CS 558 COMPUTER VISION
Lecture IX: Dimensionality Reduction
CS 558 COMPUTER VISION
Supplementary Lecture:
Single View and Epipolar Geometry
Slide adapted from S. Lazebnik
OUTLINE
Single view geometry
 Epipolar geometry

SINGLE-VIEW GEOMETRY
Odilon Redon, Cyclops, 1914
OUR GOAL: RECOVERY OF 3D STRUCTURE
•
Recovery of structure from one image is
inherently ambiguous
X?
x
X?
X?
OUR GOAL: RECOVERY OF 3D STRUCTURE
•
Recovery of structure from one image is
inherently ambiguous
OUR GOAL: RECOVERY OF 3D STRUCTURE
•
Recovery of structure from one image is
inherently ambiguous
AMES ROOM
http://en.wikipedia.org/wiki/Ames_room
OUR GOAL: RECOVERY OF 3D STRUCTURE
•
We will need multi-view geometry
RECALL: PINHOLE CAMERA MODEL
•
•
Principal axis: line from the camera center
perpendicular to the image plane
Normalized (camera) coordinate system: camera
center is at the origin and the principal axis is the zaxis
RECALL: PINHOLE CAMERA MODEL
( X ,Y , Z )  ( f X / Z , f Y / Z )
x  PX
X
   f X f
 
Y  
Z  fY 
   Z  
1
 
f
X
0  
Y 

0  
Z
1 0 
1
PRINCIPAL POINT
py
px
•
•
•
•
Principal point (p): point where principal axis intersects
the image plane (origin of normalized coordinate system).
Normalized coordinate system: origin is at the principal
point.
Image coordinate system: origin is in the corner.
How to go from normalized coordinate system to image
coordinate system?
PRINCIPAL POINT OFFSET
principal point:
py
( px , p y )
px
( X , Y , Z )  ( f X / Z  px , f Y / Z  p y )
X
   f X  Z px   f
 
Y  
 Z    f Y  Z py   
 
  
Z
 
1
 
f
px
py
1
X
0  
Y 

0  
Z
0 
1
PRINCIPAL POINT OFFSET
principal point:
 f X  Zp x   f

 
 f Y  Zp y   

 
Z

 
f
K  

f
P  KI | 0
f
( px , p y )
X
p x  1
0  
Y 



py  1
0  
Z
1  
1 0 
1
px 
p y  calibration matrix
1 
PIXEL COORDINATES
1
1

Pixel size:
mx m y

mx pixels per meter in horizontal direction,
my pixels per meter in vertical direction
 mx
K  

my
pixels/m
 f


1 
f
m
px   x
p y   
y
1  
pixels
x 
 y 
1 
CAMERA ROTATION AND TRANSLATION
•

~
~ ~
Xcam  R X - C
coords. of point
in camera frame
In general, the camera
coordinate frame will
be related to the world
coordinate frame by a
rotation and a
translation

coords. of camera center
in world frame
coords. of a point
in world frame (nonhomogeneous)
CAMERA ROTATION AND TRANSLATION
In non-homogeneous
coordinates:

~
~ ~
Xcam  R X - C
X cam
x  KI | 0Xcam
R

0

~ ~
 RC   X   R
   
1  1   0

~
 K R | RC X

~
 RC
X
1 
P  KR | t ,
~
t  RC
Note: C is the null space of the camera projection matrix (PC=0)
CAMERA PARAMETERS
•
Intrinsic parameters





Principal point coordinates
Focal length
Pixel magnification factors
Skew (non-rectangular pixels)
Radial distortion
 mx
K  

my
 f


1 
f
px   x
p y   
y
1  
x 
 y 
1 
CAMERA PARAMETERS
•
Intrinsic parameters





•
Principal point coordinates
Focal length
Pixel magnification factors
Skew (non-rectangular pixels)
Radial distortion
Extrinsic parameters

Rotation and translation relative to world coordinate
system
CAMERA CALIBRATION
x  KR t  X
  x  * * * *
 y   * * * *
  

   * * * *
X 
Y 
 
Z 
 
1
Source: D. Hoiem
CAMERA CALIBRATION
•
Given n points with known 3D coordinates Xi and
known image projections xi, estimate the camera
parameters
Xi
xi
P?
CAMERA CALIBRATION: LINEAR METHOD
x i  PXi  0
 x i  PXi
 0

T
X
i

 yi XTi

 XTi
0
xi XTi
T
 xi  P1 X i 
 y   PT X   0
 i  2 i
 1  P3T X i 
yi XTi  P1 
 
T
 xi X i  P2   0
0  P3 
Two linearly independent equations
CAMERA CALIBRATION: LINEAR METHOD
 0T
 T
X1

 T
0
XT
 n
•
•
•
•
T
1
T
X
0

XTn
0T
 y1X 

 x1X  P1 
 
  P2   0
T 
 yn X n  P3 
 xn XTn 
T
1
T
1
Ap  0
P has 11 degrees of freedom (12 parameters, but
scale is arbitrary)
One 2D/3D correspondence gives us two linearly
independent equations
Homogeneous least squares
6 correspondences needed for a minimal solution
CAMERA CALIBRATION: LINEAR METHOD
 0T
 T
X1

 T
0
XT
 n
•
T
1
T
X
0

XTn
0T
 y1X 

 x1X  P1 
 
  P2   0
T 
 yn X n  P3 
 xn XTn 
T
1
T
1
Ap  0
Note: for coplanar points that satisfy ΠTX=0,
we will get degenerate solutions (Π,0,0), (0,Π,0), or
(0,0,Π)
CAMERA CALIBRATION: LINEAR METHOD
•
•
Advantages: easy to formulate and solve
Disadvantages
Doesn’t directly tell you camera parameters
 Doesn’t model radial distortion
 Can’t impose constraints, such as known focal length
and orthogonality

•
Non-linear methods are preferred
Define error as difference between projected points
and measured points
 Minimize error using Newton’s method or other nonlinear optimization

Source: D. Hoiem
MULTI-VIEW GEOMETRY PROBLEMS
•
Structure: Given projections of the same 3D point in
two or more images, compute the 3D coordinates of that
point
?
Camera 1
R1,t1
Camera 2
R2,t2
Camera 3
R3,t3
Slide credit:
Noah Snavely
MULTI-VIEW GEOMETRY PROBLEMS
•
Stereo correspondence: Given a point in one of the
images, where could its corresponding points be in the
other images?
Camera 1
R1,t1
Camera 2
R2,t2
Camera 3
R3,t3
Slide credit:
Noah Snavely
MULTI-VIEW GEOMETRY PROBLEMS
•
Motion: Given a set of corresponding points in two or
more images, compute the camera parameters
Camera 1
R1,t1
?
Camera 2
R2,t2
?
?
Camera 3
R3,t3
Slide credit:
Noah Snavely
TRIANGULATION
•
Given projections of a 3D point in two or more
images (with known camera matrices), find the
coordinates of the point
X?
x1
O1
x2
O2
TRIANGULATION
•
We want to intersect the two visual rays
corresponding to x1 and x2, but because of noise
and numerical errors, they don’t meet exactly
R1
R2
X?
x1
O1
x2
O2
TRIANGULATION: GEOMETRIC APPROACH
•
Find shortest segment connecting the two
viewing rays and let X be the midpoint of that
segment
X
x1
O1
x2
O2
TRIANGULATION: LINEAR APPROACH
1 x1  P1X
2 x 2  P2 X
x1  P1X  0
[x 1 ]P1X  0
x 2  P2 X  0
[x 2 ]P2 X  0
Cross product as matrix multiplication:
 0

a  b   az
 a y

 az
0
ax
a y  bx 
 
 a x  by   [a ]b
0  bz 
TRIANGULATION: LINEAR APPROACH
1 x1  P1X
2 x 2  P2 X
x1  P1X  0
[x 1 ]P1X  0
x 2  P2 X  0
[x 2 ]P2 X  0
Two independent equations each in terms of
three unknown entries of X
TRIANGULATION: NONLINEAR APPROACH

Find X that minimizes
d ( x1 , P1 X )  d ( x2 , P2 X )
2
2
X?
x’1
x1
O1
x’2
x2
O2
TWO-VIEW GEOMETRY
EPIPOLAR GEOMETRY
X
x
x’
• Baseline – line connecting the two camera centers
• Epipolar Plane – plane containing baseline (1D family)
• Epipoles
= intersections of baseline with image planes
= projections of the other camera center
THE EPIPOLE
Photo by Frank Dellaert
EPIPOLAR GEOMETRY
X
x
x’
• Baseline – line connecting the two camera centers
• Epipolar Plane – plane containing baseline (1D family)
• Epipoles
= intersections of baseline with image planes
= projections of the other camera center
• Epipolar Lines - intersections of epipolar plane with image
planes (always come in corresponding pairs)
EXAMPLE: CONVERGING CAMERAS
EXAMPLE: MOTION PARALLEL TO IMAGE PLANE
EXAMPLE: MOTION PERPENDICULAR TO IMAGE PLANE
EXAMPLE: MOTION PERPENDICULAR TO IMAGE PLANE
EXAMPLE: MOTION PERPENDICULAR TO IMAGE PLANE
e’
e
Epipole has same coordinates
in both images.
Points move along lines
radiating from e: “Focus of
expansion”
EPIPOLAR CONSTRAINT
X
x
•
x’
If we observe a point x in one image, where can the
corresponding point x’ be in the other image?
EPIPOLAR CONSTRAINT
X
X
X
x
x’
x’
x’
• Potential matches for x have to lie on the corresponding
epipolar line l’.
• Potential matches for x’ have to lie on the corresponding
epipolar line l.
EPIPOLAR CONSTRAINT EXAMPLE
EPIPOLAR CONSTRAINT: CALIBRATED CASE
X
x
•
•
•
x’
Assume that the intrinsic and extrinsic parameters of the
cameras are known
We can multiply the projection matrix of each camera (and
the image points) by the inverse of the calibration matrix to
get normalized image coordinates
We can also set the global coordinate system to the coordinate
system of the first camera. Then the projection matrix of the
first camera is [I | 0].
EPIPOLAR CONSTRAINT: CALIBRATED CASE
X = RX’ + t
x’
x
t
R
The vectors x, t, and Rx’ are coplanar
EPIPOLAR CONSTRAINT: CALIBRATED CASE
X
x
x  [t  ( R x)]  0
x’
xT E x  0 with
E  [t ]R
Essential Matrix
(Longuet-Higgins, 1981)
The vectors x, t, and Rx’ are coplanar
EPIPOLAR CONSTRAINT: CALIBRATED CASE
X
x
x  [t  ( R x)]  0
•
•
•
•
•
x’
xT E x  0 with
E  [t ]R
E x’ is the epipolar line associated with x’ (l = E x’)
ETx is the epipolar line associated with x (l’ = ETx)
E e’ = 0 and ETe = 0
E is singular (rank two)
E has five degrees of freedom
EPIPOLAR CONSTRAINT: UNCALIBRATED CASE
X
x
•
•
x’
The calibration matrices K and K’ of the two
cameras are unknown
We can write the epipolar constraint in terms of
unknown normalized coordinates:
ˆxT E xˆ  0
x  K xˆ, x  K xˆ
EPIPOLAR CONSTRAINT: UNCALIBRATED CASE
X
x’
x
xˆ E xˆ  0
T
x F x  0 with
T
1

F  K EK
T
ˆx  K 1 x
xˆ   K  x
1
Fundamental Matrix
(Faugeras and Luong, 1992)
EPIPOLAR CONSTRAINT: UNCALIBRATED CASE
X
x’
x
xˆ E xˆ  0
T
•
•
•
•
•
x F x  0 with
T
1

F  K EK
T
F x’ is the epipolar line associated with x’ (l = F x’)
FTx is the epipolar line associated with x (l’ = FTx)
F e’ = 0 and FTe = 0
F is singular (rank two)
F has seven degrees of freedom
THE EIGHT-POINT ALGORITHM
x = (u, v, 1)T, x’ = (u’, v’, 1)T
Minimize:
N
T
2

(
x
F
x
)
 i i
i 1
under the constraint
F33 = 1
THE EIGHT-POINT ALGORITHM
•
Meaning of error
N
T
2

(
x
F
x
)
 i i :
i 1
•
sum of Euclidean distances between points xi and
epipolar lines F x’i (or points x’i and epipolar lines
FTxi) multiplied by a scale factor
Nonlinear approach: minimize
 d ( x , F x)  d ( x, F
N
2
i 1
2
i
i
i
T
xi )

PROBLEM WITH EIGHT-POINT ALGORITHM
PROBLEM WITH EIGHT-POINT ALGORITHM
Poor numerical conditioning
 Can be fixed by rescaling the data

THE NORMALIZED EIGHT-POINT ALGORITHM
(Hartley, 1995)
•
•
•
•
Center the image data at the origin, and scale it so the
mean squared distance between the origin and the data
points is 2 pixels
Use the eight-point algorithm to compute F from the
normalized points
Enforce the rank-2 constraint (for example, take SVD of
F and throw out the smallest singular value)
Transform fundamental matrix back to original units: if
T and T’ are the normalizing transformations in the two
images, than the fundamental matrix in original
coordinates is TT F T’
COMPARISON OF ESTIMATION ALGORITHMS
8-point
Normalized 8-point
Nonlinear least squares
Av. Dist. 1
2.33 pixels
0.92 pixel
0.86 pixel
Av. Dist. 2
2.18 pixels
0.85 pixel
0.80 pixel
FROM EPIPOLAR GEOMETRY TO CAMERA CALIBRATION
•
•
•
Estimating the fundamental matrix is known as
“weak calibration”
If we know the calibration matrices of the two
cameras, we can estimate the essential matrix: E
= KTFK’
The essential matrix gives us the relative
rotation and translation between the cameras, or
their extrinsic parameters