Single-view geometry
Odilon Redon, Cyclops, 1914
Geometric vision
• Goal: Recovery of 3D structure
• What cues in the image allow us to do this?
Visual cues
Shading
Merle Norman Cosmetics, Los Angeles
Slide credit: S. Seitz
Visual cues
Focus
From The Art of Photography, Canon
Slide credit: S. Seitz
Visual cues
Perspective
Slide credit: S. Seitz
Visual cues
Motion
Slide credit: S. Seitz
Our goal: Recovery of 3D structure
• We will focus on perspective and motion
• We need multi-view geometry because
recovery of structure from one image is
inherently ambiguous
X?
x
X?
X?
Our goal: Recovery of 3D structure
• We will focus on perspective and motion
• We need multi-view geometry because
recovery of structure from one image is
inherently ambiguous
Our goal: Recovery of 3D structure
• We will focus on perspective and motion
• We need multi-view geometry because
recovery of structure from one image is
inherently ambiguous
Recall: Pinhole camera model
( X ,Y , Z )  ( f X / Z , f Y / Z )
X
   f X f
 
Y  
Z  fY 
   Z  
1
 
f
X
0  
Y 

0  
Z
1 0 
1
x  PX
Pinhole camera model
 f X f

 
 fY 
 Z  

 
x  PX
f
X
0  
 1
Y 
 1

0  

Z
1 
1 0 
1
P  diag ( f , f ,1)I | 0
Camera coordinate system
• 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 z-axis
• Principal point (p): point where principal axis intersects
the image plane (origin of normalized coordinate system)
Principal point offset
principal point:
( px , p y )
• Camera coordinate system: origin is at the
prinicipal point
• Image coordinate system: origin is in the corner
Principal point offset
principal point:
( px , p y )
( 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
f
( px , p y )
X
p x  1
0  
Y 



py  1
0  
Z
1  
1 0 
1
px 
p y  calibration matrix
1 
P  KI | 0
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
• In general, the camera
coordinate frame will
be related to the world
coordinate frame by a
rotation and a
translation

~
~ ~
Xcam  R X - C
coords. of point
in camera frame

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
 mx
 f

Focal length
K  
my

Pixel magnification factors

1 
Skew (non-rectangular pixels)
Radial distortion
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
• Given n points with known 3D coordinates Xi
and known image projections xi, estimate the
camera parameters
Xi
xi
P?
Camera calibration
T
 xi  P1 
 
  yi   P2T  X i
 1  P3T 
 x i  PXi
 0

T
X
i

 yi XTi

 XTi
0
T
xi X i
x i  PXi  0
yi XTi  P1 
 
T
 xi X i  P2   0
0  P3 
Two linearly independent equations
Camera calibration
 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
 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
• Once we’ve recovered the numerical form of
the camera matrix, we still have to figure out
the intrinsic and extrinsic parameters
• This is a matrix decomposition problem, not
an estimation problem (see F&P sec. 3.2, 3.3)
Two-view geometry
•
Scene geometry (structure): Given
projections of the same 3D point in two or
more images, how do we compute the 3D
coordinates of that point?
•
Correspondence (stereo matching):
Given a point in just one image, how does it
constrain the position of the corresponding
point in a second image?
•
Camera geometry (motion): Given a set of
corresponding points in two images, what
are the cameras for the two views?
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