The Chern-Mather class of the multiview variety 7 1 Corey Harrisand Daniel Lowengrub 0 2 January26, 2017 n a J Abstract 4 2 The multiview variety associated to a collection of N cameras records which sequences of imagepointsinP2N canbeobtainedbytakingpicturesofagivenworldpointx ∈ P3 with ] the cameras. In order to reconstruct a scene from its picture under the different cameras it G isimportanttobeabletofindthecriticalpointsofthefunctionwhichmeasuresthedistance A between a general point u ∈ P2N and the multiview variety. In this paper we calculate a . specificdegree3polynomialthatcomputesthenumberofcriticalpointsasafunctionofN. In h ordertodothis,weconstructaresolutionofthemultiviewvariety,anduseittocomputeits t a Chern-Matherclass. m [ 1 Introduction 1 v Suppose that a collection of cameras are used to generate images of a scene. The problem of 9 7 triangulationistodeducetheworldcoordinatesofanobjectfromitspositionineachofthecamera 0 images. Ifweassume thatthe imagepointsaregivenwithinfinite precision, thentwocameras 7 sufficetodeterminetheworldpoint. However,duetothemanysourcesofnoiseinrealimages 0 such as pixelization and distortion, there typically will not be an exact solution and we will . 1 insteadtrytofindaworldpointwhosepictureis“ascloseaspossible”totheimagepoints. 0 More precisely, suppose the cameras are C1,...,CN and the image points are p1,...,pn ∈ 7 R2. Thegoalistofindaworldpointq∈R3 thatminimizestheleastsquareserror 1 : v N Xi error(q)= X(Ci(q)−pi)2. i=1 r a Oneapplicationistheproblemofreconstructingthe3Dstructureofatouristattractionbased on millions of online pictures. It is difficult to obtain the precise configuration of any single camera,soitwouldnotmakesensetouseonlyasmallsubsetofthemanddisregardtherest. A betterapproachistosolveanoptimizationproblemwhichincorporatesasmanyofthecameras aspossible. Thistechniquewasusedin[1]toreconstructtheentirecityofRomefromtwomillion onlineimages. Since the camera function C : R3 R2 is not linear, the standard method for solving the i triangulation problem is to first find th→e critical points of error(q) (e.g, with gradient descent), andthenselecttheonewiththesmallesterror.Inordertogaugethedifficultyofthisproblem,it isimportanttobeabletopredictthenumberofcriticalpointsthatweexpecttofindforagiven configurationofcameras. The goal of this paper is to give an explicit expression for the number of critical points of error(q) as a function of the number of cameras N. In fact, we compute this expression for a variation of the problem in which we allow the world points to take complex values, and we allowthesepointstobeintheprojectivespaceP3 asopposedtotheaffinespaceC3. Ourmain C resultisthatthenumberofcriticalpointsoferror(q)ispolynomialinthenumberofcameras. 1 Theorem1. Thenumberofcriticalpointsoferror(q)onP3 isequalto C p(N)=6N3−15N2+11N−4 whereNisthenumberofcameras. Note that our reformulation of the problem only increases the number of possible critical points. Onecansolvetheoriginalproblembyfirstfindingthesepoints,andthendiscardingthe onesthatarenotinR3. Forasimilarreason,thepolynomialp(N)isanupperboundonthenumberofcriticalpoints intheclassicaltriangulationproblem. In[10],adetailedinvestigationoftheLagrangemultiplier equationswhichdefinethecriticalpointsisusedtocomputethenumberofsuchpointsforN≤7. Basedontheseresults,itwasconjecturedin[4,Conjecture3.4]thatthenumberofpointsshould growasthefollowingpolynomial: 9 21 q(N)= N3− N2+8N−4. 2 2 Wenotethatourupperboundp(N)isfairlyclose. Inordertocomputethenumberp(N),wetakeaslightlydifferentperspectiveonthefunction error(q). BycombiningthecamerasC :R3 R2 weobtainarationalmap i → φ:R3 R2N. → Afterpassingtothecomplexnumbersandtakingtheprojectiveclosureweobtainarational map φ:P3C P2CN. → The image of this map is a three-dimensional variety MV ⊂ P2N which is known as the N multiviewvariety. Wecannowinterprettheerrorfunctionerror(q)asmeasuringthedistancebe- tweenapointq∈ P2N andMV . Withthisformulation,thenumberofcriticalpointsisknown N astheEuclideandistancedegreeofthevarietyMV .ThenotionofEDdegreewasintroducedin[4], N andtheauthorsremarkin[4,ex3.3]thatthetriangulationproblemwastheiroriginalmotivation forthisconcept. Inparticular,byusingresultsfrom[4]weproveinsection5thatthisnumbercanbecomputed intermsoftheChern-MatherclasscM(MV ). Ingeneral,theChern-Matherclassonlyprovides N an upper bound on the ED degree, but in the proof of theorem 4 we show that this inequality canbepromoted toanequalityfor reasonsspecific to the multiviewvariety. One advantageof thisapproachisthatitdependsonlyonthegeometricpropertiesofMV ,andnotonthespecific N featuresofthedefiningequations. Anotheradvantageisthatitreducesmostofthedifficultyto localcalculationsonMV . N OnecommonwayofcalculatingtheChern-MatherclassofasingularvarietyXistofirstfind aresolution X˜ −f X → and then analyze the singularities of f in order to compare the Chern class c(X˜) to the Chern- MatherclasscM(X) Inoursituation,itisnaturaltobuildaresolutionofMV byresolvingtherationalmapφ. In N section3,weconstructsucharesolution φ˜ :P˜3 MV N → andcalculateitsChowringandChernclass. In order to compare the Chern class of P˜3 to the Chern-Mather class of MV , we use the N theory of higher discriminants which was introduced in [9]. One aspect of this theory is that it specifieswhichpartsofthesingularlocusofXweneedtounderstandinordertorelatec(X˜)to cM(X). Aprecisestatementisgiveninproposition5. 2 Asweshowinproposition6,thehigherdiscriminantsofφ˜ aresurprisinglynice. Specifically, itturnsoutthatinordertocalculatecM(MV ),weonlyhavetocomputetheEulerobstruction N ofasinglepointx∈MV . N Moreover, in section 5.2 we show that after intersecting MV with a hyperplane at x, the N resultingsurfacesingularity(S,x)istaut. Inparticular,theEulerobstructionEu (x)isdeter- MVN minedbytheresolutiongraphofxinS. ThisallowsustousetheenumerativepropertiesofP˜3 thatareworkedoutinsection3tocomputeEu (x). MVN Inthefinalsection,weputthesepiecestogetherandobtainthepolynomialp(N). 2 Definitions and notation Let P be a 3×4 matrix with values in R. We consider each row l as an affine function on R3. Explicitly, l sends a vector v = (x,y,z) to the dot product of l and (x,y,z,1). We denote these functionsbyf,gandh. ThematrixPdefinesarationalmapφ :R3 99KR2: P v7 (f(v)/h(v),g(v)/h(v)) → which corresponds to the operation of mapping the “world coordinates” R3 to the “image coordinates” R2. In other words, it describes the process of taking a picture of the world with a camerawhoseparametersareencodedinP. It is not hard to prove that this description of a camera is equivalent to the pinhole camera model. Inparticular,thecamerahasapositioncalledthecameracenterandispointinginacertain direction.Theplanedefinedbythecameracenteranddirectioniscalledthecameraplane. Itturns outthatwiththeabovenotation,thecameraplaneistheplanedefinedbytheideal(h),andthe camera center is the point defined by (f,g,h). For the purposes of this paper, this observation willbetakenasadefinition. Now, suppose that we have a collection of cameras P ,...,P . By taking a picture of the 1 N worldwitheachofthecameras,weobtainarationalmap: φ ×···×φ :R3 99KR2×···×R2 =∼ R2N P1 PN Thismapclearlyextendstothecomplexnumbers,givingusarationalmapfromC3 99KC2N. Furthermore,byclearingthedenominatorsinthedefinitionofthemapsφ weobtainarational Pi map φ:P3 99KP2N C C definedby φ([x:y:z:w])=(f1h2...hN :g1h2...hN :···:h1...hN−1gN :h1...hN). (1) Theschemetheoreticimageofthismapiscalledthemultiviewvarietyassociatedtothecameras P ,...,P . 1 N Example1. Considerthefollowingthreecameras: 1 0 0 1 0 1 0 1 1 0 0 1 P1 =0 0 1 1, P2 =0 0 1 1, P3 =0 1 0 1. 0 1 0 0 1 0 0 0 0 0 1 0       Theassociatedrationalmapis φ([x:y:z:w])=[(x−w)xz:(z−w)xz:(y−w)yz:(z−w)yz:(x−w)xy:(y−w)xy:xyz]). 3 L 12 H 2 H 1 q 2 q 1 p 123 L 23 q 3 L 13 H 3 Figure1: Schematicofthreecameras Wesaythatacollectionofcamerasisingeneralpositionifthehyperplanesdefinedbythelinear functions{f ,g ,h ,...,f ,g ,h }associatedtotherowsofthecameramatricesareingeneral 1 1 1 N N N position. Finally, we will use the following notation throughout the paper (see figure 1). The camera planeofthei-thcamerawillbedenotedbyH andthecenterofthei-thcamerawillbedenoted i by q . Also, we define L = H ∩H for all 1 ≤ i < j ≤ N, and p = H ∩H ∩H for all i ij i j ijk i j k 1≤i0andα=1. Byproposition2theblowupatq willcontribute 0 k i α =(1−Q )(1+Q )3−1. i i i Similarly,γ representsthecontributionfromtheblowupofthepointp . ijk ijk Finally, we compute the contribution from the blowup along a line f : L ֒ Y . Since L ij 1 ij passesthrough N−2 of the blown up points in Y , a quick calculationshows t→hat we cantake 1 c = 1, c = −2(N−3)h, and the rest to be zero. In addition, since L =∼ P1, we can take 0 1 ij α=(1+h)2. ThisimpliesthatthecontributioncomingfromL is ij β =(1+h)2·[(1−T )((1−η)(−2(N−3)h)+(1+T )2)−(1−2(N−3)h)]. ij ij ij Wenowcomputethepullbackofc1(OP2n(1))inA•(P˜3)alongthemapφ˜. Lemma1. Thepullbackofc1(OP2n(1))toP˜3is φ˜∗(c1(OP2n(1)))∩[P˜3]=N·h+2· X Pijk+ X Qi+ X Tij. 1≤i

The Chern-Mather class of the multiview variety

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