1.07M

triangulation

1.

Course Schedule
Session I
Introduction
The Mathematics of 3D Triangulation
3D Scanning with Swept-Planes
Camera and Swept-Plane Light Source Calibration
Reconstruction and Visualization using Point Clouds

2.

3D Triangulation: plane-ray intersection
plane
intersection point
ray
projector /
coordinate
systems

3.

Representation of lines and rays
line
point
scale
parameter
parameter
is positive
p q v
reference
point
v
q
ray=“1/2 line”
vector
v
q
p q v

4.

Representation of planes
parametric
2 vectors
reference
point
q
implicit
reference
point
v2
p
v1
point
n
q
p
P
normal
vector
p q 1v1 2v2
2 scale
parameters
P
nt ( p q ) 0
1 implicit
equation

5.

Triangulation by line-plane intersection
object being
scanned
illuminated
point on object
projected light plane
P { p : nt ( p q p ) 0}
p
qp
n
v
same
coordinate
system
intersection
of light plane
with object
camera ray
qL
L { p qL v}

6.

Triangulation by line-line intersection
object being
scanned
lines may not intersect !
p
projected light ray
L1 { p q1 1v1}
q1
v1
v2
camera ray
q2
L2 { p q2 2v2 }

7.

Approximate line-line intersection
Midpoint of segment joining
arbitrary points in the two lines
p1 q1 1v1
q1
v1 p12 ( 1, 2 )
Least-squares approach
p12 ( 1, 2 )
p2 q2 2v2
q1
v2
v1
v2
q2
q2
Find parameters which minimize

8.

Approximate line-line intersection
p1 q1 1v1
p2 q2 2v2

9.

Perspective projection under the pinhole model
center of projection
image plane
image
point
light direction
for a projector
light direction
for a camera
3D point
Calibration: mapping from image points to rays

10.

The ideal pinhole camera
camera coordinate system = world coordinate system
q 0
v3
v2
v1
f 1
u1
u u2
1
p1
p p2
3
p

11.

The general pinhole model
camera coordinate system
world coordinate system
C
3
2
p
u
3
1
W
C RX W T
1
2
Ideal assumptions
• Image lengths = world lengths
• Focal length = 1
• Image origin = optical center
• Image plane spanned by two basis vectors

12.

The general pinhole model
camera coordinate system
world coordinate system
C
3
2
p
u
3
1
W
C RX W T
1
2

13.

Plane defined by image line and center of projection
Implicit equation of line in image coordinates
L {u : l t u 0}
center of projection
q
n
image plane
P { p : nt ( p q) 0}

14.

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