Graphics Reference
In-Depth Information
(a)
(b)
(c)
(d)
Figure 5.1.
An original image (a) and the results of various affine and projective transformations.
(b) A similarity transformation (i.e., translation, rotation, and uniform scale). (c) A vertical shear.
(d) A general projective transformation. While (b) and (c) can bewritten as affine transformations,
(d) cannot.
It's usually convenient to represent image coordinates and transformations in
homogeneous coordinates
for parameter estimation. This means that we represent
an image location
can be converted
back to an image coordinate by dividing by its third element to get an image location
z
,
z
. In this way we can rewrite Equation (
5.2
)as
(
x
,
y
)
as a triple
(
x
,
y
,1
)
, and that any triple
(
x
,
y
,
z
)
x
y
1
h
11
h
12
h
13
h
21
h
22
h
23
h
31
h
32
h
33
x
y
1
∼
(5.3)
The symbol
in Equation (
5.3
) means that the two vectors are equivalent up to
a scalar multiple; that is, to obtain actual pixel coordinates on the left side of
Equation (
5.3
), we need to divide the vector on the right side of Equation (
5.3
)by
its third element.
Hartley and Zisserman [
188
] described how to obtain an initial estimate of the
parameters of a projective transformation given a set of feature matches using the
normalized
direct linear transform
,or
DLT
. The steps are as follows:
∼
1. The input is two sets of features
{
(
x
1
,
y
1
)
,
...
,
(
x
n
,
y
n
)
}
in the first image plane
x
1
,
y
1
)
x
n
,
y
n
)
}
and
in the second image plane. We normalize each set
o
f
feature matches to have zero mean and average distance from the origin
√
2.
This can be accomplished by a pair of similarity transformations, represented
as 3
{
(
,
...
,
(
3 matrices
T
and
T
applied to the homogeneous coordinates of the
points.
2. Construct a 2
n
×
×
9 matrix
A
, where each feature match generates two rows of
A
, that is, the 2
×
9 matrix
000
x
i
y
i
x
i
y
i
y
i
y
i
y
i
1
−
−
−
A
i
=
(5.4)
x
i
x
i
x
i
y
i
x
i
x
i
y
i
1000
−
−
−
Note that Equation (
5.3
) for feature match
i
is equivalent to
]
=
A
i
[
h
11
h
12
h
13
h
21
h
22
h
23
h
31
h
32
h
33
0
(5.5)
UDV
.
D
will be a 9
3. Compute the singular value decomposition of
A
,
A
9
diagonal matrix with positive entries that decrease from upper left to lower
right. Let
h
be the last column of
V
(a 9
=
×
×
1 vector).
