# Multi-View Geometry (Cont.)

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"Multi-View Geometry (Cont.)"

```					Multi-View Geometry (Cont.)
Stereo Constraints (Review)

p                          p’ ?

Given p in left image, where can the corresponding point p’
in right image be?
Stereo Constraints (Review)
M
Image plane             Epipolar Line

Y1            p
p’
Y2
X2
O1
Z1                X1

O2         Z2
Focal plane
Epipole
Epipolar Constraint (Review)
From Geometry to Algebra (Review)

P

p
p’

O                         O’
From Geometry to Algebra (Review)

P

p
p’

O                         O’
Linear Constraint:
Should be able to express as matrix
multiplication.
The Essential Matrix (Review)
The Essential Matrix

• Based on the Relative Geometry of the Cameras
• Assumes Cameras are calibrated (i.e., intrinsic
parameters are known)
• Relates image of point in one camera to a second
camera (points in camera coordinate system).
• Is defined up to scale
• 5 independent parameters
The Essential Matrix

  T
Similarly p is the epipolar line corresponding to p in the
right camera
The Essential Matrix

 e  t Re  0


Similarly,  e  R t  e   R t e  0
T      T       T   T
           

Essential Matrix is singular with rank 2

e’
Small Motions and Epipolar Constraint
Motion Models (Review)

 X   1                X  TX 
Y          1          Y   TY 
                                             3D Rigid Motion
 Z    
                        1   Z  TZ 
   

 X    0                 1 0 0   X  TX 
Y                         0 1 0    Y    T 
  
              0                           Y         VX 
 Z     
                        0  0 0 1   Z  TZ 
                       V   Velocity Vector
 Y
VZ 
 
X   X   0                   X  TX 
 Y Y                                                   VTX 
                     0          Y   TY 
                             Translational
 Z   Z   
                            0   Z  TZ 
                       VTY   Component of Velocity
VT 
 Z
VX   0         Z         Y   X  VT                  X 
V                                        
  X   Y   VT 
x

0                                             Angular Velocity
 Y  Z                                    Y
 Y
VZ   Y
              X            0   Z  VTZ 
                         Z 
 
Small Motions

t  tv
R  I  t  
p  p  tp
VX 

p T p  0                                  p  VY   Velocity Vector

 
VZ 
 

p v I  t   p  tp   0
T                                             VTX 
                   
v  VTY   Translational
VT     Component of Velocity
 Z

p   T
v   p   p  p .v  0
                X 
  Y  
Angular Velocity
 
 Z 
 
Translating Camera

pT v   p   p  p .v  0


 0
 p  p .v  0



p, p, and v are coplanar

Focus of expansion (FOE): Under pure translation, the motion
field at every point in the image points toward the focus of
expansion
FOE for Translating Camera
FOE from Basic Equations of Motion
VTZ x  VTX f                       X xy Y x 2
px 
                       Y f   Z y          
Z                              f       f
VTZ y  VTY f                      Y xy  X y 2
py 
                        X f  Z x          
Z                              f       f
 0                                     q
VTZ x  VTX f                                                       
q   p
px 

Z                                                                      
p
VTZ y  VTY f
py 

Z
O       v
VTX
e  x0 , y0 
x0  f                                                   VTZ
VTZ                          p x   x  x0 

Z
VTY                                             VTZ
y0  f                                p y   y  y0 

VTZ                                              Z
What if Camera Calibration is not known
Review: Intrinsic Camera Parameters

Y                    M  X C  , Y C  , Z C  
Image plane

i  ku I                j               Z                     C
v                                                                 X
j  kv J        J
i
I
u m     u   , v  , f 
C        C                     Focal plane
u   , v  
I       I

 X C  
U new   f u              0     u0       0  C  
 new                                       Y                f u  fku
V        0                 fv   v0       0  C 
Z 
 S   0                                     0                  f v  fkv
                            0     1          
 1 
        
P
Fundamental Matrix


p T p  0         p and p are in cameracoordinatesystem

If u and u’ are corresponding image coordinates then we have

uPp 1
p  P 1u
1

u  P2 p              p  P21u 

P
u T P T
1       2 u  0
1

 uT Fu  0                                  
F  P T P21
1
Fundamental Matrix

u Fu  0
T


F  P T P21
1

Fundamental Matrix is singular with rank 2
In principal F has 7 parameters up to scale and can be estimated
from 7 point correspondences
Direct Simpler Method requires 8 correspondences
Estimating Fundamental Matrix
The 8-point algorithm
u Fu  0
T

Each point correspondence can be expressed as a linear equation
 F11   F12   F13  u 
u v 1 F21
       F22   F23   v   0
 
 F31
       F32   F33   1 
 
 F11 
F 
 12 
 F13 
 
 F21 
uu uv u uv vv v u v 1 F22   0
 
 F23 
F 
 31 
 F32 
F 
 33 
The 8-point Algorithm
Shape from Stereo
Pinhole Camera Model

X
xf
Z
Basic Stereo Derivations

Derive expression for Z as a function of x1, x2, f and B
Basic Stereo Derivations

X              X B          B
x1   f       x2   f       x1  f
Z                Z           Z
fB
Z
x1  x2
Basic Stereo Derivations

Define the disparity: d  x1  x2

fB
Z
d

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