ENGN2502 Lecture 3

The Mathematics of 3D Triangulation

What is a Camera ?

What is a Camera ?

3D Triangulation: Ray-Plane Intersection

ray

plane

intersection point

projector /

coordinate

systems

Representation of Lines and Rays

line

q

v

vqp reference

point

vector

scale parameter

point

q

v

ray=“1/2 line” vqp

parameter is positive

Representation of Planes

2211 vvqp

1v

2vq

p

P

parametric

n

q

p

0)( qpntP

implicit

2 scale parameters

reference point

2 vectors

point

reference point

normal vector

1 implicit equation

Triangulation by Line-Plane Intersection

object being scanned

pqn

}0)(:{ pt qpnpP

projected light plane

p

illuminated point on object

}{ vqpL L

camera ray

Lq

v

intersection of light plane

with object

same

coordinate

system

Triangulation by Line-Line Intersection

object being scanned

}{ 2222 vqpL

camera ray

2q

2v

projected light ray

1q1v

}{ 1111 vqpL

p

lines may not intersect !

Approximate Line-Line Intersection

1q1v

2q

2v

1111 vqp

2222 vqp

),( 2112 p

Midpoint of segment joining arbitrary points in the two lines

1q1v

2q

2v

),( 2112 p

Least-squares approach

Find parameters which minimize

Approximate Line-Line Intersection

1111 vqp

2222 vqp

Perspective Projection (Pinhole Model)

center of projection

image point

image plane

3D point

light direction for a projector

light direction for a camera

Calibration: mapping from image points to rays

The Ideal Pinhole Camera

camera coordinate system = world coordinate system

1v

3v2v

1

2

1

u

u

u

0q

3

2

1

p

p

p

p

1f

The General Pinhole Model

world coordinate system camera coordinate system

W

C

1 2

31

2 3 u

p

TRXWC

Ideal assumptions • Image lengths = world lengths • Focal length = 1 • Image origin = optical center • Image plane spanned by two basis vectors

The General Pinhole Model

world coordinate system camera coordinate system

W

C

1 2

31

2 3 up

TRXWC

Plane Defined by Image Line and Projection Center

center of projection q

n

image plane

}0)(:{ qpnpP t

}0:{ uluL t

Implicit equation of line in image coordinates