Multiple View Geometry

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Multiple View Geometry

• Marc Pollefeys
• University of North Carolina at Chapel Hill

Modified by Philippos Mordohai
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Epipolar geometry
Underlying structure in set of matches for rigid
scenes

• Computable from corresponding points
• Simplifies matching
• Allows to detect wrong matches
• Related to calibration

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Outline

• Trifocal and quadrifocal tensors
• 2-D Projective geometry
• 3-D Projective geometry
• Chapters 1 and 2 of Multiple View Geometry in
  Computer Vision by Hartley and Zisserman

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The trifocal tensor
Three back-projected lines have to meet in a
single line Incidence relation provides
constraint on lines (impossible in two-view
case) Constraints contained in 3x3x3 trifocal
tensor
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Line-line-line relation
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Point-line-line relation
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Point-line-point relation
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Point-point-point relation
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Point-point-point relation
Given point correspondence in two images, the
point cannot always be determined in third image
(if it is on trifocal plane)
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The quadrifocal tensor
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Outline

• Trifocal and quadrifocal tensors
• 2-D Projective geometry
• 3-D Projective geometry

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Projective 2D Geometry

• Points, lines conics
• Transformations invariants (next week)

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Homogeneous coordinates
Homogeneous representation of lines
equivalence class of vectors, any vector is
representative Set of all equivalence classes in
R3?(0,0,0)T forms P2
The point x lies on the line l if and only if
xTllTx0
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Points from lines and vice-versa
Intersections of lines
The intersection of two lines and is
Example
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Ideal points and the line at infinity
Intersections of parallel lines
Note that in P2 there is no distinction between
ideal points and others
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Duality
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Outline

• Trifocal and quadrifocal tensors
• 2-D Projective geometry
• 3-D Projective geometry

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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
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Lines
Span of WT is pencil of points
Span of W is pencil of planes
(4dof 2 for each point on the planes)
Example X-axis
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Outline

• Trifocal and quadrifocal tensors
• 2-D Projective geometry
• 3-D Projective geometry
• 2-D Projective geometry revisited

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Conics
Curve described by 2nd-degree equation in the
plane
5DOF
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Five points define a conic
For each point the conic passes through
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Tangent lines to conics
The line l tangent to C at point x on C is given
by lCx
l
x
C
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Dual conics
A line tangent to the conic C satisfies
Dual conics line conics conic envelopes
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Outline

• Trifocal and quadrifocal tensors
• 2-D Projective geometry
• 3-D Projective geometry
• Back to 3-D

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Quadrics and dual quadrics
(Q 4x4 symmetric matrix)

• 9 d.o.f.
• in general 9 points define quadric
• det Q0 ? degenerate quadric
• (plane n quadric)conic
• transformation

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Quadric classification
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Quadric classification
Projectively equivalent to sphere
sphere
ellipsoid
paraboloid
hyperboloid of two sheets
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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

• canical position
• contains directions
• two planes are parallel ? line of intersection in
  p8
• 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

• O8 is only fixed as a set
• Circles intersect O8 in two points
• Spheres intersect p8 in O8

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The absolute conic
Given plane at infinity and 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

• 8 dof
• plane at infinity p8 is the nullvector of O8
• Angles

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Next week

• Projective transformations
• Why do we need the Absolute Conic, the Absolute
  Quadric and their images