Grayscale Implementation - Logic-based Nonlinear Image Processing - page 102

falls within the envelope, the original value is retained. However, where it falls
above , its value is trimmed to the envelope's extremities. This is set

out in the equation below.

β or below α

α if ψ < α ,

,

ψ = ψα if ≤ ψβ ≤ and (8.1)

con

β if ψ > β

The envelope represents the upper and lower bounds of the expected output from
the filter with the optimum filter output opt ideally lying somewhere in between.

As will be seen in later examples, the envelope is usually formed from a simple
combination of filters. The effect of an envelope constraint may either reduce or in-
crease the filter error. The search space is made smaller, so the filter output may be
better estimated from a smaller training set. However, there is an increase in con-
straint error since the filter is limited in its range of outputs. Where it lies outside the
envelope there will be an error introduced by restricting the output to the closest
edge of the envelope, either

ψ

. The reader should remember that in general the
quantity being trimmed is not the output from the optimum filter

β or α

ψ opt , but an esti-

mate derived from a finite number of training samples opt, N where N is the number

of samples. A well-designed envelope will prevent very large errors from occur-
ring.
Brun et al. 5 have produced mathematical proofs to show that the optimum filter
with output lying between

ψ

may be obtained by determining the optimum
filter without the envelope constraint and trimming it to the envelope. This simpli-

β and α

Figure 8.1 Envelope constraint. The constrained version ψ con of the output ψ is formed by re-
stricting its value to lie within an envelope having lower bound α and upper bound β .