Image Processing Reference

In-Depth Information

Fig. 10.6
The coordinate system of the image
(x, y)
and the coordinate system of the projector

(x
,y
)
seen from the image's point of view. (
b
) The circles are projected from the projector in

order to find corresponding points in the two coordinate systems

purpose of the game could then be to see how many circles you can touch in a prede-

fined time period. For such a system to work you need, among other things, to know

what a detected pixel coordinate (the position of the finger) corresponds to in the

image projected onto the table. If both camera and projector are tilted with respect

to the table, then two keystone errors are actually present. In general, the geometric

transformation which maps from one plane (camera image) to another (projected

image) is known as a
projective transformation
or
homography
. It can be calculated

in the following way using the
Direct Linear Transform
[10].

First have a look at Fig.
10.6
(a) to see what we are dealing with. To the left you

see an illustration of two coordinate systems. The
(x, y)
coordinate system is the

coordinate system of the image and the
(x
,y
)
coordinate system is the coordinate

system of the projector seen from the image's point of view. Or in other words, if

you make the projector project two perpendicular arrows onto a plane (for example

a table) and capture a picture of the table, then the perpendicular arrows will look

like the
x
and
y
arrows. So the transformation we are after should map from
(x, y)

to
(x
,y
)
.

The use of a homography is not limited to finding the correspondence between

an image and a projector. Imagine we have a robot arm that should pick something

up from a table. A camera captures an image of the table, finds the object of interest

and send its position to the robot. The table is the robot's coordinate system meaning

that the origin is one of the corners and the
x
and
y
axes are two perpendicular

edges of the table. The image's coordinate system is now
x
and
y
, and we need to

find a transformation from
(x, y)
to
(x
,y
)
. So the exact same situation as with the

projector and hence the exact same solution.

From the theory of homography we know that the mapping between the two

coordinate systems is

⎡

⎤

⎡

⎤

⎡

⎤

x

h

·

a
1

a
2

a
3

x

y

1

⎣

⎦
=

⎣

⎦
·

⎣

⎦

y

h

h

·

b
1

b
2

b
3

(10.12)

c
1

c
2

1

From this it follows that