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total square error for the conventional unweighted least square approximation
may be less than that for the weighted least square, the approximation pro-
duced by the latter may be psychovisually more appealing than that by the
former, provided weights are chosen judiciously. For an image, edge points are
more informative than the homogeneous regions because edges are the dis-
tinct features of an image. Thus, edges should be given more emphasis while
approximating an image patch and this can be done through weighted least
square. Thus, the weighted squared error can be written as
E
2
=
u
s
pq
(
u, v
))]
2
[
W
(
u, v
)(
g
(
u, v
)
−
v
(4.10)
p
q
=
u
φ
rp
(
u
)
φ
zq
(
v
)
V
rz
]
2
,
[
W
(
u, v
)(
g
(
u, v
)
−
v
r
=0
z
=0
where
W
(
u, v
) is the weight associated with the pixel corresponding to (
u, v
).
For
p
=
q
, the surface
s
pq
(
u, v
) is defined on a square support. Since
W
(
u, v
)
is the weight associated with each pixel, it can be considered constant for that
pixel. Therefore, one needs to find out the weight matrix before solving equa-
tions for the weighted least square. Once
W
(
u, v
) is known, these equations
reduce to a system of linear equations and can be solved by any conventional
technique.
We emphasize that for order determination, we use the unweighted ap-
proximation scheme. In the weighted least square approximation of regions,
special weights are given to boundary pixels so that the error, in the mean
square sense, over the boundary is less than that in the unweighted least
square approximation. For this, we have considered the gradients of boundary
pixels as their weights. One can also consider higher power of gradients. The
gradients of the boundary pixels,
G
(
u, v
) and hence the weights
W
(
u, v
)in
equation (4.10), can be calculated using the following equation.
W
(
u, v
)=(
G
v
2
+
G
u
2
)
1
/
2
,
(4.11)
where
G
u
=
g
(
u
+1
,v
)
−
2
g
(
u, v
)+
g
(
u
−
1
,v
)and
G
v
=
g
(
u, v
+1)
−
2
g
(
u, v
)+
g
(
u, v
1).
Image compression in our scheme is a two-stage process. In stage 2, for
encoding we approximate the subimages minimizing a
weighted
least square
error with a polynomial of the same order as determined in stage 1 (for seg-
mentation). The same order is used because the order (global and also local)
of a subimage or the nature of approximation is not expected to change due
to merging of small regions. However, one can once again find the order of
approximation before encoding. The reason is, the best fit surface does not nec-
essarily psychovisually represent the most appealing (informative) surface. If
we try to find the optimal order of the polynomial using weighted least square,
then that optimal order is expected to be more than that for the unweighted
least square. Consequently, the compression ratio will go down. Of course, the
two orders cannot be widely different. Thus, there is a need to compromise.
−
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