Projective 2D geometry course 2ContentMultiple 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 transformationsProjection along rays through a common point, (center of projection) defines a mapping from one plane to anotherRemoving projective distortionMore examples Transformation of lines and conicsA hierarchy of transformationsClass I: Isometries: preserve Euclidean distanceClass II: Similarities: isometry composed with an isotropic scalingClass III: Affine transformations: non singular linear transformation followed by a translationClass IV: Projective transformations: general non singular linear transformation of homogenous coordinatesAction of affinities and projectivities on line at infinityDecomposition of projective transformationsOverview transformationsNumber of invariants?Projective 2D geometry course 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, 23 Projective 3D Geometry Parameter EstimationJan. 28, 30 Parameter Estimation Algorithm EvaluationFeb. 4, 6 Camera Models Camera CalibrationFeb. 11, 13 Single View Geometry Epipolar GeometryFeb. 18, 20 3D reconstruction Fund. Matrix Comp.Feb. 25, 27 Structure Comp. Planes & HomographiesMar. 4, 6 Trifocal Tensor Three View ReconstructionMar. 18, 20 Multiple View Geometry MultipleView ReconstructionMar. 25, 27 Bundle adjustment PapersApr. 1, 3 Auto-Calibration PapersApr. 8, 10 Dynamic SfM PapersApr. 15, 17 Cheirality PapersApr. 22, 24 Duality 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,c=l()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 l l'Line joining two pointsThe line through two points and is x'xl×=xx'Example1=x1=yIdeal points and the line at infinity()T0,,l'l ab −=×Intersections of parallel lines ()()TTand ',,l' ,,l cbacba ==Example1=x 2=xIdeal 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 Î 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=+++++ fxxexxdxcxxbxax3231,xxyxxx aaor homogenized0xx =CTor in matrix form⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=fedecbdba2/2/2/2/2/2/Cwith{}fedcba :::::5DOF:Five points define a conicFor each point the conic passes through022=+++++ feydxcyybxaxiiiiiior()0,,,,,22=cfyxyyxxiiiiii()Tfedcba ,,,,,=c0111115525552544244424332333232222222211211121=⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡cyxyyxxyxyyxxyxyyxxyxyyxxyxyyxxstacking 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; conicC is the envelope of linesl x0=CxxT0*=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)Tll=CllmDegenerate 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:P2→P2 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 H⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛321333231232221131211321'''xxxhhhhhhhhhxxxxx' H=or8DOF•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
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