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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)

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

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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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21

22

23

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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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