Image Processing Reference
In-Depth Information
Figure 5.2
Lattice representation of a 3-input function.
determine a constraint that reduces the search space to allow training on a realisti-
cally sized training set but that allows sufficient flexibility to produce an accurate
solution. As with filter window size, it is a trade off between estimation and con-
straint error. The amount by which the imposition of the increasingness constraint
limits the filter should not be underestimated.
Consider the lattice representation of a function of three variables shown in
Fig. 5.2.
The lattice has the value of
x
= (1, 1, 1) at the top and
x
= (0, 0, 0) at the bottom
and all the values in between. The partial ordering is conveyed by the connecting
lines, indicating that some values of
x
are above (or below) others, as defined by
Eq. 5.3. This lattice structure can be extended to any number of variables, though it
becomes increasingly complex to illustrate.
An increasing function causes the lattice to be cut into two sections—top and
bottom. All of the inputs in the top section have a corresponding filter output of 1.
All those in the bottom section have an output of 0. When an input is encountered
for which the output is 1, then every input above that in the lattice can be assumed
to have an output of 1. Similarly, when an input is found for which the output is 0,
then every input beneath it has an output of 0. The entire function may be specified
by identifying the minimum inputs for which the output is 1 as shown in Eq. 5.5.
These inputs are known as the basis inputs.
ψ
inc
(
x
) = 1
if there exists
i
such that
x
x
bi
and
ψ
inc
(
x
)=0
for all
x
<
x
bi
for all
i
.
(5.5)
inc
] is known as the basis. In Fig. 5.2 the basis contains two
inputs
x
b1
= (1, 1, 0),
x
b2
= (0, 0, 1). This completely defines the function for all in-
The set of inputs B[
ψ
