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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)
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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
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, ,i j k
, , :x y z
x
y
z
p
coordinates of point P
: position vector of point P
O: origin;
: basis vectors
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◎ 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
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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
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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
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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
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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
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○ 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: (?)
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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
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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
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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
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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
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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
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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)
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,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
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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
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。 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
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2.2.1. Intrinsic parameters
Start with ideal perspective projection equations
: scale
: skew
: shift
parameters
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○ The relationship between the physical image frame and the normalized one
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,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
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A LQ(Only if) – If , A: nonsingular
A can always be factorized intoQ: orthonormal matrixL : right upper triangular matrix
det( ) 0A
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( ) ( ) ( ),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
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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
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• Paraperspective Projection Models -- Objects lie near the optical axis
• Weak Perspective Projection Models -- Objects lie on the optical axis and their reliefs are ignored
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Consider object reliefs in weak perspective projection
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