1. Multiple View Geometry
Marc Pollefeys
University of North Carolina at Chapel Hill
Modified by Philippos Mordohai

2. Fundamental matrix (3x3 rank 2 matrix)
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 C1 C2 l2 P l1 e1 e2 C1 C2 l2 P l1 e1 e2 m1 L1 m2 L2 M l2 C1 m1 L1 m2 L2 M C2 m1 m2
lT1 l2
Fundamental matrix (3x3 rank 2 matrix)

3. Outline The trifocal tensor 2-D Projective geometry
Chapters 2 and 3 of “Multiple View Geometry in Computer Vision” by Hartley and Zisserman

4. 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

5. Line-line-line relation

6. Point-line-line relation

7. Point-line-point relation

8. Point-point-point relation

9. 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)

10. Outline The trifocal tensor 2-D Projective geometry

11. Projective 2D Geometry Points, lines & conics
Transformations & invariants (next week)

12. 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
Homogeneous representation of points on
if and only if
The point x lies on the line l if and only if xTl=lTx=0
Homogeneous coordinates
Inhomogeneous coordinates
but only 2DOF

13. Points from lines and vice-versa
Intersections of lines
The intersection of two lines and is
Line joining two points
The line through two points and is
Example

14. Ideal points and the line at infinity
Intersections of parallel lines
Example
tangent vector
normal direction
Ideal points
Line at infinity
Note that in P2 there is no distinction
between ideal points and others

15. Duality 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 theorem

16. Outline The trifocal tensor 2-D Projective geometry

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

18. 3D points 3D point in R3 in P3 projective transformation
(4x4-1=15 dof)

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

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

21. Points from planes
(solve as right nullspace of )

22. Lines (4dof: 2 for each point on the planes) Example: X-axis
Span of WT is pencil of points:
Span of W*T is pencil of planes:
(4dof: 2 for each point on the planes)
Example: X-axis

23. Conics Curve described by 2nd-degree equation in the plane
or homogenized
or in matrix form
with
5DOF:

24. Five points define a conic
For each point the conic passes through or
stacking constraints yields

25. Tangent lines to conics
The line l tangent to C at point x on C is given by l=Cx l x C

26. Dual conics A line tangent to the conic C satisfies
In general (C full rank):
Dual conics = line conics = conic envelopes

27. Quadrics and dual quadrics
(Q : 4x4 symmetric matrix)
9 d.o.f.
in general 9 points define quadric
det Q=0 ↔ degenerate quadric
(plane ∩ quadric)=conic
transformation
3. and thus defined by less points
4. 5. Derive X’QX=x’M’QMx=0
relation to quadric (non-degenerate)
transformation

28. Quadric classification
Rank
Sign.
Diagonal
Equation
Realization 4
(1,1,1,1)
X2+ Y2+ Z2+1=0
No real points 2
(1,1,1,-1)
X2+ Y2+ Z2=1
Sphere
(1,1,-1,-1)
X2+ Y2= Z2+1
Hyperboloid (1S) 3
(1,1,1,0)
X2+ Y2+ Z2=0
Single point 1
(1,1,-1,0)
X2+ Y2= Z2
Cone
(1,1,0,0)
X2+ Y2= 0
Single line
(1,-1,0,0)
X2= Y2
Two planes
(1,0,0,0)
X2=0
Single plane
Signature sigma= sum of diagonal,e.g =2,always more + than -, so always positive…

29. Quadric classification
Projectively equivalent to sphere:
sphere
ellipsoid
hyperboloid of two sheets
paraboloid
Ruled quadrics:
hyperboloids of one sheet
Ruled quadric: two family of lines, called generators.
Hyperboloid of 1 sheet topologically equivalent to torus!
Degenerate ruled quadrics:
cone
two planes

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

31. 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

32. The plane at infinity
The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity
Represents 3DOF between projective and affine
canonical position
contains directions
two planes are parallel  line of intersection in π∞
line // line (or plane)  point of intersection in π∞

33. The absolute conic
The absolute conic Ω∞ is a (point) conic on π.
In a metric frame:
or conic for directions:
(with no real points)
The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity
Represent 5 DOF between affine and similarity
Ω∞ is only fixed as a set
Circles intersect Ω∞ in two points
Spheres intersect π∞ in Ω∞

34. The absolute conic Euclidean: Projective: (orthogonality=conjugacy)
Given plane at infinity and absolute conic
Euclidean:
Projective:
(orthogonality=conjugacy)
normal
Orthogonality is conjugacy with respect to Absolute Conic
plane
Pole-polar relationship. (out of scope for now)

35. The absolute dual quadric
The absolute conic Ω*∞ is a fixed conic under the projective transformation H iff H is a similarity
1, not equation like abs conic
8 dof
plane at infinity π∞ is the nullvector of Ω∞
Angles:

36. Next week Projective transformations
Why do we need the Absolute Conic, the Absolute Quadric and their images