Weighted Myriad Filters - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 170

where M is the set of most repeated values, and r is the number of times a member of

M is repeated in the sample.

Proof: Following the steps of the proof for the unweighted version, it is
straightforward that

N

ˆ

2

β = arg min

X j ∈M

W i ( X i − X j ) . (5.34)

0

=

1 ,X i =

i X j

Dividing by i = 1 W i , and applying square root to the expression to be mini-
mized, the desired result is obtained.

PROPERTY 7 (Shift and Sign Invariance)
Let Z i

= +

X i b. Then, for any K and W ,

(i) ˆ

ˆ

β ( Z 1 , ... ,Z N ) = β ( X 1 , ... ,X N ) + b;

K K

(ii) ˆ

ˆ

β ( − Z 1 , ... , − Z N ) =− β ( Z 1 , ... ,Z N ) .

K K

Proof: Follows from the definition of the weighted myriad in Equation 5.26.

PROPERTY 8 (Unbiasedness)
Let X 1 ,X 2 , ,X N be all independent and symmetrically distributed around the point

of symmetry c. Then, ˆ

...

β

K is also symmetrically distributed around c. In particular, if

E ˆ

K exists, then E ˆ

β β =

c.

K

Proof: If X i is symmetric about c , then 2 c − X i has the same distribution as X i .

It follows that ˆ has the same distribution as ˆ β K ( X 1 ,X 2 , ... ,X N ) β K ( 2 c − X 1 , 2 c −

ˆ

X 2 , ... , 2 c − X N ) , which from Property 7, is identical to 2 c − β K ( X 1 ,X 2 , ... ,X N ) .

It follows that ˆ

β K ( X 1 ,X 2 , ... ,X N ) is symmetrically distributed about c .

5.5.1 Geometrical Interpretation

Weighted myriads as defined in Equation 5.27 can be interpreted in a more
intuitive manner. Allow a vertical bar to run horizontally through the real
line as depicted in Figure 5.9a. Then the sample myriad, ˆ

β K , indicates the
position of the bar for which the product of distances from point A to the
sample points X 1 ,X 2 , ,X N is minimum. If weights are introduced, each

sample point X i is assigned a different point A i in the bar, as illustrated in
Figure 5.9b.
The geometrical interpretation of the myriad is intuitively insightful. When
K approaches 0, it gives a conceptually simple pictorial demonstration of the
mode-myriad formula in Equation 5.13.

...