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
