Weighted Myriad Filters - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 168

Q (θ

x

^

x (N)

θ

θ

(1)

FIGURE 5.8
Sketch of a typical weighted myriad objective function Q (β) .

The output is thus one of the local minima of Q (β) :

ˆ

Q (

β) = 0 . (5.30)

Figure 5.8 depicts a typical objective function Q , for various values of K .

Note in the figure that the number of local minima in the objective function
Q

(β)

depends on the value of the parameter K .Inparticular, when K is very
large only one extremum exists.
As K becomes larger, the number of local minima of G

(β)

decreases. In
fact, it can be proved (by examining the second derivative G (β)

(β)

) that a suffi-

cient (but not necessary ) condition for G (β) , and log ( G (β)) ,tobeconvex and,

therefore, have a unique local minimum is that

max

j

K > { W j } ( X

) −

X

) ).

( N ( 1

= 1

This condition is, however, not necessary; the onset of convexity could be at
a much lower K .
As stated in the next property, in the limit as K →∞ , with the weights { W i }

held constant, it can be shown that Q exhibits a single local extremum. The

proof is a generalized form of that used to prove the linear property of the
unweighted sample myriad.

(β)

PROPERTY 4 (Linear Property)
In the limit as K →∞ , the weighted myriad reduces to the normalized linear estimate

i = 1 W i X i

ˆ

lim
K

β K =

i = 1 W i .

(5.31)

→∞