Projective 3D geometry. Singular Value Decomposition

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Transcript of Projective 3D geometry. Singular Value Decomposition

  • Slide 1
  • Projective 3D geometry
  • Slide 2
  • Singular Value Decomposition
  • Slide 3
  • Homogeneous least-squares Span and null-space Closest rank r approximation Pseudo inverse
  • Slide 4
  • Projective 3D Geometry Points, lines, planes and quadrics Transformations , and
  • Slide 5
  • 3D points in R 3 in P 3 (4x4-1=15 dof) projective transformation 3D point
  • Slide 6
  • Planes Dual: points planes, lines lines 3D plane Euclidean representation Transformation
  • Slide 7
  • Planes from points (solve as right nullspace of ) Or implicitly from coplanarity condition
  • Slide 8
  • Points from planes (solve as right nullspace of ) Representing a plane by its span
  • Slide 9
  • Lines Example: X -axis (4dof) two points A and B two planes P and Q
  • Slide 10
  • Points, lines and planes
  • Slide 11
  • Plcker matrices Plcker 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
  • Slide 12
  • Plcker matrices Dual Plcker 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)
  • Slide 13
  • Plcker line coordinates (Plcker internal constraint) (two lines intersect)
  • Slide 14
  • 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
  • Slide 15
  • 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
  • Slide 16
  • Quadric classification Projectively equivalent to sphere: Ruled quadrics: hyperboloids of one sheet hyperboloid of two sheets paraboloid sphere ellipsoid Degenerate ruled quadrics: conetwo planes
  • Slide 17
  • Hierarchy of transformations Projective 15dof Affine 12dof Similarity 7dof Euclidean 6dof Intersection and tangency Parallellism of planes, Volume ratios, centroids, The plane at infinity The absolute conic Volume
  • Slide 18
  • 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
  • Slide 19
  • The plane at infinity The plane at infinity 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 4.line // line (or plane) point of intersection in
  • Slide 20
  • The absolute conic The absolute conic is a fixed conic under the projective transformation H iff H is a similarity The absolute conic is a (point) conic on . In a metric frame: or conic for directions: (with no real points) 1. is only fixed as a set 2.Circle intersect in two points 3.Spheres intersect in
  • Slide 21
  • The absolute conic Euclidean : Projective: (orthogonality=conjugacy) plane normal
  • Slide 22
  • The absolute dual quadric The absolute conic * is a fixed conic under the projective transformation H iff H is a similarity 1.8 dof 2.plane at infinity is the nullvector of 3.Angles: