计算机视觉课件(8)

1) Solve rotation with quaternion:
Orthonornal constraint of rotation matrix is absorbed in quaternion expression.
In absolute coordinate system, a set of points at a 3D object
One should remember the constraint of rotation matrix that is orthogonal matrix, i.e.
R R(1) | R(2) | R(3)
R(i)

R( j)
1 0
for i j for i j
This adds 6 additional equations in solving for rotation.
Course 12 Calibration
Course 12 Calibration
1.Introduction
In theoretic discussions, we have assumed: ----- Camera is located at the origin of coordinate system
of scene. ----- Optic axis of camera is pointing in z-direction of scene
coordinates. ----- Image plane is to perpendicular to z-axis with origin at
(f, 0, 0) ----- X and Y axes of image coordinates are parallel to x
are :
{ pa1,
pa2 ,
,
paN
}
In model coordinate system, the same set of points are
correspondingly
measuredas: { pm1, pm2 ,
,
pmN
}
they satisfy
pai R(q) pmi p0
----- Relative orientation: between two camera systems.
----- Exterior orientation: between camera and scene systems
----- Interior orientation: within a camera, such as camera constant, principal point, lens distortion, etc.
2) Coordinate systems:
----- scene coordinates (global coordinates, world
coordinates, absolute coordinates).
----- Camera coordinates.
----- Image coordinate systems (X ,Y )
----- Pixel coordinates [i, j]
For a image of size m×n, image center
cˆx
m 1 2
cˆy
n 1 2
X
sx (
j
cˆx )
sx (
j
m 1) 2
Y
sy (i
cˆy )
sy (i
n 1) 2
2. Absolute orientation:
In absolute coordinate system:
Centroid of the point set is
And
the
ray
Pa
from
1 N
set
N
Pai
i 1
centroid
to
pointpai
is:
rai pai pa
In the same way, in model coordinate system:
absolute
coordinates;
p0
is
the
origin
of
model
coordinate
system in the absolute coordinate system.
Given:
pa,i ,
pm,i
(i
1,2,3,)
We want to find:
R and p0
It is the same expression as 3D motion estimates from 3D points. Therefore, all the algorithms of motion estimation (using 3D points) can be used to solve for absolute orientation.
pc (xm , ym , zm ) pa ( xa , ya , za )
in model coordinate system. in absolute coordinate system.
Then:
pa Rpm p0
where R is rotation of model coordinate corresponding to
and y axes of scene coordinates, respectively. ----- No any distortions for camera, i.e. pinhole camera
model.
1) What we need to calibrate:
----- Absolute orientation: between two coordinate systems, e.g.,robot coordinates and model coordinates.
To find out relationship between two coordinate systems from 3 or more 3D points that have expressed in the two coordinate systems.
Let a 3Dpoint p is expressed by