Slide 1ContentMultiple View Geometry course schedule (tentative)Projective 2D GeometryHomogeneous coordinatesPoints from lines and vice-versaIdeal points and the line at infinityA model for the projective planeDualityConicsFive points define a conicTangent lines to conicsDual conicsDegenerate conicsProjective transformationsSlide 16Removing projective distortionMore examplesTransformation of lines and conicsA hierarchy of transformationsClass I: Isometries: preserve Euclidean distanceSlide 22Slide 23Slide 24Action of affinities and projectivities on line at infinityDecomposition of projective transformationsOverview transformationsNumber of invariants?Projective 2D geometrycourse 2Multiple View GeometryComp 290-089Marc PollefeysContent•Background: Projective geometry (2D, 3D), Parameter estimation, Algorithm evaluation.•Single View: Camera model, Calibration, Single View Geometry.•Two Views: Epipolar Geometry, 3D reconstruction, Computing F, Computing structure, Plane and homographies.•Three Views: Trifocal Tensor, Computing T.•More Views: N-Linearities, Multiple view reconstruction, Bundle adjustment, auto-calibration, Dynamic SfM, Cheirality, DualityMultiple View Geometry course schedule(tentative)Jan. 7, 9 Intro & motivation Projective 2D GeometryJan. 14, 16 (no course) Projective 2D GeometryJan. 21, 23Projective 3D Geometry Parameter EstimationJan. 28, 30Parameter Estimation Algorithm EvaluationFeb. 4, 6 Camera Models Camera CalibrationFeb. 11, 13Single View Geometry Epipolar GeometryFeb. 18, 203D reconstruction Fund. Matrix Comp.Feb. 25, 27Structure Comp. Planes & HomographiesMar. 4, 6 Trifocal Tensor Three View ReconstructionMar. 18, 20Multiple View Geometry MultipleView ReconstructionMar. 25, 27Bundle adjustment PapersApr. 1, 3 Auto-Calibration PapersApr. 8, 10 Dynamic SfM PapersApr. 15, 17Cheirality PapersApr. 22, 24Duality Project Demos•Points, lines & conics•Transformations & invariants•1D projective geometry and the Cross-ratioProjective 2D GeometryHomogeneous coordinates0 cbyax Ta,b,c0,0)()( kkcykbxka TTa,b,cka,b,c ~Homogeneous representation of linesequivalence class of vectors, any vector is representativeSet of all equivalence classes in R3(0,0,0)T forms P2Homogeneous representation of points0 cbyax Ta,b,cl Tyx,x onif and only if 0l 11 x,y,a,b,cx,y,T 0,1,,~1,, kyxkyxTTThe point x lies on the line l if and only if xTl=lTx=0Homogeneous coordinatesInhomogeneous coordinates Tyx, T321,, xxxbut only 2DOFPoints from lines and vice-versal'lx Intersections of lines The intersection of two lines and is ll'Line joining two pointsThe line through two points and is x'xl xx'Example1x1yIdeal points and the line at infinity T0,,l'l ab Intersections of parallel lines TTand',,l' ,,l cbacba Example1x2xIdeal points T0,,21xxLine at infinity T1,0,0l l22RPtangent vectornormal direction ab , ba,Note that in P2 there is no distinction between ideal points and othersTheir inner product is zeroIn projective plane, two distinct lines meet in a single point andTwo distinct points lie on a single line not true in R^2A model for the projective planePoints in P^2 are represented by rays passing through origin in R^3.Lines in P^2 are represented by planes passing through originPoints and lines obtained by intersecting rays and planes by plane x3 = 1Lines lying in the x1 – x2 plane are ideal points; x1-x2 plane is l_{infinity}Dualityxl0xl T0lx Tl'lx x'xl Duality principle:To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theoremRoles of points and lines can be interchangedConicsCurve described by 2nd-degree equation in the plane022 feydxcybxyax0233231222121 fxxexxdxcxxb xax3231,xxyxxx or homogenized 0xx CTor in matrix formfedecbdba2/2/2/2/2/2/Cwith fedcba :::::5DOF:Five points define a conicFor each point the conic passes through022 feydxcyybxaxiiiiiior 0,,,,,22cfyxyyxxiiiiii Tfedcba ,,,,,c0111115525552544244424332333232222222211211121cyxyyxxyxyyxxyxyyxxyxyyxxyxyyxxstacking constraints yieldsTangent lines to conicsThe line l tangent to C at point x on C is given by l=CxlxCDual conics0ll*CTA line tangent to the conic C satisfies Dual conics = line conics = conic envelopes1* CCIn general (C full rank):Conic C, also called, “point conic” defines an equation on pointsApply duality: dual conic or line conic defines an equation on linesC* is the adjoint of Mtrix C; defined in Appendix 4 of H&ZPoints lie on a point lines are tangent conic to the point conic C; conic C is the envelope of lines l x0CxxT0*lClTDegenerate conicsA conic is degenerate if matrix C is not of full rankTTmllm Ce.g. two lines (rank 2)e.g. repeated line (rank 1)TllCllmDegenerate line conics: 2 points (rank 2), double point (rank1) CC **Note that for degenerate conicsProjective transformationsA projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3 lie on the same line if and only if h(x1),h(x2),h(x3) do. ( i.e. maps lines to lines in P2)Definition: A mapping h:P2P2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2 reprented by a vector x it is true that h(x)=HxTheorem:Definition: Projective transformation: linear transformation on homogeneous 3 vectors represented by a non singular matrix H321333231232221131211321'''xxxhhhhhhhhhxxxxx' Hor8DOF•projectivity=collineation=projective transformation=homography•Projectivity form a group: inverse of projectivity is also a projectivity; so is a composition of two projectivities.Projection along rays through a common point, (center of projection) defines a mapping from one plane to anothercentral projection may be expressed by x’=Hx(application of theorem)Central projection maps points on one plane to points on another
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