1. No category

Projective Geometry
Lecture slides by Steve Seitz (mostly)
Lecture presented by Rick Szeliski
CSE 576, Spring 2008
Projective Geometry
1
Final project ideas
Discussion by Steve Seitz and Rick Szeliski
…
CSE 576, Spring 2008
Projective Geometry
2
Projective geometry
Ames Room
Readings
•
Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix:
Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992,
(read 23.1 - 23.5, 23.10)
– available online: http://www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf
CSE 576, Spring 2008
Projective Geometry
3
Projective geometry—what’s it good for?
Uses of projective geometry
•
•
•
•
•
•
•
Drawing
Measurements
Mathematics for projection
Undistorting images
Focus of expansion
Camera pose estimation, match move
Object recognition
CSE 576, Spring 2008
Projective Geometry
4
Applications of projective geometry
Vermeer’s Music Lesson
Reconstructions by Criminisi et al.
CSE 576, Spring 2008
Projective Geometry
5
Measurements on planes
4
3
2
1
1
2
Approach: unwarp then measure
What kind of warp is this?
CSE 576, Spring 2008
Projective Geometry
3
4
6
Image rectification
p’
p
To unwarp (rectify) an image
• solve for homography H given p and p’
• solve equations of the form: wp’ = Hp
– linear in unknowns: w and coefficients of H
– H is defined up to an arbitrary scale factor
– how many points are necessary to solve for H?
CSE 576, Spring 2008
Projective Geometry
7
work out on board
Solving for homographies
CSE 576, Spring 2008
Projective Geometry
8
Solving for homographies
A
2n × 9
h
9
0
2n
Defines a least squares problem:
• Since h is only defined up to scale, solve for unit vector ĥ
• Solution: ĥ = eigenvector of ATA with smallest eigenvalue
• Works with 4 or more points
CSE 576, Spring 2008
Projective Geometry
9
The projective plane
Why do we need homogeneous coordinates?
• represent points at infinity, homographies, perspective
projection, multi-view relationships
What is the geometric intuition?
• a point in the image is a ray in projective space
-y
(sx,sy,s)
(x,y,1)
(0,0,0)
-z
x
image plane
• Each point (x,y) on the plane is represented by a ray (sx,sy,s)
– all points on the ray are equivalent: (x, y, 1)  (sx, sy, s)
CSE 576, Spring 2008
Projective Geometry
10
Projective lines
What does a line in the image correspond to in
projective space?
• A line is a plane of rays through origin
– all rays (x,y,z) satisfying: ax + by + cz = 0
in vector notation :
 x
0  a b c  y 
 z 
l
p
• A line is also represented as a homogeneous 3-vector l
CSE 576, Spring 2008
Projective Geometry
11
Point and line duality
• A line l is a homogeneous 3-vector
• It is  to every point (ray) p on the line: l p=0
l
p1
p2
l1
p
l2
What is the line l spanned by rays p1 and p2 ?
• l is  to p1 and p2  l = p1  p2
• l is the plane normal
What is the intersection of two lines l1 and l2 ?
• p is  to l1 and l2  p = l1  l2
Points and lines are dual in projective space
CSE 576,• Spring
2008any
given
Projective
Geometry
formula, can
switch
the meanings of points and
lines to get another formula
12
Ideal points and lines
-y
-z
(a,b,0)
-y
(sx,sy,0)
x
image plane
-z
image plane
x
Ideal point (“point at infinity”)
• p  (x, y, 0) – parallel to image plane
• It has infinite image coordinates
Ideal line
• l  (a, b, 0) – parallel to image plane
• Corresponds to a line in the image (finite coordinates)
– goes
CSE 576, Spring
2008
through imageProjective
origin Geometry
(principle point)
13
Homographies of points and lines
Computed by 3x3 matrix multiplication
• To transform a point: p’ = Hp
• To transform a line: lp=0  l’p’=0
– 0 = lp = lH-1Hp = lH-1p’  l’ = lH-1
– lines are transformed by postmultiplication of H-1
CSE 576, Spring 2008
Projective Geometry
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3D projective geometry
These concepts generalize naturally to 3D
• Homogeneous coordinates
– Projective 3D points have four coords: P = (X,Y,Z,W)
• Duality
– A plane N is also represented by a 4-vector
– Points and planes are dual in 3D: N P=0
• Projective transformations
– Represented by 4x4 matrices T: P’ = TP,
CSE 576, Spring 2008
Projective Geometry
N’ = N T-1
15
3D to 2D: “perspective” projection
Matrix Projection:
 wx * * * *  X 
p  wy  * * * *  YZ   ΠP
 w  * * * *
 1 
What is not preserved under perspective projection?
What IS preserved?
CSE 576, Spring 2008
Projective Geometry
16
Vanishing points
image plane
vanishing point v
camera
center
C
line on ground plane
Vanishing point
• projection of a point at infinity
CSE 576, Spring 2008
Projective Geometry
17
Vanishing points
image plane
vanishing point v
camera
center
C
line on ground plane
line on ground plane
Properties
• Any two parallel lines have the same vanishing point v
• The ray from C through v is parallel to the lines
• An image may have more than one vanishing point
– in fact every pixel is a potential vanishing point
CSE 576, Spring 2008
Projective Geometry
18
Vanishing lines
v1
v2
Multiple Vanishing Points
• Any set of parallel lines on the plane define a vanishing point
• The union of all of vanishing points from lines on the same
plane is the vanishing line
– For the ground plane, this is called the horizon
CSE 576, Spring 2008
Projective Geometry
19
Vanishing lines
Multiple Vanishing Points
• Different planes define different vanishing lines
CSE 576, Spring 2008
Projective Geometry
20
Computing vanishing points
V
P0
D
 PX
P
Pt   Y
 PZ


CSE 576, Spring 2008
P  P0  tD
 tDX 
 tDY 
 tDZ 

1

Projective Geometry
21
Computing vanishing points
V
P0
D
 PX
P
Pt   Y
 PZ


Properties
 tDX   PX
 tDY   PY

 tDZ   PZ
 
1
 
/ t  DX 
/ t  DY 
/ t  DZ 

1/ t

t 
P  P0  tD
 DX 
D 
P   Y 
 DZ 
 
 0 
v  ΠP
• P is a point at infinity, v is its projection
• They depend only on line direction
• Parallel lines P0 + tD, PProjective
at P
1 + tD intersect
CSE 576, Spring 2008
Geometry
22
Computing the horizon
C
l
ground plane
Properties
• l is intersection of horizontal plane through C with image plane
• Compute l from two sets of parallel lines on ground plane
• All points at same height as C project to l
– points higher than C project above l
• Provides way of comparing height of objects in the scene
CSE 576, Spring 2008
Projective Geometry
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CSE 576, Spring 2008
Projective Geometry
24
Fun with vanishing points
CSE 576, Spring 2008
Projective Geometry
25
Perspective cues
CSE 576, Spring 2008
Projective Geometry
26
Perspective cues
CSE 576, Spring 2008
Projective Geometry
27
Perspective cues
CSE 576, Spring 2008
Projective Geometry
28
Comparing heights
Vanishing
Point
CSE 576, Spring 2008
Projective Geometry
29
Measuring height
5
4
3
5.4
Camera height
3.3
2.8
2
1
What is the height of the camera?
CSE 576, Spring 2008
Projective Geometry
30
Computing vanishing points (from lines)
v
q2
q1
p2
p1
Intersect p1q1 with p2q2
Least squares version
• Better to use more than two lines and compute the “closest” point of
intersection
• See notes by Bob Collins for one good way of doing this:
– http://www-2.cs.cmu.edu/~ph/869/www/notes/vanishing.txt
CSE 576, Spring 2008
Projective Geometry
31
Measuring height without a ruler
C
Z
ground plane
Compute Z from image measurements
• Need more than vanishing points to do this
CSE 576, Spring 2008
Projective Geometry
32
The cross ratio
A Projective Invariant
• Something that does not change under projective transformations
(including perspective projection)
The cross-ratio of 4 collinear points
P3
P2
P1
P4
P3  P1 P4  P2
P3  P2 P4  P1
Xi 
Y 
Pi   i 
 Zi 
 
1
P1  P3 P4  P2
Can permute the point ordering
P1  P2 P4  P3
• 4! = 24 different orders (but only 6 distinct values)
This is the fundamental invariant of projective geometry
CSE 576, Spring 2008
Projective Geometry
33
Measuring height

TB R
R B T
H

R
scene cross ratio
T
(top of object)
t  b vZ  r
t
r
C
vZ
b
R
H
(reference point)
R
B
r  b vZ  t
H

R
image cross ratio
(bottom of object)
ground plane
X 
 x
Y 
scene points represented as P    image points as p   y 
Z 
 1 
CSE 576, Spring 2008
Projective Geometry

1
34
Measuring height
vz
r
vanishing line (horizon)
t0
vx
t
vy
v
H
R H
b0
b
t  b vZ  r
H

r  b vZ  t
R
image cross ratio
CSE 576, Spring 2008
Projective Geometry
35
Measuring height
vz
r
vanishing line (horizon)
t0
t0
vx
vy
v
m0
t1
b1
b0
b
What if the point on the ground plane b0 is not known?
• Here the guy is standing on the box, height of box is known
• Use one side of the box to help find b0 as shown above
CSE 576, Spring 2008
Projective Geometry
36
Computing (X,Y,Z) coordinates
CSE 576, Spring 2008
Projective Geometry
37
3D Modeling from a photograph
CSE 576, Spring 2008
Projective Geometry
38
Camera calibration
Goal: estimate the camera parameters
• Version 1: solve for projection matrix
 wx * * * *  X 
x  wy  * * * *  YZ   ΠX
 w  * * * *
 1 
• Version 2: solve for camera parameters separately
– intrinsics (focal length, principle point, pixel size)
– extrinsics (rotation angles, translation)
– radial distortion
CSE 576, Spring 2008
Projective Geometry
39
Vanishing points and projection matrix
*
Π  *
*
π1
*
*
*
π2
*
*
*
π3
*
*  π1
*
π4
π1  Π 1 0 0 0
T
π2
π3
π4 
= vx (X vanishing point)
similarly, π 2  v Y , π 3  v Z
π 4  Π0 0 0 1  projection of world origin
T
Π  v X
vY
vZ
o
Not So Fast! We only know v’s and o up to a scale factor
Π  a v X
• Spring
Need2008
a bit
CSE 576,
bvY
cv Z
do
more workProjective
to getGeometry
these scale factors…
40
Finding the scale factors…
Let’s assume that the camera is reasonable
• Square pixels
• Image plane parallel to sensor plane
• Principal point in the center of the image
 r11 r12
1
0
0
1
0
0
0






Π  0 1 0  0 1 0 0 r21 r22
r
r
0 0 1 / f  0 0 1 0  31 32
0 0
 r11
  r21
r31 / f
 a v X
CSE 576, Spring 2008
r12
r13
r22
r32 / f
r23
r33 / f
bvY
cv Z
r13 0 1 0
r23 0 0 1
r33 0 0 0

0 1  0 0
t '1 
t '2 
t '3 / f 
0 t1 
0 t 2 
1 t3 

0 1
 t '1   r11 r12
t '   r
 2   21 r22
t '3  r31 r32
r13   t1 
r23  t 2 
r33  t3 
do
Projective Geometry
41
Solving for f
 v x1

v x 2
vx 3

vX
 v x1

 vx 2
 fvx 3

o1   r11 / a

o2    r21 / a
o3  r31 / af
v y1
vy2
vy3
v z1
vz 2
vz 3
vY
vZ
v y1
vy2
fvy 3
r12 / b
r22 / b
r32 / bf
o
v z1
vz 2
fvz 3
t '1 / d 
t '2 / d 
t '3 / df 
o1   r11 / a r12 / b r13 / c t '1 / d 

o2   r21 / a r22 / b r23 / c t '2 / d 
fo3  r31 / a r32 / b r33 / c t '3 / d 
Orthogonal
vectors
Orthogonal
vectors
vx1v y1 vx 2v y 2  f vx3v y 3  0
2
CSE 576, Spring 2008
r13 / c
r23 / c
r33 / cf
Projective Geometry
f 
v x1v y1 v x 2 v y 2
 v x 3v y 3
42
Solving for a, b, and c
 v x1 / f

v x 2 / f
 vx3

v y1 / f
vy2 / f
vy3
v z1 / f
vz 2 / f
vz 3
Norm
= 1/a
o1 / f   r11 / a r12 / b r13 / c t '1 / d 

o2 / f   r21 / a r22 / b r23 / c t '2 / d 
o3   r31 / a r32 / b r33 / c t '3 / d 
Norm
= 1/a
Solve for a, b, c
• Divide the first two rows by f, now that it is known
• Now just find the norms of the first three columns
• Once we know a, b, and c, that also determines R
How about d?
• Need a reference point in the scene
CSE 576, Spring 2008
Projective Geometry
43
Solving for d
 wu   r11 r12
 wv   r
   21 r22
 w  r31 r32
r13
r23
r33
0
t '1 / d   
0


t '2 / d 
H 
t '3 / d   
1
Suppose we have one reference height H
• E.g., we known that (0, 0, H) gets mapped to (u, v)
r13 H  t '1 / d
u
r33 H  t '3 / d
t '1  ut '3
d
ur33 H  r13 H
Finally, we can solve for t
 t1   r11 r12
t   r
 2   21 r22
t3   r31 r32
CSE 576, Spring 2008
Projective Geometry
r13  T  t '1 
r23  t '2 
r33  t '3 
44
Calibration using a reference object
Place a known object in the scene
• identify correspondence between image and scene
• compute mapping from scene to image
Issues
• must know geometry very accurately
CSE 576,
Projective Geometry
• Spring
must2008
know 3D->2D correspondence
45
Chromaglyphs
Courtesy of Bruce Culbertson, HP Labs
http://www.hpl.hp.com/personal/Bruce_Culbertson/ibr98/chromagl.htm
CSE 576, Spring 2008
Projective Geometry
46
Estimating the projection matrix
Place a known object in the scene
• identify correspondence between image and scene
• compute mapping from scene to image
CSE 576, Spring 2008
Projective Geometry
47
Direct linear calibration
CSE 576, Spring 2008
Projective Geometry
48
Direct linear calibration
Can solve for mij by linear least squares
• use eigenvector trick that we used for homographies
CSE 576, Spring 2008
Projective Geometry
49
Direct linear calibration
Advantage:
• Very simple to formulate and solve
Disadvantages:
•
•
•
•
Doesn’t tell you the camera parameters
Doesn’t model radial distortion
Hard to impose constraints (e.g., known focal length)
Doesn’t minimize the right error function
For these reasons, nonlinear methods are preferred
• Define error function E between projected 3D points and image positions
– E is nonlinear function of intrinsics, extrinsics, radial distortion
• Minimize E using nonlinear optimization techniques
– e.g., variants of Newton’s method (e.g., Levenberg Marquart)
CSE 576, Spring 2008
Projective Geometry
50
Alternative: multi-plane calibration
Images courtesy Jean-Yves Bouguet, Intel Corp.
Advantage
•
•
•
Only requires a plane
Don’t have to know positions/orientations
Good code available online!
–
Intel’s OpenCV library: http://www.intel.com/research/mrl/research/opencv/
–
Matlab version by Jean-Yves Bouget:
http://www.vision.caltech.edu/bouguetj/calib_doc/index.html
CSE 576, Spring 2008
Projective Geometry
51
– Zhengyou Zhang’s web site: http://research.microsoft.com/~zhang/Calib/
Some Related Techniques
Image-Based Modeling and Photo Editing
• Mok et al., SIGGRAPH 2001
• http://graphics.csail.mit.edu/ibedit/
Single View Modeling of Free-Form Scenes
• Zhang et al., CVPR 2001
• http://grail.cs.washington.edu/projects/svm/
Tour Into The Picture
• Anjyo et al., SIGGRAPH 1997
• http://koigakubo.hitachi.co.jp/little/DL_TipE.html
CSE 576, Spring 2008
Projective Geometry
52