Image Processing Reference

In-Depth Information

Figure 5.1
Assignment of output to unknown input patterns. If the patterns at the top and bot-

tom are assigned a value of 1, it is reasonable to assume that the middle patterns should also

be assigned a value of 1.

Increasingness implies a partial
ordering
of the input values of a function. That is,

F
(
x
)

≥

F
(
y
)

for

x

≥

y
,

(5.3)

where
x
y
implies that
X
i
Y
i
for every component of
x
and
y
. For example, if
x
=

(011) and
y
= (001) then
x
>
y
and therefore it follows that
F
(
x
)

F
(
y
) for any in-

creasing function
F
. However for
x
= (010) and
y
= (001) there is no ordering of
x

and
y
and therefore nothing can be inferred about the ordering of
F
(
x
) and
F
(
y
)
.

A filter based on an
increasing
function is known as an
increasing
filter (

≥

inc
).

Increasingness is simply a further constraint on the filter. It will cause an increase in

constraint error unless the optimal filter happens to be an increasing filter, i.e.,

ψ

MAE(

ψ

opt
)

≤

MAE(

ψ

inc
).

(5.4)

Even though the best possible increasing filter may be inferior to the best filter

overall, it will be easier to train because its search space will be significantly re-

duced. This means that the estimation error of the increasing filter will be much

lower than for the filter without this constraint. The key to good filter design is to