计算机视觉课件12

xai yai zai
xai 0 zai
yai
yai zai
0 xai

za
i

yai
xa 0
i

i.e.
qTNq max
where N N NmiTNai i1
This equation can be solved by linear least squares,
2

N
raiRrm
imax
i1
Using quaternion to express rotation and recall that
R(q)rqrq*
we have:
2

N
raiR(q)rmi
N i1
rai

qrm
iq*
i1
N

N
(Nmqi )T(Na
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.
In the same way, in model coordinate system:
1 N
Pm N i1 Pmi ;
rm i pm i pm
Since
rai
and
Rrci
are parallel, We can solve for
rotation by least-squares.
In absolute coordinate system, a set of points at a 3D object
are :
{p a1,p a2, ,p a} N
In model coordinate system, the same set of points are
correspondingly me{ ap sum r1 e,dp am s2 :, ,p m} N
To find out relationship between two coordinate systems from 3 or more 3D points that have expressed in the two coordinate systems.
Let a 3D point p is expressed by
with
P~Pr,P~Pr
as
Use 8-point method , one can solve for rotation and translation b .
2)
Let
Irtlearnadtirvrebemtheethdioredc:tion
from
camera
centers
1) Solve relative orientation problem by motion estimation.
In motion estimation, camera is stationary. Object
is movt t 1 i2 n: : g. p p 1 2 ( ( x x ,, y y ,, z z ) ) P P 1 2 ( ( X X ,Y ,Y ) )
Course 12 Calibration
Course 12 Calibration
1.Introduction
In theoretic discussions, we have assumed: ----- Camera is located at the origin of coordinate system
pasR pmp0
If one notices the fact that the distances between points of 3D scene are not affected by choice of coordinate systems, we can easily solve for scale factor:


t 1 :p ( x ,y ,z ) P l( X l,Y l)
t 2 :p ( x ,y ,z ) P r ( X r ,Y r )



Use Pl Pr to stationary camera
find case
Randb , it is the same
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
Biblioteka Baidu
sx(
j
cˆx)
sx(
j
m1) 2
Y
sy(i
cˆy)
sy(i
n1) 2
2. Absolute orientation:



Using P1 P2 to find RandT
In stereo case, we can imagine that a camera first take an image of scene at position O l , and then moves to O r to take the second image of the same scene. The scene is stationary. Camera is moving.
to
scene
point, respectively.
X l rl Y l
f l
X r rr Y r
f r
Since baseline of stereo system and the normal of epipolar
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
p a s ( R p m ) p 0 R ( s p m ) p 0
3. Relative Orientation
To determine the relationship between two camera coordinate systems from the projections of corresponding points in the two camera. (e.g., in stereo cases). This is to say, given pairs of image point correspondences, to find rotation and translation of the camera from one position to the other position.
absolute
coordinates;
p0
is
the
origin
of
model
coordinate
system in the absolute coordinate system.
Given:
p a,i,p m ,i(i1 ,2,3 , )
We want to find:
Randp0
such as SVD. After R is found, camera position
can be easily calculated by
p 0p aR (q)p m
2) Scale Problem
If absolute coordinate system and model coordinate system may have different measurements, scale problem is introduced.
they satisfy
p a i R (q)p m ip 0
In absolute coordinate system:
Centroid of the point set is
And
the
ray
Pa
from
1 N
set
N
Pai
i1
centroid
to

pointpai
is:
raip aipa
p p c a ((x xm a,,y ya m ,,zzam ))iinn
model coordinate system. absolute coordinate system.
Then: p aRp mp 0
where R is rotation of model coordinate corresponding to
----- 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.
This adds 6 additional equations in solving for rotation.
1) Solve rotation with quaternion:
Orthonornal constraint of rotation matrix is absorbed in quaternion expression.

iq)
i1
qT NmT iNaiq
i1
N
qT(
NmTiNai)qqTNq
i1
Where
0 xm i ym i zm i
Nm
ixzym m m
i i i
0 zm i ym i
zm i 0 xm i
ym i
xm 0
i

0

Na
i


s
n i1
pai pa
2
1/2

n i1
pmipm
2
or
s



i j
pai paj
2
1/ 2

i j
pmi

pmj
2

Once scale factor is found, the problem becomes ordinary absolute orientation problem
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
f orij f orij
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.