(p) and the plane at the back of
thecamera.Noticethat,asaphotographiccamera,therearemanypinholecameras
corresponding to a given focal point. All of these cameras are equivalent up to a
projectivetransformation.Theessentialpartofapinholecameraisthemappingof
thescenepointstoviewingrays,i.e.itsmodelingasageometriccamera.
p
φ(p)
focalpoint
Fig.1 Apinholecamera.
Amulti-imagevarietyistheZariskiclosureoftheimageofamap
φ :P3(cid:57)(cid:57)(cid:75)Gr(1,P3)n
p(cid:55)→(φ (p),...,φ (p)),
1 n
whereeachφ isageometriccamera.Themulti-viewvarietyisamulti-imagevariety
i
suchthateachφ isapinholecamera.LetC betheclosureoftheimageofφ inside
i i i
thei-thGr(1,P3).Thenundersomeassumptions,[11,Theorem5.1]showsthat
(C ×···×C )∩V =φ(P3),
1 n n
whereV istheconcurrentlinesvarietyconsistingoforderedn-tuplesoflinesinP3
n
thatmeetinapointx.Theconcurrentlinesandmulti-imagevarietiesareembedded
into(P5)nusingthePlu¨ckerembedding.
Themultidegreeofavarietyembeddedintoaproductofprojectivespacesisthe
polynomialwhosecoefficientsgivethenumbers(whenfinite)ofintersectionpoints
Themultidegreeofthemulti-imagevariety 3
in the variety intersected with a product of general linear subspaces. The present
paper verifies the conjectured formula [11, Equation (11)] for the multidegree of
V andcomputesthemultidegreeofthemulti-imagevariety.Todoso,wedescribe
n
V usingaprojectionofapartialflagvarietyanduseSchubertcalculustocompute
n
the cohomology classes ofV and (C ×···×C )∩V in the cohomology ring of
n 1 n n
Gr(1,P3)n. We then push forward these formulæ into (P5)n to obtain the multide-
grees.
We now describe the organization of this paper. In §2 we define the main ob-
jectsofstudy:themulti-imagevarietyandtheconcurrentlinesvariety.Wepresent
themaintheoremwhichcomputesthemultidegreesoftheseobjects.Themaintool
toprovethemaintheoremisSchubertcalculus.In§3wegiveabriefintroduction
toSchubertcalculusforGr(1,P3).In§4wecomputethecohomologyclassofthe
multi-imagevarietyandtheconcurrentlinesvarietyintermsoftheSchubertcycles
inGr(1,P3).Weprovethemaintheorembytakingthepushforwardoftheseequa-
tionstothecohomologyringofP5.In§5werefinetheseresultstoacomputationof
theK-classfortheconcurrentlinesvariety.
2 Themulti-imagevariety
TheGrassmannianGr(k,Pd)consistsofk-dimensionalplanesinsidePd.Acongru-
ence is a two-dimensional family of lines in P3, i.e. a surfaceC in Gr(1,P3). The
bidegree (α,β) of a congruenceC is a pair of nonnegative integers such that the
cohomologyclassofCinGr(1,P3)hastheform
[C]=α[L: Lcontainsafixedpoint]+β[L: Lliesinafixedplane]. (1)
Thefirstintegerα iscalledtheorderandcountsthenumberoflinesinCthatpass
through a general point of P3, and β is called the class and counts the number of
linesinCthatlieinageneralplaneofP3.ThefocallocusofCconsistsofthepoints
inP3thatdonotbelongtoα distinctlinesofC.
Example2.1.A congruence with bidegree (1,0) consists of all lines in Gr(1,P3)
that contain a fixed point. A geometric camera for such a congruence represents a
pinholecamerawherethefixedpointisthefocalpoint.
Example2.2.A two-slit camera assigns to p∈P3 the unique line passing through
pandintersectingtwofixedlinesL ,L ∈Gr(1,P3),seeFigure2.Itsfocallocusis
1 2
{L ,L }.Thesecamerascorrespondtothecongruenceswithbidegrees(1,1).
1 2
The study of congruences started with [7] which classified those of order one.
Theywerestudiedbymanymathematiciansduringthesecondhalfofthe19thcen-
tury;seethebook[5]forsomeoftheseresults.
Remark2.3.Congruencesarestudiedthearticle[6].Inparticular,[6,§2]discusses
congruencesandtheirbidegrees.ThebidegreesofcurvesinP1×P1 arediscussed
inthearticle[10].
4 LauraEscobarandAllenKnutson
φ(p)
* L1
*
L
2
p
Fig.2 Atwo-slitcamera.
Considerarationalmap
φ :P3(cid:57)(cid:57)(cid:75)C ×···×C ⊂ Gr(1,P3)n
1 n
x(cid:55)→(φ (x),...,φ (x))
1 n
wheretheclosureorφ(P3)equalsC,eachC isacongruenceandx∈φ(x)foralli.
i i i i
Eachmapφ isdefinedeverywhereexceptonthefocallocusofC.Inthelanguage
i i
ofalgebraicvisionsuchamapmeanstakingpictureswithnrationalcameraswhere
eachC isthei-thimageplane.Animportantassumptionisthatlighttravelsalong
i
rayswhichmeansthatC is2-dimensional.
i
LetC ,...,C be congruences of bidegree (1,β). Note that φ(x) is the unique
1 n i i
lineinC passingthrough x.The multi-imagevariety of (C ,...,C )istheZariski
i 1 n
closure of φ(P3). The concurrent lines variety V consists of ordered n-tuples of
n
lines in P3 that meet in a point x. By [11, Theorem 5.1], if the focal loci of the
congruencesarepairwisedisjointthenthemulti-imageequalstheintersection
(C ×···×C )∩V ⊂ Gr(1,P3)n.
1 n n
Mostofthecamerasstudiedincomputervisionareassociatedwithcongruences
of order 1. However, cameras of higher order also appear in computer vision, see
[13] and [11, §7]. As an example of a camera associated to a congruence of bide-
gree(2,2)letusdiscussanon-centralpanoramiccamera,seeFigure3.Considera
circleX obtainedbyrotatingapointaboutaverticalaxisL.Therearetwolinesin
Gr(1,P3)passingthroughageneralpointp∈P3andintersectingbothLandX.The
congruenceCconsistingofalllinesintersectingbothX andLhasbidegree(2,2).A
physicalrealizationofanon-centralpanoramiccameraconsistsofasensoronthe
circletakingmeasurementspointingoutwards.Thisorientationofthesensoryields
amapφ :P3(cid:57)(cid:57)(cid:75)Gr(1,P3),i.e.itassignsonlyonelineφ(p)toapoint p∈P3.
The concurrent lines and multi-image varieties are embedded into (P5)n using
thePu¨ckerembedding
Gr(1,P3)(cid:44)→P5.
ThemultidegreeofavarietyX embeddedintoaproductofprojectivespacesPa1×
···×Pan is a homogeneous polynomial whose term qzr1···zrn indicates that there
1 n
Themultidegreeofthemulti-imagevariety 5
φ(p) L
p
*
*
*
X
*
Fig.3 Anon-centralpanoramiccamerapicksoneofthetwolinesintersectingthecircleXandthe
lineL.
are q intersection points when X is intersected with the product H ×···×H ⊂
1 n
Pa1×···×Pan ofgenerallinearsubspaces,wheredim(Hi)=ri.Thedegreeofthis
polynomialequalsthecodimensionofX inPa1×···×Pan.Anequivalentdefinition
ofthemultidegreeofXisitscohomologyclassinthecohomologyringofPa1×···×
Pan.Thebuilt-incommandmultidegreeinthesoftwareMacaulay2[4]computesthe
multidegree of X from its defining ideal. We refer to the book [9, §8.5] for more
detailsonmultidegrees.
The main theorem of this paper computes the multidegree of the multi-image
variety:
Theorem2.4.ThemultidegreeoftheconcurrentlinesvarietyV in(P5)nequals
n
 
(z z ···z )34∑z−2z−1+8 ∑ z−1z−1z−1. (2)
1 2 n  i j i j k 
(i,j) {i,j,k}
i(cid:54)=j
Let(α,β)bethebidegreeofC fori=1,...,n.Themultidegreeof(C ×···×C )∩
i i i 1 n
V in(P5)nequals
n
(cid:32)
(α +β)(α +β )
(α α ···α )(z z ···z )5 ∑ i i j j z−2z−1+
1 2 n 1 2 n αα i j
(i,j) i j
i(cid:54)=j
(cid:33)
(α +β)(α +β )(α +β )
∑ i i j j k k z−1z−1z−1 ,
αα α i j k
{i,j,k} i j k
wherewedistributeaccordinglywheneverα =0.
i
6 LauraEscobarandAllenKnutson
Inparticular,themultidegreeofthemulti-imagevarietyof(C ,...,C )wherethe
1 n
bidegreeofC is(1,β)equals
i i
(cid:32)
(z z ···z )5 ∑(1+β)(1+β )z−2z−1+
1 2 n i j i j
(i,j)
i(cid:54)=j
(cid:33)
∑ (1+β)(1+β )(1+β )z−1z−1z−1 .
i j k i j k
{i,j,k}
Remark2.5.Ponce-Sturmfels-Trager [11, equation (11)] had conjectured Formula
2forthemultidegreeofV ,basedonexperimentalevidencefromMacaulay2.
n
Remark2.6.Aholt-Sturmfels-Thomas [1, Corollary 3.5] gives an equation for the
multidegree of the multi-image variety for the congruences with bidegree (1,0).
ThisequationcoincideswiththeequationofTheorem2.4.
Toprovethistheorem,wefirstuseSchubertcalculustoobtainthecohomology
classesofV andthemulti-imagevarietyinthecohomologyringofGr(1,P3)n;see
n
Theorems4.2and4.5.Usingthisformulawethendescribethemultidegreesofthese
varietiesandproveTheorem2.4.Inthenextsectionweintroduceournotationfor
SchubertvarietiesandreviewthepartofSchubertcalculusthatweneed.
3 Schubertvarieties
In this section we review Schubert varieties in Gr(k,Pd) keeping Gr(1,P3) as the
main example. For further details we recommend the book [3]. Fix a coordinate
systemforPd.LetP(i,i+1,...,j) denotethecoordinatesubspaceofPn spannedbythe
coordinates(i,i+1,...,j)andconsiderthestandardflag
E :=P(0)⊂P(0,1)⊂···⊂P(0,1,...,d)=Pd.
•
The Schubert variety in Gr(k,Pd) corresponding to the subset {j ,...,j } ⊂
1 k+1
{1,2,...,d+1}is
(cid:110) (cid:111)
X := p∈Gr(k,Pd):dim(p∩P(0,1,...,jl−1))≥l−1forl=1,...,k+1 .
{j1,...,jk+1}
(3)
Example3.1.The(cid:0)d+1(cid:1)=(cid:0)4(cid:1)SchubertvarietiesinGr(1,P3)are
k+1 2
X ={P(0,1)},
{1,2}
X ={L∈Gr(1,P3):P(0)⊂L⊂P(0,1,2)},
{1,3}
X ={L∈Gr(1,P3):P(0)⊂L},
{1,4}
Themultidegreeofthemulti-imagevariety 7
X ={L∈Gr(1,P3):L⊂P(0,1,2)},
{2,3}
X ={L∈Gr(1,P3):dim(L∩P(0,1))≥1}, and
{2,4}
X =Gr(1,P3).
{3,4}
ConsidertheSchubertcellsdefinedbyreplacing≥by=inEquation(3).Since
Gr(1,P3) is a disjoint union of the Schubert cells, and they are each contractible,
the classes [X ] for J ⊂{1,2,...,d+1} form a basis for the cohomology ring of
J
Gr(k,Pd).Theringoperationisthecupproduct
[X ](cid:96)[X ]:=[X (Eop)∩X ],
I J I • J
where XI(Eo•p) is defined by using P(d−jl+1,d−jl+2,...,d) instead of P(0,1,...,jl−1) in
Equation3.(UnlikeX ,thisvarietyintersectsX transversely,whilehavingthesame
I J
cohomologyclassasX .)Onealsogetsabasisforthecohomologyringofproducts
I
ofGrassmanniansviatheKu¨nnethisomorphism.
Example3.2.Notethat[X ](cid:96)[X ]=0foranyJ(cid:54)={3,4}sinceP(2,3)∈/X .Any
J {1,2} J
lineLcontainingP(0)isnotcontainedinP(1,2,3)andtherefore[X ](cid:96)[X ]=0.
{1,4} {2,3}
Wehavethat[X ](cid:96)[X ]=[X ]sincethereisauniquelinecontainingthe
{1,4} {1,4} {1,2}
pointsP(0)andP(3).OngeneralGrassmannians,computing[X ](cid:96)[X ]canbedone
I J
using the “Pieri rule” for special classes, see Equation (4), and the “Littlewood-
Richardsonrule”[8]forarbitrarydimensions.
SchubertvarietiesstratifyGr(k,Pd).Theposetoftheirinclusionsismosteasily
describedwhenSchubertvarietiesareindexedusingpartitions.Apartitionofninto
k+1partsisalistλ =(λ1≥λ2≥···≥λk+1>0)suchthatn=∑iλi.Thereisa
bijectionbetween(k+1)-subsetsof[d+1]andpartitionsλ withatmostk+1parts
suchthatλ ≤d−kwhichisgivenby
1
j={j ,...,j } ↔ λ =(d−k+1−j , d−k+2−j ,..., d−j ,d+1−j ).
1 k+1 1 2 k k+1
Partitionscanbevisualizedinthefollowingway.Givenλ ≥λ ≥...≥λ ,we
1 2 k+1
drawafiguremadeupofsquaressharingedgesthathasλ squaresinthefirstrow,
1
λ squaresinthesecondrowstartingoutrightbelowthebeginningofthefirstrow,
2
andsoon.ThisfigureiscalledaYoungdiagram.SeeFigure4foranexample.We
saythatλ ⊂λ(cid:48)ifλ ≤λ(cid:48)foralli,i.e.ifthediagramofλ liesinsidetheoneforλ(cid:48).
i i
Fig.4 TheYoungdiagramofthepartitionλ=(5,3,3,2)
FromnowonwewillindexSchubertcellsandvarietiesbypartitions.Withthis
indexingsetwehavethefollowingfacts:
8 LauraEscobarandAllenKnutson
• dim(Xλ)=(k+1)(d−k)−∑iλi,
• X ⊃X iffµ ⊃λ,and
λ µ
• ThePierirule:supposethatλ =(λ ,0,...,0)andµ =(µ ,µ ,...,µ )isany
1 1 2 k+1
partition.Then
[X ](cid:96)[X ]=∑[X ], (4)
λ µ ν
wherethesumisoverallpartitionsν =(ν ,ν ,...,ν )suchthatν ≤d−k,
1 2 k+1 1
µi≤νi≤µi−1,and∑νi=∑(λi+µi).
Example3.3.TheSchubertvarietiesinGr(1,P3)areorderedbycontainmentinthe
posetinFigure5.ThisposetisrankedbythedimensionsoftheSchubertvarieties.
X∅=Gr(1,P3)
X ={L : L∩P(0,1)(cid:54)=∅}
1
X1,1={L : L⊂P(0,1,2)} X2={L : P(0)⊂L}
X2,1={L : P(0)⊂L⊂P(0,1,2)}
X2,2={L=P(0,1)}.
Fig.5 ThecontainmentposetoftheSchubertvarietiesinGr(1,P3).
BythePierirule,wehavethat[X ](cid:96)[X ]=[X ]and[X ](cid:96)[X ]=[X ].
1 2 2,1 1 1,1 2,1
Remark3.4.WecanrewriteEquation1forthecohomologyclassofacongruence
as
[C]=α[X ]+β[X ].
2 1,1
Remark3.5.In the article [6, §6], Schubert varieties and the intersection theory of
Gr(1,P3)arealsodiscussed.
4 Computingthemultidegrees
LetM⊂Gr(0,P3)×Gr(1,P3)bethepartialflagmanifoldconsistingofpairs(P,L)
wherePisapointinP3 andLisalinethroughP.Suchapairiscalleda(partial)
Themultidegreeofthemulti-imagevariety 9
flaginP3.DefineMn tobethesubvarietyofMn consistingoflistsofnflagssuch
∆
thatthepointPisthesameinallofthem.Considerthediagonal
(P3)n :={(P,...,P):P =···=P}; (5)
∆ 1 n 1 n
thenMn isthepreimageof(P3)n undertheprojectionMn→(P3)ninducedby
∆ ∆
M−→P3, (P,L)(cid:55)−→P.
Forexample,M2 ⊆M2 consistsofpairsofflagsoftheform((P,L ),(P,L )).The
∆ 1 2
followingisstraightforward:
Proposition4.1.TheconcurrentlinesvarietyV istheimageofMn underthepro-
n ∆
jectionMn−→p Gr(1,P3)ninducedby
M(cid:44)→Gr(0,P3)×Gr(1,P3)−→Gr(1,P3).
Theintersection(cid:0)Mi−1×M2×Mn−i−1(cid:1)∩(cid:0)Mi×M2×Mn−i−2(cid:1)consistsofthelists
∆ ∆
of flags ((P,L ),...,(P,L )) such that P =P =P , and this intersection is
1 1 n n i i+1 i+2
transverse.WecanwriteMn asthetransverseintersection
∆
n−1
Mn = (cid:92)(cid:0)Mi−1×M2×Mn−i−1(cid:1). (6)
∆ ∆
i=1
FromthisdescriptionwededucethefollowingTheoremswhichgivethecohomol-
ogyclassesofV andthemulti-imagevariety.
n
Theorem4.2.Theclass[V ]oftheconcurrentlinesvarietyinthecohomologyring
n
ofGr(1,P3)nis
[Vn]= ∑ (cid:79)n−1(cid:104)L:L∩P(vi,vi+1,...,vi+1)(cid:54)=∅(cid:105). (7)
0=v0≤v1≤...≤vn=3 i=0
vj+1−vj<3
Moreover,[V ]canbewrittenas
n

[X ] ifv −v =0
n  2 i i−1
(cid:79)
[Vn]= ∑ [X1] ifvi−vi−1=1. (8)
0=v0≤v1≤...≤vn=3 i=1[Gr(1,P3)] ifv −v =2
vj+1−vj<3 i i−1
Remark4.3.Equation(7)ismorenaturalthanEquation(8),inthat(7)iscorrectin
T-equivariantcohomology,whereas(8)isonlycorrectinordinarycohomology.
10 LauraEscobarandAllenKnutson
Example4.4.Considern=3.InT-equivariantcohomologywehavethat
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
[V ]= L:L∩P(0)(cid:54)=∅ ⊗ L:L∩P(0,1)(cid:54)=∅ ⊗ L:L∩P(1,2,3)(cid:54)=∅ +
n
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
L:L∩P(0)(cid:54)=∅ ⊗ L:L∩P(0,1,2)(cid:54)=∅ ⊗ L:L∩P(2,3)(cid:54)=∅ +
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
L:L∩P(0,1)(cid:54)=∅ ⊗ L:L∩P(1)(cid:54)=∅ ⊗ L:L∩P(1,2,3)(cid:54)=∅ +
(4otherterms).
Inordinarycohomologywehavethat
[V ]=[X ]⊗[X ]⊗[Gr(1,P3)] + [X ]⊗[Gr(1,P3)]⊗[X ]+
n 2 1 2 1
[X ]⊗[X ]⊗[Gr(1,P3)] + [X ]⊗[X ]⊗[X ] + [X ]⊗[Gr(1,P3)]⊗[X ]+
1 2 1 1 1 1 2
[Gr(1,P3)]⊗[X ]⊗[X ] + [Gr(1,P3)]⊗[X ]⊗[X ].
2 1 1 2
Proof. Fromequation(6)wehavethat
n−1
[Mn]=∏(cid:0)1⊗···⊗1⊗[M2]⊗1⊗···⊗1(cid:1)
∆ ∆
i=1
inthecohomologyof(Gr(0,P3)×Gr(1,P3))n.IdentifyingH∗((P3)2)∼=H∗(P3)⊗
H∗(P3)usingtheKu¨nnethisomorphism,wehave
(cid:2)(P3)2(cid:3)= ∑ (cid:104)P(0,1,...v)(cid:105)⊗(cid:104)P(v,v+1,...3)(cid:105)
∆
0≤v≤3
andhence
(cid:2)(P3)n(cid:3)= ∑ (cid:104)P(v0,...,v1)(cid:105)⊗···⊗(cid:104)P(vn−1,...,vn)(cid:105).
∆
0=v0≤v1≤...≤vn=3
PullingthisbacktoMn,wegetessentiallythesameformula
(cid:104) (cid:105) (cid:104) (cid:105)
[Mn]= ∑ (P,L):P∈P(v0,...,v1) ⊗···⊗ (P,L):P∈P(vn−1,...,vn) .
∆
0=v0≤v1≤...≤vn=3
PushingforwardthisclassundertheprojectionmapMn−→p Gr(1,P3)n,weget
p [Mn]=[V ]. Not all terms of the [Mn] formula above survive when we project
∗ ∆ n ∆
Mn toGr(1,P3)nbyforgettingP:
∆
• The image of {(P,L) : P ∈ P(0,1,2,3)} under p equals Gr(1,P3). Since for
v −v =3thedimensionof{(P,L): P∈P(0,1,2,3)}dropsunderp,anyfactor
i i−1
(cid:104) (cid:105)
(P,L): P∈P(0,1,2,3) pushesdownto0.
(cid:104) (cid:105) (cid:104) (cid:105)
• Theotherterms (P,L): P∈P(i,i+1,...,j) pushdownto L: L∩P(i,i+1,...,j)(cid:54)=∅ .