Projective 3D geometry class 4

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Projective 3D geometryclass 4

• Multiple View Geometry
• Slides modified from Marc Pollefeys Comp 290-089

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Last week
line at infinity (affinities)
circular points (similarities)
(orthogonality)
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Last week
cross-ratio
pole-polar relation
conjugate points lines
Chasles theorem
projective conic classification
affine conic classification
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Fixed points and lines
(eigenvectors H fixed points)
(?1?2 ? pointwise fixed line)
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Singular Value Decomposition
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Singular Value Decomposition

• 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
• ?8, ?8 and O 8

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3D points
3D point
in R3
in P3
projective transformation
(4x4-115 dof)
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Planes
3D plane
Dual points ? planes, lines ? lines
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Planes from points
Or implicitly from coplanarity condition
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Points from planes
Parameterizing points on a plan by representing a
plane by its span
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Lines
(4dof)
Example X-axis
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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
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Plücker line coordinates
(Plücker internal constraint)
(two lines intersect)
(two lines intersect)
(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 Q0 ? degenerate quadric
4. pole polar
5. (plane n quadric)conic
6. transformation

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Quadric classification
Rank Sign. Diagonal Equation Realization
4 4 (1,1,1,1) X2 Y2 Z210 No real points
2 (1,1,1,-1) X2 Y2 Z21 Sphere
0 (1,1,-1,-1) X2 Y2 Z21 Hyperboloid (1S)
3 3 (1,1,1,0) X2 Y2 Z20 Single point
1 (1,1,-1,0) X2 Y2 Z2 Cone
2 2 (1,1,0,0) X2 Y2 0 Single line
0 (1,-1,0,0) X2 Y2 Two planes
1 1 (1,0,0,0) X20 Single plane
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Quadric classification
Projectively equivalent to sphere
sphere
ellipsoid
paraboloid
hyperboloid of two sheets
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Twisted cubic

1. 3 intersection with plane (in general)
2. 12 dof (15 for A 3 for reparametrisation (1 ?
   ?2?3)
3. 2 constraints per point on cubic, defined by 6
   points
4. projectively equivalent to (1 ? ?2?3)
5. Horopter degenerate case for reconstruction

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Hierarchy of transformations
Projective 15dof
Intersection and tangency
Parallellism of planes, Volume ratios,
centroids, The plane at infinity p8
Affine 12dof
Similarity 7dof
The absolute conic O8
Euclidean 6dof
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.
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The plane at infinity
The plane at infinity p? 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
   p8
4. line // line (or plane) ? point of intersection
   in p8

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The absolute conic
The absolute conic O8 is a (point) conic on p?.
In a metric frame
or conic for directions (with no real points)
The absolute conic O8 is a fixed conic under the
projective transformation H iff H is a similarity

1. O8 is only fixed as a set
2. Circle intersect O8 in two points
3. Spheres intersect p8 in O8

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The absolute conic
Euclidean
Projective
(orthogonalityconjugacy)
normal
plane
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The absolute dual quadric
The absolute conic O8 is a fixed conic under
the projective transformation H iff H is a
similarity

1. 8 dof
2. plane at infinity p8 is the nullvector of O8
3. Angles

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