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Projective 3D geometry class 4 Multiple View Geometry Comp 290-089 Marc Pollefeys

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Content 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, Duality

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Multiple View Geometry course schedule (subject to change) Jan. 7, 9Intro & motivationProjective 2D Geometry Jan. 14, 16(no course)Projective 2D Geometry Jan. 21, 23Projective 3D GeometryParameter Estimation Jan. 28, 30Parameter EstimationAlgorithm Evaluation Feb. 4, 6Camera ModelsCamera Calibration Feb. 11, 13Single View GeometryEpipolar Geometry Feb. 18, 203D reconstructionFund. Matrix Comp. Feb. 25, 27Structure Comp.Planes & Homographies Mar. 4, 6Trifocal TensorThree View Reconstruction Mar. 18, 20Multiple View GeometryMultipleView Reconstruction Mar. 25, 27Bundle adjustmentPapers Apr. 1, 3Auto-CalibrationPapers Apr. 8, 10Dynamic SfMPapers Apr. 15, 17CheiralityPapers Apr. 22, 24DualityProject Demos

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Last week … (orthogonality) circular points (similarities) line at infinity (affinities)

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Last week … pole-polar relation conjugate points & lines projective conic classification affine conic classification A B C D X Chasles theorem cross-ratio

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Fixed points and lines (eigenvectors H =fixed points) (eigenvectors H - T =fixed lines) ( 1 = 2 pointwise fixed line)

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Singular Value Decomposition

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Homogeneous least-squares Span and null-space Closest rank r approximation Pseudo inverse

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Projective 3D Geometry Points, lines, planes and quadrics Transformations П, ω and Ω

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3D points in R 3 in P 3 (4x4-1=15 dof) projective transformation 3D point

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Planes Dual: points planes, lines lines 3D plane Euclidean representation Transformation

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Planes from points (solve as right nullspace of ) Or implicitly from coplanarity condition

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Points from planes (solve as right nullspace of ) Representing a plane by its span

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Lines Example: X -axis (4dof)

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Points, lines and planes

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Plücker matrices Plücker matrix (4x4 skew-symmetric homogeneous matrix) 1.L has rank 2 2.4dof 3.generalization of 4. L independent of choice A and B 5.Transformation Example: x -axis

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Plücker matrices Dual Plücker matrix L * Correspondence Join and incidence (plane through point and line) (point on line) (intersection point of plane and line) (line in plane) (coplanar lines)

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Plücker line coordinates on Klein quadric

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Plücker line coordinates (Plücker internal constraint) (two lines intersect)

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Quadrics and dual quadrics ( Q : 4x4 symmetric matrix) 1.9 d.o.f. 2.in general 9 points define quadric 3.det Q=0 degenerate quadric 4.pole – polar 5.(plane quadric)=conic 6.transformation 1.relation to quadric (non-degenerate) 2.transformation

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Quadric classification RankSign.DiagonalEquationRealization 44(1,1,1,1)X 2 + Y 2 + Z 2 +1=0No real points 2(1,1,1,-1)X 2 + Y 2 + Z 2 =1Sphere 0(1,1,-1,-1)X 2 + Y 2 = Z 2 +1Hyperboloid (1S) 33(1,1,1,0)X 2 + Y 2 + Z 2 =0Single point 1(1,1,-1,0)X 2 + Y 2 = Z 2 Cone 22(1,1,0,0)X 2 + Y 2 = 0Single line 0(1,-1,0,0)X 2 = Y 2 Two planes 11(1,0,0,0)X 2 =0Single plane

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Quadric classification Projectively equivalent to sphere: Ruled quadrics: hyperboloids of one sheet hyperboloid of two sheets paraboloid sphere ellipsoid Degenerate ruled quadrics: conetwo planes

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Twisted cubic conictwisted cubic 1.3 intersection with plane (in general) 2.12 dof (15 for A – 3 for reparametrisation (1 θ θ 2 θ 3 ) 3.2 constraints per point on cubic, defined by 6 points 4.projectively equivalent to (1 θ θ 2 θ 3 ) 5.Horopter & degenerate case for reconstruction

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Hierarchy of transformations Projective 15dof Affine 12dof Similarity 7dof Euclidean 6dof Intersection and tangency Parallellism of planes, Volume ratios, centroids, The plane at infinity π The absolute conic Ω Volume

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Screw decomposition Any particular translation and rotation is equivalent to a rotation about a screw axis and a translation along the screw axis. screw axis // rotation axis

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The plane at infinity The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity 1.canical position 2.contains directions 3.two planes are parallel line of intersection in π 4.line // line (or plane) point of intersection in π

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The absolute conic The absolute conic Ω is a fixed conic under the projective transformation H iff H is a similarity The absolute conic Ω is a (point) conic on π. In a metric frame: or conic for directions: (with no real points) 1.Ω is only fixed as a set 2.Circle intersect Ω in two points 3.Spheres intersect π in Ω

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The absolute conic Euclidean : Projective: (orthogonality=conjugacy) plane normal

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The absolute dual quadric The absolute conic Ω * is a fixed conic under the projective transformation H iff H is a similarity 1.8 dof 2.plane at infinity π is the nullvector of Ω 3.Angles:

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Next classes: Parameter estimation Direct Linear Transform Iterative Estimation Maximum Likelihood Est. Robust Estimation

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