1 chapter 2: geometric camera models objective: formulate the geometrical relationships between...
Post on 21-Jan-2016
214 Views
Preview:
TRANSCRIPT
1
Chapter 2: Geometric Camera Models Objective: Formulate the geometrical relationships between image and scene measurements
Scene: a 3-D function, g(x,y,z)
Image: a 2-D function, f(x,y)
2
Contents: (i) Homogeneous coordinates(ii) Geometric transformations(iii) Intrinsic and extrinsic camera parameters(iv) Affine projection models
2.1. Elements of Analytical Euclidean Geometry
2.1.1. Coordinate Systems
○ Right-handed coordinate system
3
, ,i j k
, , :x y z
x
y
z
p
coordinates of point P
: position vector of point P
O: origin;
: basis vectors
4
◎ Homogeneous Coordinates
Advantages: (a) Some nonlinear systems can be transformed into linear ones (b) Equations written in terms of homogeneous coordinates become more compact. (c) A transformation, comprising rotation, translation, scaling, and perspective projection, can be written in a single matrix
5
x
y
z
p
1
x
y
z
p
0ax by cz d
0T p a
b
c
d
○ Point:
○ Plane equation:
or p
cx
cy
cz
c
where
6
2 2 2 2x y z r
0T S p p
2
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0
S
r
○ Sphere equation:
where
7
2 2 2200 110 020 011 002
101 100 010 001 000 0
a x a xy a y a yz a z
a xz a x a y a z a
0TQ p p
200 110 101 100
110 020 011 010
101 011 002 001
100 010 001 000
1 1 1
2 2 21 1 1
2 2 21 1 1
2 2 21 1 1
2 2 2
a a a a
a a a a
a a a a
a a a a
Q
○ Quadric surface equation:
where
8
2.1.2 Coordinate System Changes and Rigid Transformations
Two subjects: (a) Coordinate system changes (b) Rigid transformations
Consider two coordinate systems, A and B
9
○ Coordinate System Changes
Position vectors:
( , , )B B BP x y z ( , , )A A AP x y z
,pA
AA
A
x
y
z
pB
BB
B
x
y
z
B AMp pCoordinate transformation: (?)
10
BAT
。 Translation vector
: the vector translates the origin of coordinate system A to that of system B
○ Rigid Transformations
Tx
BA y
z
t
t
t
11
BA R
。 Rotation matrix
: the 3 by 3 matrix rotates coordinate system A to coincide with system B
11 12 13
21 21 23
31 31 33
BA
r r r
R r r r
r r r
A B A B A B
A B A B A B
A B A B A B
i i j i k i
i j j j k j
i k j k k k
12
BA R
Ai ( , )
B B Bi j ,k
BA R
BA R
Bi ( , )
A A A i j ,k
BA R
The 1st column of is formed by projecting onto
The columns of form frame A described in terms of frame B
is formed by projecting onto
The rows of form frame B described in terms of frame A
The 1st row of
A B A B A BBA A B A B A B
A B A B A B
R
i i j i k i
i j j j k j
i k j k k k
13
1( ) ( )B B TA AR R
det( ) 1BA R ( )B A T
A BR R
, :B AA BR R
* Properties:
(a)
(b)
: unitary matrix
(c)
(d) orthonormal matrices
○ Rigid Transformation: A rigid transformation preserves: (1) the distance between two point
s (2) the angle between two vectors
B B A BA AR p p T
14
be their corresponding
points in frame B, i.e.,
Proof: Let be two points in frame A
Then,
,A Ap q
,B Bp q
B B A BA AR p p T B B A B
A AR q q T
( )B B B A AA R p q p q
|| || || ||p q p qB B B A B AA AR R
(1) Distance preservation
1/ 2[( ) ( )] p q p qB A B A T B A B AA A A AR R R R
15
1/ 2
[( ) ( ) ( ) ( )
( ) ( ) ( ) ( )]
p p q p
p q q q
B A T B A B A T B AA A A A
B A T B A B A T B AA A A A
R R R R
R R R R
1/ 2
[( ) ( ) ( )( ) ( ) ( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( )( )]
p p q p
p q
q q
A T B T B A A T B T B AA A A A
A T B T B AA A
A T B T B AA A
R R R R
R R
R R
1/ 2
1/ 2
[( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )]
[( ) ( )] || ||
p p q p p q q q
p q p q p q
A T A A T A A T A A T A
A A T A A A A
(2) Angle preservation (Assignment)
164 4
,1
B BA AB
A T
RH
0
T(0,0,0)T T0
11 12
21 22
,A A
AA A
11 12
21 22
B BB
B B
11 11 12 21 11 12 12 22
21 11 22 21 21 12 22 22
A B A B A B A BAB
A B A B A B A B
det( ) det( )det( )AB A B
○ Matrices can be multiplied in blocks
○ ○ In homogeneous coordinates:
-- (2.7)
where
then
1 1
B ABA H
p p,B B A BA AR p p T
17
2.2 Camera Parameters。 Intrinsic parameters -- Relate the actual camera coordinate system to the idealized camera coordinate system (1) the focal length of the lens f (2) the size and shape of the pixels (3) the position of the principal point (4) the angle between the two image axes
Idealized camera Actual camera
18
。 Extrinsic parameters -- Relate the idealized camera coordinate system to a real world coordinate system (1) translation and (2) rotation parameters
。 Camera calibration -- estimates the intrinsic and extrinsic parameters of a camera
Idealized cameracoordinate system
Real worldCoordinate system
1919
2.2.1. Intrinsic parameters
Start with ideal perspective projection equations
: scale
: skew
: shift
parameters
20
21
22
23
24
○ The relationship between the physical image frame and the normalized one
25
26
27
,1 1
p pC WCW H
1
TC CW WC
W T
RH
0
,1 1
p pP P
C WC W
,P PC C WW H
where
28
29
30
31
A LQ(Only if) – If , A: nonsingular
A can always be factorized intoQ: orthonormal matrixL : right upper triangular matrix
det( ) 0A
32
( ) ( ) ( ),W A LQ L Q b b b L b b
( )C CW WK R T
,b CWT
Compared with
L, K : right upper triangular matrices
Q, : orthonomal matrices
: vectors
W is a perspective projection matrix
CW R
33
34
35
2.3. Affine Projection Models• Orthographic Projection Models -- Objects are far from the camera
• Parallel Projection Models -- Objects are far and lie off the optical axis of the camera
36
• Paraperspective Projection Models -- Objects lie near the optical axis
• Weak Perspective Projection Models -- Objects lie on the optical axis and their reliefs are ignored
37
Consider object reliefs in weak perspective projection
38
39
40
41
42
43
44
45
46
47
48
top related