Databases Reference
In-Depth Information
We have used
R
k
instead of
R
k
on the left-hand side of (
40
) because it is generally not possible
to find an exact functional relationship between domain and range blocks. Therefore, we have
to settle for some degree of loss of information. Generally, this loss is measured in terms of
mean squared error.
The effect of all these functions together can be represented as the transformation
f
(
·
)
.
Mathematically, this transformation can be viewed as a union of the transformations
f
k
:
f
=
f
k
(41)
k
Notice that while each transformation
f
k
maps a block of different size and location to the
location of
R
k
, looking at it from the point of view of the entire image, it is a mapping from
the image to the image. As the union of
R
k
is the image itself, we could represent all the
transformations as
I
(
I
(42)
wherewehaveused
I
instead of
I
to account for the fact that the reconstructed image is an
approximation to the original.
We can now pose the encoding problem as that of obtaining
D
k
,
g
k
, and
m
k
such that the
difference
d
=
f
)
R
k
,
R
k
)
R
k
,
R
k
)
(
is minimized, where
d
(
can be the mean squared error between
the blocks
R
k
and
R
k
.
Let us first look at how we would obtain
g
k
and
m
k
assuming that we already know which
domain block
D
k
we are going to use. We will then return to the question of selecting
D
k
.
Knowing which domain block we are using for a given range block automatically specifies
the amount of displacement required. If the range blocks
R
k
are of size
M
×
M
, then the
domain blocks are usually taken to be of size 2
M
×
2
M
. In order to adjust the size of
D
k
to be
the same as that of
R
k
, we generally replace each 2
2 block of pixels with their average value.
Once the range block has been selected, the geometric transformation is easily obtained.
Let's define
T
k
×
=
g
k
(
D
k
)
, and
t
ij
as the
ij
th pixel in
T
k
,
i
,
j
=
0
,
1
,...,
M
−
1. The
massic transformation
m
k
is then given by
m
k
(
t
ij
)
=
i
(α
k
t
ij
+
k
)
(43)
where
i
denotes a shuffling or rearrangement of the pixels with the block. Possible rear-
rangements (or
isometries
) include the following:
(
·
)
1.
Rotation by 90 degrees,
i
(
t
ij
)
=
t
j
(
M
−
1
−
i
)
(
t
ij
)
=
2.
Rotation by 180 degrees,
i
t
(
M
−
1
−
i
)(
M
−
1
−
j
)
−
(
t
ij
)
=
3.
Rotation by
t
(
M
−
1
−
i
)
j
4.
Reflection about midvertical axis,
i
90 degrees,
i
(
t
ij
)
=
t
i
(
M
−
1
−
j
)
5.
Reflection about midhorizontal axis,
i
(
t
ij
)
=
t
(
M
−
1
−
i
)
j
6.
Reflection about diagonal,
i
(
t
ij
)
=
t
ji
7.
Reflection about cross diagonal,
i
(
t
ij
)
=
t
(
M
−
1
−
j
)(
M
−
1
−
i
)
8.
Identity mapping,
i
(
t
ij
)
=
t
ij
α
k
,
k
, and
an isometry. For a given range block
R
k
, in order to find the mapping that gives us the
Therefore, for each massic transformation
m
k
, we need to find values of
