3D photography

1
3D photography

• Marc Pollefeys
• Fall 2007
• http//www.inf.ethz.ch/personal/pomarc/courses/3dp
  hoto/

2
3D photography course schedule(tentative)
Lecture Exercise
Sept 26 Introduction -
Oct. 3 Geometry Camera model Camera calibration
Oct. 10 Single View Metrology Measuring in images
Oct. 17 Feature Tracking/matching (Friedrich Fraundorfer) Correspondence computation
Oct. 24 Epipolar Geometry F-matrix computation
Oct. 31 Shape-from-Silhouettes (Li Guan) Visual-hull computation
Nov. 7 Stereo matching Project proposals
Nov. 14 Structured light and active range sensing Papers
Nov. 21 Structure from motion Papers
Nov. 28 Multi-view geometry and self-calibration Papers
Dec. 5 Shape-from-X Papers
Dec. 12 3D modeling and registration Papers
Dec. 19 Appearance modeling and image-based rendering Final project presentations
3
Projective Geometry and Camera model Class 2

• points, lines, planes
• conics and quadrics
• transformations
• camera model
• Read tutorial chapter 2 and 3.1
• http//www.unc.edu/courses/2004fall/comp/290/089/
  

4
Homogeneous coordinates
Homogeneous representation of 2D points and lines
The point x lies on the line l if and only if
Note that scale is unimportant for incidence
relation
equivalence class of vectors, any vector is
representative Set of all equivalence classes in
R3?(0,0,0)T forms P2
5
Points from lines and vice-versa
Intersections of lines
The intersection of two lines and is
Example
Note
with
6
Ideal points and the line at infinity
Intersections of parallel lines
Note that in P2 there is no distinction between
ideal points and others
7
3D points and planes
Homogeneous representation of 3D points and planes
The point X lies on the plane p if and only if
The plane p goes through the point X if and only
if
8
Planes from points
9
Points from planes
Representing a plane by its span
10
Lines
Representing a line by its span
(4dof)
Example X-axis
(Alternative Plücker representation, details see
e.g. HZ)
11
Points, lines and planes
12
Plücker coordinates

• Elegant representation for 3D lines

(with A and B points)
(Plücker internal constraint)
(two lines intersect)
(for more details see e.g. HZ)
13
Conics
Curve described by 2nd-degree equation in the
plane
5DOF
14
Five points define a conic
For each point the conic passes through
or
15
Tangent lines to conics
The line l tangent to C at point x on C is given
by lCx
l
x
C
16
Dual conics
A line tangent to the conic C satisfies
Dual conics line conics conic envelopes
17
Degenerate conics
A conic is degenerate if matrix C is not of full
rank
e.g. two lines (rank 2)
e.g. repeated line (rank 1)
Degenerate line conics 2 points (rank 2), double
point (rank1)
18
Quadrics and dual quadrics
(Q 4x4 symmetric matrix)

• 9 d.o.f.
• in general 9 points define quadric
• det Q0 ? degenerate quadric
• tangent plane

• relation to quadric
  (non-degenerate)

19
2D projective transformations
Definition
A projectivity is an invertible mapping h from P2
to itself such that three points x1,x2,x3 lie on
the same line if and only if h(x1),h(x2),h(x3) do.
projectivitycollineationprojective
transformationhomography
20
Transformation of 2D points, lines and conics
For a point transformation
Transformation for lines
21
Fixed points and lines
(eigenvectors H fixed points)
(?1?2 ? pointwise fixed line)
22
Hierarchy of 2D transformations
transformed squares
invariants
Concurrency, collinearity, order of contact
(intersection, tangency, inflection, etc.), cross
ratio
Projective 8dof
Parallellism, ratio of areas, ratio of lengths on
parallel lines (e.g midpoints), linear
combinations of vectors (centroids). The line at
infinity l8
Affine 6dof
Ratios of lengths, angles. The circular points
I,J
Similarity 4dof
Euclidean 3dof
lengths, areas.
23
The line at infinity
The line at infinity l? is a fixed line under a
projective transformation H if and only if H is
an affinity
Note not fixed pointwise
24
Affine properties from images
projection
rectification
25
Affine rectification
v1
v2
l8
l1
l3
l2
l4
26
The circular points
The circular points I, J are fixed points under
the projective transformation H iff H is a
similarity
27
The circular points
circular points
28
Conic dual to the circular points
l8
29
Angles
Euclidean
30
Transformation of 3D points, planes and quadrics
For a point transformation
(cfr. 2D equivalent)
Transformation for lines
Transformation for conics
Transformation for dual conics
31
Hierarchy of 3D transformations
Projective 15dof
Intersection and tangency
Parallellism of planes, Volume ratios,
centroids, The plane at infinity p8
Affine 12dof
Similarity 7dof
Angles, ratios of length The absolute conic O8
Euclidean 6dof
Volume
32
The plane at infinity
The plane at infinity p? is a fixed plane under
a projective transformation H iff H is an
affinity

1. canonical position
2. contains directions
3. two planes are parallel ? line of intersection in
   p8
4. line // line (or plane) ? point of intersection
   in p8

33
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 circular points
3. Spheres intersect p8 in O8

34
The absolute dual quadric
The absolute dual quadric 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

35
Camera model

• Relation between pixels and rays in space

?
36
Pinhole camera
37
Pinhole camera model
linear projection in homogeneous coordinates!
38
Pinhole camera model
39
Principal point offset
principal point
40
Principal point offset
calibration matrix
41
Camera rotation and translation

42
CCD camera
43
General projective camera
11 dof (533)
intrinsic camera parameters extrinsic camera
parameters
44
Radial distortion

• Due to spherical lenses (cheap)
• Model

R
R
straight lines are not straight anymore
http//foto.hut.fi/opetus/260/luennot/11/atkinson_
6-11_radial_distortion_zoom_lenses.jpg
45
Camera model

• Relation between pixels and rays in space

?
46
Projector model

• Relation between pixels and rays in space
• (dual of camera)
• (main geometric difference is vertical principal
  point offset
• to reduce keystone effect)

?
47
Meydenbauer camera
vertical lens shift to allow direct
ortho-photographs
48
Affine cameras
49
Action of projective camera on points and lines
projection of point
forward projection of line
back-projection of line
50
Action of projective camera on conics and quadrics
back-projection to cone
projection of quadric
51
Image of absolute conic
52
A simple calibration device

• compute H for each square
• (corners ? (0,0),(1,0),(0,1),(1,1))
• compute the imaged circular points H(1,i,0)T
• fit a conic to 6 circular points
• compute K from w through cholesky factorization

( Zhangs calibration method)
53
Exercises Camera calibration
54
Next classSingle View Metrology
Antonio Criminisi